Left feature vector calculation method, system and device suitable for LTP system and storage medium

Abstract

The invention discloses a left feature vector calculation method, system and device suitable for an LTP system, and a storage medium. The method comprises the following steps: deducing based on a Floquet theory to obtain a time domain expression of a left feature vector of a linear periodic time-varying system; converting the left feature vector of the linear periodic time-varying system of the time domain into a frequency domain, and establishing a relation between the left feature vector of the linear periodic time-varying system of the frequency domain and the left feature vector of the harmonic state space; calculating a harmonic state space left feature vector, and converting the harmonic state space left feature vector to a time domain to obtain a time domain linear periodic time-varying system left feature vector; the method does not need to depend on the inversion of the ill-conditioned STM, avoids the demand of the inversion of the ill-conditioned STM, obtains the more reliable LTP left feature vector through calculation, can improve the solving reliability of the LTP left feature vector, and lays a foundation for the modal analysis of an LTP system.

Description

Technical Field

[0001] This invention relates to the technical field of power electronic equipment control, and in particular to a method, system, device and storage medium for calculating the left eigenvector of an LTP system. Background Technology

[0002] The large-scale grid connection of power electronic equipment has led to the transformation of traditional synchronous machine-dominated power systems into power electronic-dominated power systems (PEPS). The switching dynamics of power electronic equipment and the inter-sequence dynamic coupling introduced by negative-sequence control result in complex time-varying characteristics of the system, posing a significant challenge to the stable operation of PEPS. Furthermore, broadband oscillations associated with PEPS continuously occur in real power systems. Modal analysis, which assesses damping, oscillation frequency, and participation factors, has long been considered a systematic approach to solving these oscillation problems. Given the periodic time-varying nature of PEPS, the academic community has increasingly focused on the theoretical framework of linear time-periodic (LTP) systems: by linearizing the system near its periodic trajectory, an LTP model is constructed and modal analysis is conducted. Therefore, the need for modal analysis of LTP systems has become increasingly urgent in PEPS, where LTP system eigenvalues ​​and eigenvectors are the cornerstones of related analysis. However, the periodic time-varying nature of LTP eigenvectors presents challenges in their solution. Existing methods derive LTP eigenvectors using a forward modeling approach starting from time zero. The calculation of its left and right eigenvectors depends on the state transition matrix (STM) Φ(t,0) and its inverse matrix Φ-1(t,0), respectively.

[0003] Existing methods typically employ a forward derivation strategy based on initial conditions at time zero, such that the left and right eigenvectors of LTP depend on Φ(t,0) and its inverse matrix Φ, respectively. -1 (t,0). However, the ill-conditioned nature of high-dimensional STM in PEPS leads to the inverse matrix Φ -1 The unreliability of obtaining (t,0) poses a serious challenge to the engineering applicability of the traditional LTP left eigenvector calculation method. Summary of the Invention

[0004] In view of the aforementioned existing problems, this invention is proposed. Therefore, this invention provides a method, system, device, and storage medium for calculating the left eigenvector of an LTP system to solve the problems mentioned in the background art.

[0005] To solve the above-mentioned technical problems, the present invention provides the following technical solution:

[0006] In a first aspect, embodiments of the present invention provide a method for calculating the left eigenvector of an LTP system, comprising: deriving the time-domain representation of the left eigenvector of a linear periodic time-varying system based on Floquet theory;

[0007] The left eigenvector of the linear periodic time-varying system in the time domain is transformed to the frequency domain, and the relationship between the left eigenvector of the linear periodic time-varying system in the frequency domain and the left eigenvector of the harmonic state space is established.

[0008] Calculate the left eigenvector of the harmonic state space and transform it to the time domain to obtain the left eigenvector of the time-domain linear periodic time-varying system.

[0009] As a preferred embodiment of the method for calculating the left eigenvector of an LTP system as described in this invention, the time-domain expression of the left eigenvector of a linear periodic time-varying system derived based on Floquet theory includes: based on the state of the linear periodic time-varying system, combined with the eigenvector matrix and its inverse matrix of the linear periodic time-varying system, transforming the original linear periodic time-varying system to the modal space, so that the system matrix becomes a constant diagonal matrix;

[0010] Based on the transformation relationship and combined with the matrix differentiation property, a time-domain expression for describing the left eigenvector of a linear periodic time-varying system is obtained.

[0011] As a preferred embodiment of the method for calculating the left eigenvector of an LTP system according to the present invention, the transformation of the left eigenvector of the linear periodic time-varying system in the time domain to the frequency domain includes:

[0012] The periodic matrix and the left eigenvector of the linear periodic time-varying system are expressed in the form of a Fourier series, and the Fourier series form is substituted into the time-domain expression of the left eigenvector of the linear periodic time-varying system derived based on Floquet theory to obtain a new time-domain expression of the left eigenvector of the linear periodic time-varying system.

[0013] Based on the time-domain representation of the left eigenvector of a new linear periodic time-varying system, the equations corresponding to each frequency component are separated by comparing and matching different frequency components, thus completing the time-domain to frequency-domain conversion.

[0014] The beneficial effect of this preferred technical solution is that, by using Fourier series decomposition and the principle of harmonic balance, the originally complex time-dependent problem is transformed into a series of independent frequency component problems, thereby greatly simplifying the difficulty of LTP system analysis.

[0015] As a preferred embodiment of the method for calculating the left eigenvector of an LTP system as described in this invention, the connection between the left eigenvector of a frequency-domain linear periodic time-varying system and the left eigenvector of the harmonic state space includes: the frequency-domain expression holds true at any time point t; at each individual frequency component, the system reaches its internal dynamic equilibrium condition; and the Fourier coefficients obtained by Fourier decomposition of the left eigenvector of the linear periodic time-varying system are the left eigenvectors corresponding to the eigenvalues ​​in the harmonic state space, expressed as:

[0016]

[0017] Where L is the left eigenvector corresponding to the eigenvalue in the harmonic state space, and jωI is the diagonal coefficient.

[0018] The beneficial effects of this preferred technical solution are that by ensuring that the frequency domain equation holds at all time points to construct the constraint conditions, the constraint conditions are transformed into matrix form, and the similarity between the equation and the HSS model is identified, thereby establishing a direct relationship between the frequency domain LTP left eigenvector and the HSS left eigenvector, simplifying the analysis process of the LTP system and providing a more efficient and reliable solution method.

[0019] As a preferred embodiment of the method for calculating the left eigenvector of an LTP system according to the present invention, the calculation of the left eigenvector of the harmonic state space and its transformation to the time domain to obtain the left eigenvector of the time-domain linear periodic time-varying system includes: restoring the frequency domain result back to the time domain through Fourier series reconstruction to obtain the time function of the left eigenvector of the linear periodic time-varying system, expressed as:

[0020]

[0021] Among them, l i(k) For l i(t) The k-th Fourier coefficient vector, ω = 2π / T, where j is a unit complex number.

[0022] As a preferred embodiment of the method for calculating the left eigenvector of an LTP system as described in this invention, the time-domain expression of the left eigenvector of a linear periodic time-varying system is:

[0023]

[0024] Where A(t) represents an n×n state matrix with period T, L(t) is the right eigenvector matrix of the linear periodic time-varying system, and Λ is a diagonal matrix composed of the eigenvalues ​​of the linear periodic time-varying system.

[0025] As a preferred embodiment of the method for calculating the left eigenvector of an LTP system described in this invention, the method involves: based on the time-domain expression of the left eigenvector of a new linear periodic time-varying system, separating the equations corresponding to each frequency component by comparing and matching different frequency components, and completing the time-domain to frequency-domain conversion, including: converting the left eigenvector of the linear periodic time-varying system in the time domain to a frequency-domain representation as follows:

[0026]

[0027] Where h and k are the powers of the corresponding Fourier coefficients, respectively, and l i(h) and l i(h-k) Indicate l i(t) The h-th and hk-th Fourier coefficient vectors, A k Let be the k-th Fourier coefficient matrix of A(t).

[0028] Secondly, the present invention provides a system for calculating the left eigenvector of an LTP system, comprising:

[0029] The first processing module is used to derive the time-domain representation of the left eigenvector of a linear periodic time-varying system based on Floquet theory.

[0030] The second processing module is used to convert the left eigenvector of the linear periodic time-varying system in the time domain to the frequency domain and establish the connection between the left eigenvector of the linear periodic time-varying system in the frequency domain and the left eigenvector of the harmonic state space.

[0031] The third processing module is used to calculate the left eigenvector of the harmonic state space and convert it to the time domain to obtain the left eigenvector of the time-domain linear periodic time-varying system.

[0032] Thirdly, the present invention provides an electronic device, comprising:

[0033] Memory and processor;

[0034] The memory is used to store computer-executable instructions, and the processor is used to execute the computer-executable instructions, which, when executed by the processor, implement the steps of the left feature vector calculation method applicable to the LTP system.

[0035] Fourthly, the present invention provides a computer-readable storage medium storing computer-executable instructions that, when executed by a processor, implement the steps of the method for calculating the left eigenvector of an LTP system.

[0036] Compared with the prior art, the beneficial effects of the present invention are as follows: The present invention does not rely on the inversion of the ill-conditioned STM, thus avoiding the need for the inversion of the ill-conditioned STM, thereby calculating a more reliable LTP left eigenvector, which can improve the reliability of solving the LTP left eigenvector and lay the foundation for modal analysis of LTP systems. Attached Figure Description

[0037] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. Wherein:

[0038] Figure 1 This is a schematic diagram of a method for calculating the left eigenvector of an LTP system according to an embodiment of the present invention;

[0039] Figure 2 This is a schematic diagram of a practical large-scale PEPS (Protocol on Eigenvector) method applicable to LTP systems, as described in an embodiment of the present invention.

[0040] Figure 3 This is a graph showing the change of the condition number of Φ(t,0) over time for a practical large-scale PEPS of a method for calculating the left eigenvector of an LTP system according to an embodiment of the present invention.

[0041] Figure 4 This is a comparison diagram showing the effect of a method for calculating the left feature vector of an LTP system according to an embodiment of the present invention. Detailed Implementation

[0042] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the protection scope of the present invention.

[0043] Example 1, referring to Figure 1 This is one embodiment of the present invention, which provides a method for calculating the left eigenvector of an LTP system, comprising:

[0044] S100: The time-domain representation of the left eigenvector of a linear periodic time-varying system is derived based on Floquet theory;

[0045] S200: Transform the left eigenvector of the linear periodic time-varying system in the time domain to the frequency domain, and establish the connection between the left eigenvector of the linear periodic time-varying system in the frequency domain and the left eigenvector of the harmonic state space.

[0046] S300: Calculate the left eigenvector of the harmonic state space and transform it to the time domain to obtain the left eigenvector of the time-domain linear periodic time-varying system.

[0047] It should be noted that existing methods typically employ a forward derivation strategy based on initial conditions at time zero, such that the left and right eigenvectors of LTP depend on Φ(t,0) and its inverse matrix Φ, respectively. -1 (t,0). However, the ill-conditioned nature of high-dimensional STM in PEPS leads to the inverse matrix Φ -1 The determination of (t,0) is unreliable, a limitation that poses a serious challenge to the engineering applicability of traditional LTP left eigenvector calculation methods. To address this issue, this application provides a method suitable for calculating LTP left eigenvectors, aiming to avoid the need for inverting ill-conditioned STM, thereby obtaining a more reliable LTP left eigenvector.

[0048] It should be noted that Floquet theory is a mathematical tool used to analyze linear differential equations with periodic coefficients. This involves considering a system whose behavior changes over time, but this change is periodic—that is, the system's behavior repeats itself every certain time interval T. Examples include seasonal changes, the alternation of day and night, or signals in certain electronic circuits. The mathematical model used to describe the system is typically in the form of a linear differential equation, where coefficients (such as parameters like resistance and capacitance) change periodically with time.

[0049] Floquet theory states that periodic systems can be transformed into a more manageable form through a special transformation. Specifically, the original problem can be transformed into a system of equations with constant coefficients, greatly simplifying the solution process. After the transformation, a set of eigenvalues ​​and eigenvectors can be found to describe the dynamic behavior of the system. The eigenvalues ​​provide crucial information about the system's stability and response patterns.

[0050] In this embodiment of the application, the time-domain representation of the left eigenvector of the linear periodic time-varying system derived based on Floquet theory in step S100 includes:

[0051] Based on the state of the linear periodic time-varying system, and combining the eigenvector matrix and its inverse matrix of the linear periodic time-varying system, the original linear periodic time-varying system is transformed into the modal space, making the system matrix a constant diagonal matrix;

[0052] Based on the transformation relationship and combined with the matrix differentiation property, we obtain the time-domain expression for describing the left eigenvector of a linear periodic time-varying system.

[0053] In this embodiment of the application, the time-domain expression of the left eigenvector of the linear periodic time-varying system in step S100 is as follows:

[0054]

[0055] Where A(t) represents an n×n state matrix with period T, L(t) is the right eigenvector matrix of the linear periodic time-varying system, and Λ is a diagonal matrix composed of the eigenvalues ​​of the linear periodic time-varying system.

[0056] Specifically, based on Floquet theory, the derivation process of the time-domain representation of the left eigenvector of LTP includes:

[0057] The LTP system with general periodic coefficients can be represented by the following differential equation:

[0058]

[0059] Where x(t)=[x1(t),x2(t),…,x n (t)] T , represents the column vector of n state variables at time t.

[0060] The LTP eigenvector matrix, whose basis functions transform A(t) into a constant diagonal matrix, forms a coordinate transformation between the state and modal spaces, expressed as:

[0061] x(t)=R(t)z(t), z(t)=L(t)x(t), L(t)R(t)=I (2)

[0062]

[0063] Where z(t) represents the state variable corresponding to the modal space. Let Λ represent the time derivative of the state variable corresponding to the modal space, Λ=diag(λ1,…,λ n Let R(t) be a diagonal matrix composed of the eigenvalues ​​of LTP, and I be an n-order identity matrix. R(t) and L(t) = R -1 (t) represent the left and right eigenvector matrices of LTP, respectively.

[0064] Furthermore, from (3), we can see that:

[0065]

[0066] Since L(t)R(t)=I, by differentiating the left and right sides respectively and performing transformations, we can obtain:

[0067]

[0068] Substituting (5) into (4) and transforming it, we get:

[0069]

[0070] In this embodiment of the application, step S200, which transforms the left eigenvector of the linear periodic time-varying system in the time domain to the frequency domain, includes:

[0071] The periodic matrix and the left eigenvector of the linear periodic time-varying system are expressed in the form of a Fourier series. The Fourier series form is then substituted into the time-domain expression of the left eigenvector of the linear periodic time-varying system derived based on Floquet theory to obtain a new time-domain expression of the left eigenvector of the linear periodic time-varying system.

[0072] Based on the time-domain representation of the left eigenvector of a new linear periodic time-varying system, the equations corresponding to each frequency component are separated by comparing and matching different frequency components, thus completing the time-domain to frequency-domain conversion.

[0073] In this embodiment, based on the Fourier series decomposition and harmonic balance principle, the left eigenvector of the LTP in the time domain is transformed to the frequency domain, and L(t) = [l1(t), l2(t), ..., l n (t)] T , where l i (t), i = 1, 2, ..., n, represents each row vector in the left eigenvector matrix of LTP, for any eigenvalue λ in Λ. i (6) can be rewritten as:

[0074]

[0075] in, This represents the time derivative of each row vector in the left eigenvector matrix of the LTP.

[0076] Furthermore, the periodic matrix A(t) and the left eigenvector l i (t) can be expanded into a Fourier series:

[0077]

[0078] Among them, A k Let l be the k-th Fourier coefficient matrix of A(t). i(k) For l i(t) The k-th Fourier coefficient vector, ω=2π / T, j is a unit complex number. Substituting (8) into (7) yields:

[0079]

[0080] Where h and k are the powers of the corresponding Fourier coefficients, respectively, and l i(h) and li(h-k) Indicate l i(t) The h-th and hk-th Fourier coefficient vectors;

[0081] (9) The left eigenvector of the LTP in the time domain is transformed into the frequency domain.

[0082] In an optional embodiment, the analysis results in the frequency domain can also be obtained through wavelet transform. Wavelet transform can provide both time and frequency information simultaneously. Using wavelet transform can preserve the time-varying characteristics of the system to a certain extent.

[0083] Furthermore, since (9) holds true for any time t in step S200, the expression in the square brackets must be zero:

[0084]

[0085] Converting (10) into matrix form yields:

[0086]

[0087] Furthermore, (11) is transformed into a compact form:

[0088]

[0089] in, N m It is an n-order diagonal matrix with diagonal coefficients jmω.

[0090] In this embodiment of the application, establishing the relationship between the left eigenvector of the frequency domain linear periodic time-varying system and the left eigenvector of the harmonic state space in step S200 includes:

[0091] The frequency domain expression holds true at any time point t. For each individual frequency component, the system reaches its internal dynamic equilibrium condition. The Fourier coefficients obtained by Fourier decomposition of the left eigenvector of the linear periodic time-varying system are the left eigenvectors corresponding to the eigenvalues ​​in the harmonic state space, expressed as:

[0092]

[0093] Where L is the left eigenvector corresponding to the eigenvalue in the harmonic state space, and jωI is the diagonal coefficient.

[0094] It should be noted that by ensuring that the frequency domain equation holds at all time points to construct the constraints, and transforming the constraints into matrix form, and identifying the similarity between the equation and the HSS model, a direct relationship is established between the frequency domain LTP left eigenvector and the HSS left eigenvector, simplifying the analysis process of the LTP system and making the solution method more efficient and reliable.

[0095] In this embodiment of the application, the step S300 of calculating the left eigenvector of the harmonic state space and transforming it to the time domain to obtain the left eigenvector of the time-domain linear periodic time-varying system includes: reconstructing the frequency domain result back to the time domain through Fourier series to obtain the time function of the left eigenvector of the linear periodic time-varying system, expressed as:

[0096]

[0097] Among them, l i(k) For l i(t) The k-th Fourier coefficient vector, ω = 2π / T, where j is a unit complex number.

[0098] Furthermore, since the HSS needs to be truncated during actual calculation, the LTP left eigenvector reconstructed from the left eigenvector of the HSS under the actual m-order truncation is:

[0099]

[0100] It should be noted that this invention does not rely on the inversion of the ill-conditioned STM, thus avoiding the need for inversion of the ill-conditioned STM, thereby obtaining a more reliable LTP left eigenvector, which can improve the reliability of solving the LTP left eigenvector and lay the foundation for modal analysis of LTP systems.

[0101] Example 2 is an embodiment of the present invention. This embodiment differs from the first embodiment in that it provides a left eigenvector calculation system suitable for LTP systems, comprising:

[0102] The first processing module is used to derive the time-domain representation of the left eigenvector of a linear periodic time-varying system based on Floquet theory.

[0103] The second processing module is used to transform the left eigenvector of the linear periodic time-varying system in the time domain to the frequency domain and establish the connection between the left eigenvector of the linear periodic time-varying system in the frequency domain and the left eigenvector of the harmonic state space.

[0104] The third processing module is used to calculate the left eigenvector of the harmonic state space and transform it to the time domain to obtain the left eigenvector of the time-domain linear periodic time-varying system.

[0105] Specifically, when the various modules of the LTP system left eigenvector calculation system in this embodiment are executed, they implement the steps of the LTP system left eigenvector calculation method in Embodiment 1, for example:

[0106] In one implementation, the left eigenvector calculation system for the LTP system can perform the following steps:

[0107] Based on the state of the linear periodic time-varying system, and combining the eigenvector matrix and its inverse matrix of the linear periodic time-varying system, the original linear periodic time-varying system is transformed into the modal space, making the system matrix a constant diagonal matrix;

[0108] Based on the transformation relationship and combined with the properties of matrix differentiation, the time-domain expression for describing the left eigenvector of a linear periodic time-varying system is obtained; the time-domain expression for the left eigenvector of a linear periodic time-varying system is:

[0109]

[0110] Where A(t) represents an n×n state matrix with period T, L(t) is the right eigenvector matrix of the linear periodic time-varying system, and Λ is a diagonal matrix composed of the eigenvalues ​​of the linear periodic time-varying system.

[0111] The periodic matrix and the left eigenvector of the linear periodic time-varying system are expressed in the form of a Fourier series. The Fourier series form is then substituted into the time-domain expression of the left eigenvector of the linear periodic time-varying system derived based on Floquet theory to obtain a new time-domain expression of the left eigenvector of the linear periodic time-varying system.

[0112] Based on the time-domain representation of the left eigenvector of a novel linear periodic time-varying system, by comparing and matching different frequency components, the equations corresponding to each frequency component are separated, completing the time-domain to frequency-domain transformation. The left eigenvector of the linear periodic time-varying system in the time domain is transformed into its frequency-domain representation as follows:

[0113]

[0114] Where h and k are the powers of the corresponding Fourier coefficients, respectively, and l i(h) and l i(h-k) Indicate l i(t) The h-th and hk-th Fourier coefficient vectors, A k Let be the k-th Fourier coefficient matrix of A(t).

[0115] The frequency domain expression holds true at any time point t. For each individual frequency component, the system reaches its internal dynamic equilibrium condition. The Fourier coefficients obtained by Fourier decomposition of the left eigenvector of the linear periodic time-varying system are the left eigenvectors corresponding to the eigenvalues ​​in the harmonic state space, expressed as:

[0116]

[0117] Where L is the left eigenvector corresponding to the eigenvalue in the harmonic state space, and jωI is the diagonal coefficient.

[0118] By reconstructing the frequency domain result back to the time domain using Fourier series, we obtain the time function of the left eigenvector of the linear periodic time-varying system, expressed as:

[0119]

[0120] Among them, l i(k) For l i(t) The k-th Fourier coefficient vector, ω = 2π / T, where j is a unit complex number.

[0121] This embodiment also provides an electronic device applicable to a method for calculating the left eigenvector of an LTP system, including:

[0122] The memory and processor are used to store computer-executable instructions and execute the computer-executable instructions to implement the left feature vector calculation method for LTP systems as proposed in the above embodiments.

[0123] This embodiment also provides a storage medium storing a computer program that, when executed by a processor, implements the method for calculating the left eigenvector of an LTP system as proposed in the above embodiments.

[0124] The storage medium proposed in this embodiment and the implementation method for calculating the left feature vector of the LTP system proposed in the above embodiments belong to the same inventive concept. Technical details not described in detail in this embodiment can be found in the above embodiments, and this embodiment has the same beneficial effects as the above embodiments.

[0125] Based on the above description of the implementation methods, those skilled in the art can clearly understand that the present invention can be implemented using software and necessary general-purpose hardware, and of course, it can also be implemented using hardware, but in many cases the former is a better implementation method. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as a computer floppy disk, read-only memory (ROM), random access memory (RAM), flash memory, hard disk, or optical disk, etc., including several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods of the various embodiments of the present invention.

[0126] Example 3, referring to Figures 2-4 This is one embodiment of the present invention, which verifies the beneficial effects of the present invention through scientific experiments.

[0127] like Figure 2As shown, this embodiment takes a real large-scale PEPS system as an example. The condition number of Φ(t,0) of the system changes with time as follows: Figure 3 As shown, the condition number of Φ(t,0) increases rapidly with time, indicating that Φ(t,0) of the system is highly ill-conditioned, and inverting Φ(t,0) is unreliable. Furthermore, the norm of the corresponding left eigenvectors calculated using the proposed method and existing methods changes with time as shown in the figure. Figure 4 As shown, the proposed method can effectively calculate the left eigenvector, while existing methods suffer from divergence when calculating the left eigenvector. This invention eliminates the need for inverting the ill-conditioned STM, thus obtaining a more reliable LTP left eigenvector and improving the reliability of LTP left eigenvector calculation, laying the foundation for modal analysis of LTP systems.

[0128] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for calculating the left eigenvector of an LTP system, characterized in that, include: The time-domain representation of the left eigenvector of a linear periodic time-varying system is derived based on Floquet theory. The left eigenvector of the linear periodic time-varying system in the time domain is transformed to the frequency domain, and the relationship between the left eigenvector of the linear periodic time-varying system in the frequency domain and the left eigenvector of the harmonic state space is established. Calculate the left eigenvector of the harmonic state space and transform it to the time domain to obtain the left eigenvector of the time-domain linear periodic time-varying system.

2. The method for calculating the left eigenvector of an LTP system as described in claim 1, characterized in that, Based on Floquet theory, the time-domain representation of the left eigenvector of a linear periodic time-varying system is derived as follows: Based on the state of the linear periodic time-varying system, and combining the eigenvector matrix and its inverse matrix of the linear periodic time-varying system, the original linear periodic time-varying system is transformed into the modal space, making the system matrix a constant diagonal matrix. Based on the transformation relationship and combined with the matrix differentiation property, a time-domain expression for describing the left eigenvector of a linear periodic time-varying system is obtained.

3. The method for calculating the left eigenvector of an LTP system as described in claim 2, characterized in that, The transformation of the left eigenvector of the linear periodic time-varying system in the time domain to the frequency domain includes: The periodic matrix and the left eigenvector of the linear periodic time-varying system are expressed in the form of a Fourier series, and the Fourier series form is substituted into the time-domain expression of the left eigenvector of the linear periodic time-varying system derived based on Floquet theory to obtain a new time-domain expression of the left eigenvector of the linear periodic time-varying system. Based on the time-domain representation of the left eigenvector of a new linear periodic time-varying system, the equations corresponding to each frequency component are separated by comparing and matching different frequency components, thus completing the time-domain to frequency-domain conversion.

4. The method for calculating the left eigenvector of an LTP system as described in claim 3, characterized in that, Establishing the relationship between the left eigenvector of a frequency-domain linear periodic time-varying system and the left eigenvector of its harmonic state space includes: The frequency domain expression holds true at any time point t. For each individual frequency component, the system reaches its internal dynamic equilibrium condition. The Fourier coefficients obtained by Fourier decomposition of the left eigenvector of the linear periodic time-varying system are the left eigenvectors corresponding to the eigenvalues ​​in the harmonic state space, expressed as: Where L is the left eigenvector corresponding to the eigenvalue in the harmonic state space, and jωI is the diagonal coefficient.

5. The method for calculating the left eigenvector of an LTP system as described in claim 4, characterized in that, Calculating the left eigenvector of the harmonic state space and transforming it to the time domain to obtain the left eigenvector of the time-domain linear periodic time-varying system includes: reconstructing the frequency domain result back to the time domain through Fourier series reconstruction to obtain the time function of the left eigenvector of the linear periodic time-varying system, expressed as: Among them, l i(k) For l i(t) The k-th Fourier coefficient vector, ω = 2π / T, where j is a unit complex number.

6. The method for calculating the left eigenvector of an LTP system as described in claim 2, characterized in that: The time-domain representation of the left eigenvector of a linear periodic time-varying system is: Where A(t) represents an n×n state matrix with period T, L(t) is the right eigenvector matrix of the linear periodic time-varying system, and Λ is a diagonal matrix composed of the eigenvalues ​​of the linear periodic time-varying system.

7. The method for calculating the left eigenvector of an LTP system as described in claim 3 or 6, characterized in that, Based on the time-domain representation of the left eigenvector of a novel linear periodic time-varying system, by comparing and matching different frequency components, the equations corresponding to each frequency component are separated, completing the time-domain to frequency-domain transformation, including: transforming the left eigenvector of the linear periodic time-varying system in the time domain to its frequency-domain representation as follows: Where h and k are the powers of the corresponding Fourier coefficients, respectively, and l i(h) and l i(h-k) Indicate l i(t) The h-th and hk-th Fourier coefficient vectors, A k Let be the k-th Fourier coefficient matrix of A(t).

8. A left eigenvector calculation system suitable for LTP systems, applied to the method described in any one of claims 1-7, characterized in that, include: The first processing module is used to derive the time-domain representation of the left eigenvector of a linear periodic time-varying system based on Floquet theory. The second processing module is used to convert the left eigenvector of the linear periodic time-varying system in the time domain to the frequency domain and establish the connection between the left eigenvector of the linear periodic time-varying system in the frequency domain and the left eigenvector of the harmonic state space. The third processing module is used to calculate the left eigenvector of the harmonic state space and convert it to the time domain to obtain the left eigenvector of the time-domain linear periodic time-varying system.

9. An electronic device, comprising: Memory and processor; The memory is used to store computer-executable instructions, and the processor is used to execute the computer-executable instructions, which, when executed by the processor, implement the steps of the method for calculating the left eigenvector of an LTP system as described in any one of claims 1 to 7.

10. A computer-readable storage medium storing computer-executable instructions that, when executed by a processor, implement the steps of the method for calculating the left eigenvector of an LTP system as described in any one of claims 1 to 7.

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