Method and system for stability determination of linear periodic time-varying system based on harmonic state-space model, and medium

Abstract

This invention discloses a method, system, and medium for determining the stability of a linear periodic time-varying system based on a harmonic state-space model, belonging to the field of linear system dynamics analysis. The method includes: constructing harmonic state-space models M with high truncation order H and low truncation order L for the system, respectively. H M L ; Calculate M L The first eigenvalue and the first eigenvector; for M H Arrange its state variables in the order of to transform them; calculate M under the first eigenvalue. L The system matrix relative to the transformed M H The deviation of the system matrix is ​​used to obtain the deviation matrix; the eigenvalue deviations are calculated based on the deviation matrix and the first eigenvector; the sum of the eigenvalue deviations and the first eigenvalue is calculated to obtain M. H The second eigenvalue is used to determine the stability of the linear periodic time-varying system. This approach ensures the accuracy of eigenvalue calculation while improving computational efficiency, thus enhancing the applicability of the harmonic state-space model to large-scale systems.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of linear system dynamics analysis, and more particularly, relates to a linear periodic time-varying system stability determination method, system and medium based on a harmonic state space model. BACKGROUND

[0002] In the real physical world, systems generally have nonlinearity and time-varying nature. When analyzing the small disturbance stability of an actual system, the system can be linearized in the neighborhood of the steady-state trajectory of the nonlinear time-varying model, and then a linear periodic time-varying model is obtained. The system matrix of the linear periodic time-varying model contains parameters that change periodically with time, so the existing mature stability analysis and control theory of linear time-invariant systems cannot be directly applied. In this regard, researchers have proposed a harmonic state space model, which converts a linear periodic time-varying model into a linear time-invariant model based on Fourier series expansion and the harmonic balance principle, and then applies mature stability analysis and control theory.

[0003] The eigenvalues of the harmonic state space model can be used for stability analysis of an actual system. However, the harmonic state space model has the following problems. The truncation order determines the number of frequency components considered by the harmonic state space model, and thus determines the analysis accuracy and computational efficiency of the model. When the truncation order is high, the model analysis accuracy is high, but the calculation speed is slow and the efficiency is low; when the truncation order is low, although the computational efficiency is improved, the model analysis accuracy is poor, and even an incorrect stability analysis conclusion may be obtained. Therefore, existing research proposes an optimal truncation order selection method to reduce the truncation order as much as possible while ensuring the analysis accuracy of the harmonic state space model. However, for some large-scale actual systems, such as power electronic power systems with high-voltage direct current transmission, the truncation order must be taken to a higher value to accurately analyze the small disturbance stability, at which time the analysis accuracy and computational efficiency of the harmonic state space model are difficult to balance, and it is difficult to apply to large-scale systems. This also hinders the further development of stability mechanism analysis and control method research by applying linear time-invariant system theory. SUMMARY

[0004] In view of the defects and improvement needs of the prior art, the present application provides a linear periodic time-varying system stability determination method, system and medium based on a harmonic state space model, which aims to solve the problem that the prior art cannot simultaneously ensure the accuracy and efficiency of the eigenvalue calculation of the harmonic state space model, thereby improving the applicability of the harmonic state space model to large-scale actual systems.

[0005] To achieve the above objectives, according to one aspect of the present invention, a method for determining the stability of a linear periodic time-varying system based on a harmonic state-space model is provided, comprising: constructing a harmonic state-space model M with a first truncation order H for the linear periodic time-varying system. H And the harmonic state-space model M with the second truncation order L L H > L; Computational model M L The first eigenvalue and the first eigenvector; for model M H According to its state variables Arrange them in order to transform them. Includes the -L to Lth Fourier coefficients of the state variables in the form of a complex exponentially modulated signal. Includes the -H to -(L+1) and (L+1) to H Fourier coefficients of the state variables in the form of a complex exponentially modulated signal; calculates the model M under the first eigenvalue. L The system matrix relative to the transformed model M H The deviation of the system matrix is ​​used to obtain the deviation matrix; the eigenvalue deviation is calculated based on the deviation matrix and the first eigenvector; the sum of the eigenvalue deviation and the first eigenvalue is calculated to obtain the model M. H The second eigenvalue; the stability of the linear periodic time-varying system is determined based on the second eigenvalue.

[0006] Furthermore, the transformed model M H for:

[0007]

[0008] Where s is the Laplace operator, For the transformed model M H The system matrix, A 11 A 12 A 21 and A 22 For the transformed model M H The four submatrices in the system matrix, A 11 Includes model M L The system matrix.

[0009] Furthermore, the deviation matrix is:

[0010]

[0011] Among them, A Δ (s) is the deviation matrix, where s takes the first eigenvalue. For model M L The system matrix, For a Toplitz matrix with a second truncation order L, is a block diagonal matrix containing -L to L frequency information, and I is a unit matrix.

[0012] Further, and are respectively:

[0013]

[0014] wherein A l is the lth Fourier coefficient of the system matrix of the linear periodic time-varying model, l = -L, -(L-1),..., 0,..., L-1, L, ω0 is the fundamental frequency, and diag[·] represents a block diagonal matrix.

[0015] Further, the eigenvalue deviation is:

[0016]

[0017] wherein Δλ Li is the eigenvalue deviation, A Δ (s) is the deviation matrix, s is a Laplacian operator, λ Li is the first eigenvalue, and the first eigenvector includes l i and r i , l i is a left eigenvector, and r i is a right eigenvector.

[0018] Further, the constructing the model M H and the model M L specifically includes: establishing a linear periodic time-varying model for the linear periodic time-varying system; processing the system matrix and the state variable in the linear periodic time-varying model based on Fourier series expansion; and based on the harmonic balance principle, arranging the above processed model into a matrix form to obtain the model M H and the model M L .

[0019] Further, the method further includes: determining the influence degree of the L+1th to Hth frequency components on the second eigenvalue according to the eigenvalue deviation, and controlling the linear periodic time-varying system according to the influence degree.

[0020] According to another aspect of the present application, there is provided a linear periodic time-varying system stability determination system based on a harmonic state space model, which includes: a processor; and a memory storing a computer executable program, which, when executed by the processor, causes the processor to execute the linear periodic time-varying system stability determination method based on a harmonic state space model.

[0021] According to another aspect of the present application, a computer readable storage medium is provided, which stores a computer program, the program being executed by a processor to implement the harmonic state space model based linear periodic time-varying system stability determination method as described above.

[0022] In general, the above technical solutions conceived by the present application can achieve the following beneficial effects:

[0023] (1) A harmonic state space model based linear periodic time-varying system stability determination method is provided, which first calculates the eigenvalues of a low-truncated-order harmonic state space model, reorders the state variables of a high-truncated-order harmonic state space model, so as to subsequently calculate the deviation of the eigenvalues relative to the eigenvalues of the high-truncated-order harmonic state space model, thereby obtaining the eigenvalues of the high-truncated-order harmonic state space model, improving the calculation efficiency, reducing the demand for computer storage for solving the eigenvalues of the high-truncated-order harmonic state space model, and being able to balance the accuracy and calculation efficiency of the linear periodic time-varying system stability analysis, thereby improving the applicability of the harmonic state space model to large-scale actual systems;

[0024] (2) A specific eigenvalue deviation calculation method is provided, which can be obtained by multiplying the deviation matrix with the eigenvectors of the low-truncated-order harmonic state space model, and the calculation method is simple and the result is accurate, thereby further improving the calculation efficiency of the eigenvalues of the high-truncated-order harmonic state space model;

[0025] (3) The influence of the high-frequency components (the L+1 to H frequency components) in the linear periodic time-varying system steady-state trajectory on the eigenvalues is quantified according to the eigenvalue deviation, thereby laying a foundation for subsequent analysis and stable control of the system. BRIEF DESCRIPTION OF DRAWINGS

[0026] Figure 1 A flowchart of the harmonic state space model based linear periodic time-varying system stability determination method provided by the embodiments of the present application is shown. DETAILED DESCRIPTION

[0027] In order to make the objects, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and do not limit the present application. In addition, the technical features involved in each embodiment of the present application described below can be combined with each other as long as they do not conflict with each other.

[0028] In the present application, the terms "first", "second", etc. (if any) in the present application and the drawings are used to distinguish similar objects, and do not necessarily describe a specific order or sequence.

[0029] Example 1

[0030] A method for determining the stability of linear periodic time-varying systems based on a harmonic state-space model, see [reference]. Figure 1 The method includes operations S1-S6.

[0031] Operation S1 constructs harmonic state-space models M with the first truncation order H for the linear periodic time-varying system. H And the harmonic state-space model M with the second truncation order L L H > L.

[0032] According to an embodiment of the present invention, model M is constructed in operation S1. H and model M L Specifically, it includes the following sub-operations S11 to S13.

[0033] In suboperation S11, a linear periodic time-varying model is established for the linear periodic time-varying system. The linear periodic time-varying model of the actual system can be expressed as:

[0034]

[0035] Where x(t)=[x1(t),x2(t),…,x n (t)] T Let A(t) be the n-dimensional state variable of the linear periodic time-varying model, and let A(t) be the system matrix of the linear periodic time-varying model, where A(t) = A(t+T) and T is the minimum period of the system.

[0036] In suboperation S12, the system matrix and state variables in the linear periodic time-varying model are processed based on Fourier series expansion.

[0037] Based on the Fourier series expansion, A(t) can be expressed as:

[0038]

[0039] Among them, A m Let ω0 be the m-th Fourier coefficient of A(t), and let ω0 be the fundamental frequency.

[0040] The state variable x(t) can be represented as a complex exponential modulated signal:

[0041]

[0042] Among them, X n Let x(t) be the nth Fourier coefficient. Substituting equations (2) and (3) into equation (1), we obtain the processed model.

[0043] In sub-operation S13, the processed model is arranged in a matrix form based on the harmonic balance principle to obtain a model M H and the model M L .

[0044] the model M H is:

[0045]

[0046] the model M L is:

[0047]

[0048] wherein A l is the l-th Fourier coefficient of the system matrix of the linear periodic time-varying model of the system, l = -L, -(L-1), …, 0, …, L-1, L, and diag[·] represents a block diagonal matrix. has a dimension of (2×H+1)×n; has a dimension of (2×L+1)×n.

[0049] Operation S2, the first eigenvalue and the first eigenvector of the model M L are calculated.

[0050] The first eigenvalue and the first eigenvector are calculated in the following manner:

[0051]

[0052] wherein λ Li is the first eigenvalue, the first eigenvector includes l i and r i , l i is a left eigenvector, and r i is a right eigenvector.

[0053] Operation S3, for the model M H , its state variables are arranged in the order of to transform it, contains the -L to L-th Fourier coefficients of the state variables in the form of complex exponential modulation signals, contains the -H to -(L+1) and (L+1) to H-th Fourier coefficients of the state variables in the form of complex exponential modulation signals.

[0054] According to an embodiment of the present application, the transformed model M H is:

[0055]

[0056] wherein s is a Laplace operator, A H , A 11 , A 12 , A 21 and A 22 are four sub-matrices in the system matrix of the transformed model M H A 11 contains the system matrix of the model M L

[0057] Operation S4, the deviation of the system matrix of the model M L under the first eigenvalue relative to the system matrix of the transformed model M H is calculated to obtain a deviation matrix.

[0058] According to an embodiment of the present application, the deviation matrix is:

[0059]

[0060] wherein A Δ (s) is the deviation matrix, s is the first eigenvalue, is the system matrix of the model M L is a Toeplitz matrix with the second truncated order L, is a block diagonal matrix containing frequency information from -L to L, and I is an identity matrix.

[0061] Operation S5, the eigenvalue deviation is calculated according to the deviation matrix and the first eigenvector, and the sum of the eigenvalue deviation and the first eigenvalue is calculated to obtain the second eigenvalue of the model M H

[0062] According to an embodiment of the present application, the eigenvalue deviation is:

[0063]

[0064] wherein Δλ Li is the eigenvalue deviation.

[0065] The second eigenvalue λ H of the model M Hi = λ Li + Δλ Li . Through the above operations S1-S5, based on low-order matrix operations, the eigenvalue of the high-truncated-order harmonic state space model is finally obtained, while ensuring the calculation accuracy and calculation efficiency.

[0066] Operation S6, the stability of the linear periodic time-varying system is determined according to the second eigenvalue.

[0067] ​​​In this embodiment, operation S6 can be implemented by using the existing method for determining the stability of a linear periodic time-varying system based on eigenvalues of a harmonic state space model, which will not be described herein again.

[0068] According to an embodiment of the present application, the method further comprises quantifying the influence degree of the (L+1)th to Hth frequency components on the second eigenvalue according to the eigenvalue deviation, and performing stability control design on the linear periodic time-varying system according to the influence degree.

[0069] Embodiment Two

[0070] A system for determining the stability of a linear periodic time-varying system based on a harmonic state space model, comprising: a processor; a memory storing a computer executable program, which, when executed by the processor, causes the processor to perform the above method for determining the stability of a linear periodic time-varying system based on a harmonic state space model. The related technical solutions are the same as those in Embodiment One, which will not be described herein again.

[0071] Embodiment Three

[0072] A computer readable storage medium storing a computer program, which, when executed by a processor, implements the above method for determining the stability of a linear periodic time-varying system based on a harmonic state space model. The related technical solutions are the same as those in Embodiment One, which will not be described herein again.

[0073] Those skilled in the art can easily understand that the above description is only the preferred embodiments of the present application, and is not intended to limit the present application. Any modification, equivalent replacement and improvement made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A method for determining stability of a linear periodic time-varying system based on a harmonic state-space model, characterized by, The method comprises: constructing a harmonic state space model M with a first truncation order H for the linear periodic time-varying system H and a harmonic state space model M with a second truncation order L L , H > L The first eigenvalue and the first eigenvector of the first feature of the computational model M L ​ For the model M H its state variables are arranged in the order to transform them, the -L to Lth Fourier coefficients of the state variables in the form of complex exponential modulated signals, the -H to -(L+1)th and (L+1)th to Hth Fourier coefficients of the state variables in the form of complex exponential modulated signals; computing a deviation of the system matrix of the model M L at the first eigenvalue from the system matrix of the transformed model M H , resulting in a deviation matrix; calculating a feature value deviation according to the deviation matrix and the first feature vector, calculating a sum of the feature value deviation and the first feature value, and obtaining a second feature value of the model M H . determining stability of the linear periodic time-varying system according to the second eigenvalue; Transformed model M H is: wherein s is the Laplace operator, is the system matrix of the transformed model M H , , , and are four sub-matrices of the system matrix of the transformed model M H , comprises the system matrix of the model M L . the bias matrix is: wherein is the bias matrix, s taking the first eigenvalue, is the model M L is the system matrix, is a Toeplitz matrix with a second truncation order L, is a block diagonal matrix containing -L to Lth frequency information, is an identity matrix; the eigenvalue bias is: wherein, is the eigenvalue deviation, is the deviation matrix, is the Laplacian operator, is the first eigenvalue, the first eigenvector comprising and , is the left eigenvector, is the right eigenvector.

2. The linear periodically time-varying system stability determination method based on a harmonic state-space model according to claim 1, characterized by, and respectively. wherein is the system matrix of the system linear periodic time-varying model l the second Fourier coefficient, , is the fundamental frequency, denotes a block-diagonal matrix.

3. The linear periodically time-varying system stability determination method based on a harmonic state-space model according to claim 1, characterized by, Constructing model M H and model M L Specifically includes: establishing a linear periodic time-varying model for the linear periodic time-varying system; processing the system matrix and the state variable in the linear periodic time-varying model based on Fourier series expansion; Based on the principle of harmonic balance, the processed model is arranged in matrix form to obtain the model M H and the model M L .

4. The method for stability assessment of linear periodic time-varying system based on harmonic state-space model according to claim 1, wherein, the method further comprises: quantifying the influence degree of the L+1th to Hth frequency components on the second eigenvalue according to the eigenvalue bias, and performing stability control design on the linear periodic time-varying system according to the influence degree.

5. A system for stability determination of a linear periodically time-varying system based on a harmonic state-space model, characterized by, The method comprises: a processor; a memory storing a computer executable program, the program, when executed by the processor, causing the processor to execute the linear periodic time-varying system stability determination method based on the harmonic state space model according to any one of claims 1-4.

6. A computer-readable storage medium having stored thereon a computer program, characterized in that, The program, when executed by the processor, implements the linear periodic time-varying system stability determination method based on the harmonic state space model according to any one of claims 1-4.

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