Dynamic stability analysis control method, device and system for linear periodic time-varying system

Abstract

The application discloses a dynamic stability analysis control method, device and system of a linear periodic time-varying system, and belongs to the technical field of system stability analysis. i The partial derivative of the concerned parameter alpha with respect to the dominant eigenvalue lambda i The sensitivity of the concerned variable alpha to the dominant eigenvalue of the linear periodic time-varying system is analyzed, the influence size and direction of the system control concerned parameter on the dominant eigenvalue can be accurately described, the technical defects that the classical dominant eigenvalue sensitivity analysis method is only applicable to linear time-invariant systems are made up, and theoretical support is provided for the dominant eigenvalue sensitivity analysis of the linear periodic time-varying system. In addition, the system matrix with complex time-varying characteristics does not need to be time-invariant, and the operation is simpler.

Description

Technical Field

[0001] This invention belongs to the field of system stability analysis technology, and more specifically, relates to a dynamic stability analysis and control method, device and system for linear periodic time-varying systems. Background Technology

[0002] When studying the dominant eigenvalue sensitivity of time-varying systems in various physics and engineering problems, such as power systems, celestial motion, and attitude control, the linearization of the system's steady-state periodic trajectory reveals its linear periodic time-varying nature. Therefore, a review of the theory of linear periodic time-varying systems reveals two main analytical approaches. One approach is to transform the linear periodic time-varying system into an infinite-order linear time-invariant system based on harmonic state-space theory. Since truncation of the infinite-order matrix is ​​necessary in practical calculations, the truncation order directly affects the accuracy of the mathematical relationships and the convergence of the calculation results. Existing literature has provided effective support for the selection of the truncation order. However, although the analytical form of the dominant eigenvalue sensitivity of the linear time-invariant system can be used after truncation, the influence mechanism of the parameters of interest on the truncated harmonic state-space system matrix remains unclear, and current sensitivity analysis based on harmonic state-space theory has not been further developed. The other approach is to directly analyze the periodic time-varying system matrix based on Floquet theory. While Floquet theory reveals the general solution form of linear periodic time-varying systems and serves as the theoretical basis for calculating the dominant eigenvalues ​​of linear periodic time-varying systems, its limitations prevent further development of sensitivity analysis. However, since the information of the feature structure has not been further explored, the influence of the attention parameter on the dominant eigenvalue has not yet been analyzed based on Floquet theory.

[0003] Therefore, there is still a lack of existing methods for sensitivity analysis of dominant eigenvalues ​​of linear periodic time-varying systems. There is an urgent need to provide an effective analysis and calculation method to help improve the dynamic stability of the system by optimizing control parameters and operating modes, and to provide theoretical support for subsequent stability control. This is indeed one of the important research and development topics at present. Summary of the Invention

[0004] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides a dynamic stability analysis and control method, apparatus, and system for linear periodic time-varying systems, the purpose of which is to improve the dynamic stability analysis and control of the dominant eigenvalue λ. i The partial derivative with respect to the parameter of interest α is taken as the dominant eigenvalue λ. i The sensitivity of the variable of interest α accurately characterizes the influence mechanism of the parameter of interest on the matrix of the periodic time-varying system, so as to analyze and control the dynamic stability of the linear periodic time-varying system, thereby solving the technical problem that there is still no existing sensitivity analysis method for the dominant eigenvalue of linear periodic time-varying system.

[0005] To achieve the above objectives, according to one aspect of the present invention, a dynamic stability analysis and control method for a linear periodic time-varying system is provided, comprising:

[0006] The dominant eigenvalue λ is of interest in calculating the system matrix A of a linear periodic time-varying system. i The corresponding left eigenvector l i and the right eigenvector r i ;

[0007] The explicit representation A in the system matrix A using the attention parameter α sens1 and implicit expression part A sens2 Calculate the partial derivative of the system matrix A with respect to the parameter of interest α;

[0008] Using the partial derivative of the system matrix A with respect to the interest parameter α, and the left eigenvector l i and the right eigenvector r i Calculate the dominant eigenvalue λ i The partial derivative with respect to the parameter of interest α;

[0009] The dominant feature value λ i The partial derivative with respect to the parameter of interest α is taken as the dominant eigenvalue λ. i The sensitivity of the variable of interest α is used to determine the sensitivity of the parameter of interest α to the dominant feature value λ. i The magnitude and direction of the influence are used to analyze and control the dynamic stability of the linear periodic time-varying system.

[0010] In one embodiment, the dominant eigenvalue λ of interest in calculating the system matrix A of the linear periodic time-varying system is... i The corresponding left eigenvector l i and the right eigenvector r i include:

[0011] Using the formula R(t)=Φ(t,0)R(0)e -Λt and L(t)=R -1 (t) Calculate the left eigenvector matrix L(t) and the right eigenvector matrix R(t) of the linear periodic time-varying system, and then obtain the left eigenvector l. i and the right eigenvector r i ;

[0012] Where Φ(t,0) is the state transition matrix of the system at time 0-t; R(0) is the right eigenvector at time 0; e -Λt The matrix index is the matrix exponent corresponding to the eigenvalue Λ of the system matrix A.

[0013] In one embodiment, the display representation A in the system matrix A using the attention parameter α sens1and implicit expression part A sens2 Calculating the partial derivative of the system matrix A with respect to the parameter of interest α includes:

[0014] Calculate the partial derivatives of the state variables and algebraic variables of the linear periodic time-varying system with respect to the parameter of interest α. and To represent the partial derivative sign, using and Calculate the implicit representation A of the interest parameter α in the system matrix A. sens2 ;x ss The periodic trajectory of the state variable y under steady state represents the periodic trajectory of the state variable y under steady state. ss Represents the periodic trajectory of an algebraic variable under steady state;

[0015] Calculate the explicit representation A of the interest parameter α in the system matrix A. sens1 ;

[0016] The display expression part A sens1 The superposition of the implicit expression part and the system matrix A with respect to the interest parameter α is used as the partial derivative of the system matrix A with respect to the interest parameter α.

[0017] In one embodiment, the calculation of the partial derivatives of the state variables and algebraic variables of the linear periodic time-varying system with respect to the parameter of interest α involves... and use and Calculate the implicit representation A of the interest parameter α in the system matrix A. sens2 ,include:

[0018] The partial derivatives of the state variables and algebraic variables of the linear periodic time-varying system with respect to the parameter of interest α are calculated based on Floquet theory. and

[0019] use Calculate the implicit representation A of the interest parameter α in the system matrix A. sens2 .

[0020] In one embodiment, the partial derivatives of the state variables and algebraic variables of the linear periodic time-varying system with respect to the parameter of interest α are calculated based on Floquet theory. and include:

[0021] Using formula and Calculate the partial derivatives of the state variables and algebraic variables of the linear periodic time-varying system with respect to the parameter of interest α;

[0022] Where τ represents the integrand of the function when convolved with respect to time t, dτ represents the infinitesimal element of τ, x represents the state variable of the system, y represents the algebraic variable of the system, Φ(t,0) represents the state transition matrix of the system from 0 to t; I represents the identity matrix; Φ(T,0) represents the state transition matrix of the system from 0 to t; Φ(T,τ) represents the state transition matrix of the system from τ to t; Φ(t,τ) represents the state transition matrix of the system from τ to t; g represents the system of differential equations with algebraic variables; K α (τ) represents the time corresponding to τ. f represents a system of differential equations for the state variables.

[0023] In one embodiment, the calculation of the attention parameter α in the explicit representation portion A of the system matrix A... sens1 ,include:

[0024] Using formula Calculate the explicit representation A of the interest parameter α in the system matrix A. sens1 .

[0025] In one embodiment, the partial derivative of the system matrix A with respect to the interest parameter α, and the left eigenvector l i and the right eigenvector r i Calculate the dominant eigenvalue λ i The partial derivatives with respect to the parameter of interest α include:

[0026] Using formula Calculate the dominant eigenvalue λ i Partial derivative with respect to the parameter of interest α in, l i Let r represent the left eigenvector. i This represents the right eigenvector.

[0027] In one embodiment, the partial derivative of the parameter of interest α is calculated as follows:

[0028] Let l i r i =1, then term2 is 0;

[0029] If the derivative of time t does not contain a zeroth-order quantity of fft, then The value does not include the calculation result of term3;

[0030] The value contains term1 The zeroth order quantity of the calculated FFT is then...

[0031] According to another aspect of the present invention, a dynamic stability analysis and control device for a linear periodic time-varying system is provided, comprising:

[0032] The first calculation module is used to calculate the dominant eigenvalue λ of interest in the system matrix A of a linear periodic time-varying system. i The corresponding left eigenvector l i and the right eigenvector r i ;

[0033] The second calculation module is used to utilize the explicit representation portion A of the system matrix A with the attention parameter α. sens1 and implicit expression part A sens2 Calculate the partial derivative of the system matrix A with respect to the parameter of interest α;

[0034] The third calculation module is used to utilize the partial derivative of the system matrix A with respect to the parameter of interest α, and the left feature vector l i and the right eigenvector r i Calculate the dominant eigenvalue λ i The partial derivative with respect to the parameter of interest α;

[0035] The analysis and control module is used to process the dominant feature value λ. i The partial derivative with respect to the parameter of interest α is taken as the dominant eigenvalue λ. i The sensitivity of the variable of interest α is used to determine the sensitivity of the parameter of interest α to the dominant feature value λ. i The magnitude and direction of the influence are used to analyze and control the dynamic stability of the linear periodic time-varying system.

[0036] According to another aspect of the present invention, a dynamic stability analysis and control system for a linear periodic time-varying system is provided, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps of the above-described method.

[0037] In summary, compared with the prior art, the above-described technical solutions conceived by this invention can achieve the following beneficial effects:

[0038] (1) This invention provides a dynamic stability analysis and control method for a linear periodic time-varying system, which uses the dominant eigenvalue λ i The partial derivative with respect to the parameter of interest α is taken as the dominant eigenvalue λ. iAnalyzing the sensitivity of the variable of interest α to the dominant eigenvalue sensitivity of a linear periodic time-varying system can accurately characterize the magnitude and direction of the influence of the system's control parameters on the dominant eigenvalue. This overcomes the technical limitation of classical dominant eigenvalue sensitivity analysis methods, which are only applicable to linear time-invariant systems, and provides theoretical support for conducting dominant eigenvalue sensitivity analysis on linear periodic time-varying systems. Furthermore, it eliminates the need to perform time-invariant processing on system matrices with complex time-varying characteristics, simplifying the operation.

[0039] (2) This scheme uses the formula R(t)=Φ(t,0)R(0)e -Λt and L(t)=R -1 (t) Calculate the left eigenvector matrix L(t) and the right eigenvector matrix R(t) of the linear periodic time-varying system, and obtain the left eigenvector l from them. i and the right eigenvector r i It has low computational complexity, which can improve the computational efficiency of the entire dynamic stability analysis and control method.

[0040] (3) This scheme utilizes the partial derivatives of the state variables and algebraic variables of the linear periodic time-varying system with respect to the parameter of interest α. and Therefore, the implicit representation A of the interest parameter α in the system matrix A is calculated. sens2 It has low computational complexity, which can improve the computational efficiency of the entire dynamic stability analysis and control method.

[0041] (4) This scheme uses Floquet theory to calculate the partial derivatives of the state variables and algebraic variables of the linear periodic time-varying system with respect to the parameter of interest α. and By combining Floquet theory with the boundary conditions of trajectory sensitivity, the complex calculation of solving the partial derivatives of each state variable and algebraic variable with respect to the parameters of interest is avoided, which is beneficial for its application in high-order systems.

[0042] (5) This scheme uses the time-varying state transition matrix information of the system to calculate the partial derivatives of the state variables and algebraic variables with respect to the parameters of interest; thus filling the technical gap that the existing technology cannot directly calculate the two.

[0043] (6) Using formulas Calculate the explicit representation A of the interest parameter α in the system matrix A. sens1 It has low computational complexity, which can improve the computational efficiency of the entire dynamic stability analysis and control method.

[0044] (7) This scheme utilizes the formula Calculate the dominant eigenvalue λ i Partial derivative with respect to the parameter of interest α It can provide an analytical form for calculating eigenvalue sensitivity, realizing the relationship between the parameter of interest α and the eigenvalue λ. i It affects the accurate calculation of magnitude and direction.

[0045] (8) This scheme utilizes the formula Characterizing the dominant eigenvalue λ i Partial derivative with respect to the parameter of interest α because The value is independent of time t, and the sensitivity of the dominant eigenvalue is ignored. Time-dependent quantities, including periodic variables; time-independent quantities, specifically, the 0th order of FFT, are used to characterize the dominant eigenvalue λ. i The sensitivity of the variable of interest α has low computational complexity, which makes up for the technical limitation of the classical dominant eigenvalue sensitivity analysis method, which is only applicable to linear time-invariant systems. It provides theoretical support for carrying out dominant eigenvalue sensitivity analysis on linear periodic time-varying systems. Attached Figure Description

[0046] Figure 1 This is a flowchart of a dynamic stability analysis and control method for a linear periodic time-varying system provided by the present invention. Detailed Implementation

[0047] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0048] The dynamic evolution of the existing system is described by the following set of nonlinear differential-algebraic equations, where x(t)=[x1(t),…,x n (t)] T Let y(t) be an n-dimensional column vector formed by the state variables in the system; y(t) = [y1(t), ..., y2(t)]. m (t)] T α is an m-dimensional column vector formed by algebraic variables in the system; α is the set of parameters of interest in the system.

[0049]

[0050] Dominant eigenvalue sensitivity is an important indicator in system analysis and control, used to characterize the magnitude and direction of the influence of parameters of interest on the dominant eigenvalue of the system. It is often applied in diverse scenarios such as controller location, control / structural parameter optimization, and stability margin calculation. Taking the dominant eigenvalue sensitivity of a linear time-invariant system as an example, the following equation represents the steady-state operating point of the system before linearization:

[0051]

[0052] Linearizing the system shown in the following equation based on the steady-state operating point yields:

[0053]

[0054] Where A 11 A 12 A 21 A 22 Let the constant coefficient matrices be n×n, n×m, m×n, and m×m dimensions, respectively. Eliminating the algebraic variables yields:

[0055]

[0056] The following equation shows the definition of the dominant eigenvalue sensitivity of a linear time-invariant system:

[0057]

[0058] Where, λ i Let A be the i-th dominant eigenvalue of the system, α be the parameter of interest where the disturbance occurs, and A be the i-th dominant eigenvalue of the system. sys For the system matrix, l i and r i These are the system's corresponding dominant eigenvalues ​​λ. i The left and right eigenvectors satisfy the following relationship:

[0059] A sys r i =λ i r i ,l i A sys =λ i l i ;

[0060] The above analyses of the dominant eigenvalue sensitivity are all based on the assumption that the system possesses time-invariant characteristics. However, for many physical and engineering problems, such as power systems, celestial motion, and attitude control, the nonlinear differential-algebraic equations describing the dynamic evolution of these systems often have periodic solutions. In this case, the system exhibits time-varying characteristics, i.e., it has the periodic solutions shown below:

[0061]

[0062] When studying the dominant eigenvalue sensitivity problem of such systems, similarly, we first linearize the system based on its steady-state periodic trajectory:

[0063]

[0064] Substituting the steady-state periodic trajectory into the above equation, we obtain the periodically time-varying A.11 A 12 A 21 A 22 The coefficient matrix indicates that the system exhibits linear, periodic, and time-varying properties.

[0065] like Figure 1 As shown, this invention provides a dynamic stability analysis and control method for a linear periodic time-varying system, comprising:

[0066] S1: Calculate the dominant eigenvalue λ of the system matrix A for a linear periodic time-varying system. i The corresponding left eigenvector l i and the right eigenvector r i ;

[0067] S2: Using the explicit representation of the interest parameter α in the system matrix A, part A sens1 and implicit expression part A sens2 Calculate the partial derivative of the system matrix A with respect to the parameter of interest α;

[0068] S3: Utilizing the partial derivative of the system matrix A with respect to the parameter of interest α, and the left eigenvector l i and the right eigenvector r i Calculate the dominant eigenvalue λ i The partial derivative with respect to the parameter of interest α;

[0069] S4: The dominant eigenvalue λ i The partial derivative with respect to the parameter of interest α is used as the dominant eigenvalue λ. i The sensitivity of the parameter α to the dominant eigenvalue λ is determined. i The magnitude and direction of the influence are used for dynamic stability analysis and control of linear periodic time-varying systems.

[0070] Specifically, by linearizing the original mathematical relationships of the nonlinear system based on the steady-state periodic trajectory, a linear periodic time-varying system is obtained:

[0071]

[0072]

[0073] in,

[0074] This represents the derivative of Δx with respect to time.

[0075] Δx represents: a first-order approximation of the state variables in the system;

[0076] Δy represents: a first-order approximation of the algebraic variables in the system;

[0077] A 11 A 12 A21 A 22 The constant coefficient matrices are respectively n×n, n×m, m×n, and m×m dimensional;

[0078] x ss This represents the periodic trajectory of the state variable under steady-state conditions.

[0079] y ss Represents the periodic trajectory of an algebraic variable under steady state;

[0080] α is the system parameter of interest;

[0081] System Matrix

[0082] In one embodiment, S1 includes:

[0083] Based on the numerical integration method, the numerical solution of the system state transition matrix Φ(T) can be obtained by calculating the state value x(T) at time T when the identity matrix is ​​used as the initial state of a linear periodic time-varying system. First, the dominant eigenvalue vector matrix of the system is calculated, where eig represents the dominant eigenvalue and left and right eigenvectors of the calculated matrix.

[0084]

[0085] Further calculate the left and right eigenvectors of the LTP system:

[0086] R(t)=Φ(t,0)R(0)e -Λt L(t)=R -1 (t).

[0087] In one embodiment, another key to calculating the dominant eigenvalue sensitivity of a linear periodic time-varying system lies in determining the partial derivative of the system matrix A with respect to the parameter of interest α. In this embodiment, the explicit representation of the parameter of interest α in the system matrix A is utilized. sens1 and implicit expression part A sens2 Calculate the partial derivatives of the system matrix A with respect to the parameter of interest α. S2 includes:

[0088] S21: Calculate the partial derivatives of the state variables and algebraic variables of a linear periodic time-varying system with respect to the parameter of interest α. and Therefore, the implicit representation A of the parameter of interest α in the system matrix A is calculated. sens2 ;x ss The periodic trajectory of the state variable y under steady state represents the periodic trajectory of the state variable y under steady state. ss Represents the periodic trajectory of an algebraic variable under steady state;

[0089] S22: Calculate the explicit representation of the parameter of interest α in the system matrix A. sens1 ;

[0090] S23: The expression section A will be displayed. sens1 The superposition of the implicit expression and the implicit expression is taken as the partial derivative of the system matrix A with respect to the parameter of interest α.

[0091] In one embodiment, S2 needs to clarify the relationship between each element of the original system matrix and the parameter of interest α, and obtain the partial derivative of the system matrix based on the chain rule:

[0092]

[0093] Where A sens1 To focus on the explicit representation of parameter α in the system matrix, differentiation can be directly performed using symbolic computation in mathematical software. sens2 This involves the implicit expression of the parameter α, where the partial derivatives of the system matrix with respect to the state variables and algebraic variables can be obtained through symbolic computation using mathematical software. However, the partial derivatives of the steady-state trajectories of the state variables and algebraic variables with respect to the parameter α cannot be obtained through symbolic computation and their analytical forms need to be further derived.

[0094] Based on the above equation, we can see the change in the steady-state periodic trajectory of the system when α changes slightly:

[0095]

[0096] In one embodiment, S21 includes: obtaining the partial derivatives of the system's state variables and algebraic variables with respect to the parameters of interest based on Floquet theory, as shown in the following equations:

[0097]

[0098] Where Φ(t,0) represents the state transition matrix of the system from time 0 to t;

[0099] Φ(T,0) represents the state transition matrix of the system from 0 to T.

[0100] Φ(T,τ) represents the state transition matrix of the system as τ-T;

[0101] g represents a system of differential equations with algebraic variables;

[0102] f represents a system of differential equations for the state variables;

[0103] K α (τ) represents

[0104] Furthermore, utilizing The characterization focuses on the implicit representation of parameter α in the system matrix A. sens2 .

[0105] In one embodiment, S22 includes: utilizing the formula Characterize the explicit representation of the parameter α in the system matrix A. sens1 .

[0106] In one embodiment, according to Floquet theory, the dominant eigenvalues ​​of a linear periodic time-varying system are related to the system matrix. The relationship is shown in the equation:

[0107]

[0108] The dominant eigenvalue λ is obtained using the above formula. i The partial derivative with respect to the parameter of interest α is a time-independent value:

[0109]

[0110] Organizing can yield

[0111]

[0112] In one embodiment, S4

[0113] Because of l i r i =1, then term2 in the formula is 0;

[0114] Due to the sensitivity of dominant eigenvalues The value of is independent of time t, while l i , If all of them are periodic and their product is a periodic function, and their derivative with respect to time t does not contain a zero-order quantity of fft, then... The value does not include the calculation result of term3 in the formula;

[0115] because The value is independent of time t. The value will only include that in term1 The zeroth order quantity of the calculated FFT;

[0116] Therefore, the dominant eigenvalue sensitivity form is obtained as shown, that is, the 0th order {·}0 of the FFT containing only the first term; that is:

[0117]

[0118] For example, for small-disturbance stability control of power systems with complex periodic time-varying characteristics, the variables of interest are system operating mode, transformer ratio, equipment control parameters, etc., and the dominant characteristic is a weakly damped broadband oscillation mode; the dynamic stability analysis and control method of the linear periodic time-varying system of this invention is applied as follows:

[0119] Determine the sensitivity of the weakly damped broadband oscillation mode in the system to the eigenvalues ​​of the variables of interest; construct a parameter optimization set, including the increase or decrease in output of different equipment, the change in line voltage level under the influence of different transformer taps, and the change in control parameters of different equipment; based on the parameter optimization set, adjust the operating mode, voltage level, and control parameters to increase the stability margin of the system.

[0120] According to another aspect of the present invention, a dynamic stability analysis and control device for a linear periodic time-varying system is provided, comprising:

[0121] The first calculation module is used to calculate the dominant eigenvalue λ of interest in the system matrix A of a linear periodic time-varying system. i The corresponding left eigenvector l i and the right eigenvector r i ;

[0122] The second calculation module is used to utilize the explicit representation of the parameter of interest α in the system matrix A. sens1 and implicit expression part A sens2 Calculate the partial derivative of the system matrix A with respect to the parameter of interest α;

[0123] The third calculation module is used to utilize the partial derivative of the system matrix A with respect to the parameter of interest α, and the left eigenvector l i and the right eigenvector r i Calculate the dominant eigenvalue λ i The partial derivative with respect to the parameter of interest α;

[0124] The analysis and control module is used to analyze the dominant eigenvalue λ. i The partial derivative with respect to the parameter of interest α is used as the dominant eigenvalue λ. i The sensitivity of the parameter α to the dominant eigenvalue λ is determined. i The magnitude and direction of the influence are used for dynamic stability analysis and control of linear periodic time-varying systems.

[0125] According to another aspect of the present invention, a dynamic stability analysis and control system for a linear periodic time-varying system is provided, including a memory and a processor. The memory stores a computer program, and the processor executes the computer program to implement the steps of the above-described method.

[0126] According to another aspect of the present invention, a computer storage medium is provided, wherein the methods described in the embodiments of the present invention can be implemented in hardware or firmware, or implemented as computer code that can be recorded on a storage medium, or implemented as computer code originally stored on a remote storage medium or a non-transitory machine-readable storage medium and subsequently stored on a local storage medium after being downloaded via a network, thereby allowing the methods described herein to be processed by software stored on a storage medium using a general-purpose computer, a dedicated processor, or programmable or dedicated hardware. The storage medium may be a magnetic disk, an optical disk, read-only memory, random access memory, flash memory, hard disk, or solid-state drive, etc.; further, the storage medium may also include combinations of the above types of memory. It is understood that a computer, processor, microprocessor controller, or programmable hardware includes storage components capable of storing or receiving software or computer code, which, when accessed and executed by the computer, processor, or hardware, implements the methods shown in the embodiments described above.

[0127] Those skilled in the art will readily understand that the above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A dynamic stability analysis and control method for a linear periodic time-varying system, characterized in that, include: Calculate the system matrix of a linear periodic time-varying system A Dominant eigenvalues ​​of interest λ i The corresponding left feature vector l i and right eigenvectors r i ; Using the parameters of interest α In the system matrix A The display expression section A sens1 and implicit expression part A sens2 Calculate the system matrix A For the parameters of interest The partial derivative; Using the system matrix A For the parameters of interest The partial derivative of the left eigenvector l i and the right feature vector r i Calculate the dominant eigenvalue λ i For the parameters of interest The partial derivative; The dominant feature value λ i For the parameters of interest The partial derivative of is used as the dominant eigenvalue. λ i For the parameters of interest Sensitivity, to determine the parameters of interest For the dominant eigenvalue λ i The magnitude and direction of the influence are used to analyze and control the dynamic stability of the linear periodic time-varying system.

2. The dynamic stability analysis and control method for a linear periodic time-varying system as described in claim 1, characterized in that, The system matrix of the linear periodic time-varying system is calculated. A Dominant eigenvalues ​​of interest λ i The corresponding left feature vector l i and right eigenvectors r i include: Using formula and Calculate the left eigenvector matrix of a linear periodic time-varying system. L (t) and the right eigenvector matrix R (t), based on the left feature vector matrix L (t) and the right eigenvector matrix R (t) yields the left eigenvector l i and right eigenvectors r i ; in, Let be the state transition matrix of the system at time 0-t; The right eigenvector at time 0; The system matrix A eigenvalues Λ The corresponding matrix index.

3. The dynamic stability analysis and control method for a linear periodic time-varying system as described in claim 1, characterized in that, The use of attention parameters α In the system matrix A The display expression section A sens1 and implicit expression part A sens2 Calculate the system matrix A For the parameters of interest The partial derivatives include: Calculate the partial derivatives of the state variables and algebraic variables of the linear periodic time-varying system with respect to the parameter of interest α. and , To represent the partial derivative sign, using and Calculate the parameters of interest α In the system matrix A Implicit expression part in A sens2 ; This represents the periodic trajectory of the state variable under steady-state conditions. Represents the periodic trajectory of an algebraic variable under steady state; Calculate the parameters of interest α In the system matrix A The display expression section A sens1 ; The display expression part A sens1 The superposition of the implicit expression part and the system matrix is ​​the system matrix. A The partial derivative with respect to the parameter of interest α.

4. The dynamic stability analysis and control method for a linear periodic time-varying system as described in claim 3, characterized in that, The partial derivatives of the state variables and algebraic variables of the linear periodic time-varying system with respect to the parameter of interest α are calculated. and ,use and Calculate the parameters of interest α In the system matrix A Implicit expression part in A sens2 ,include: The partial derivatives of the state variables and algebraic variables of the linear periodic time-varying system with respect to the parameter of interest α are calculated based on Floquet theory. and ; use The parameters of interest are calculated. α In the system matrix Implicit expression part in A sens2 , Represents the system matrix right The partial derivative, Represents the system matrix right The partial derivative of .

5. The dynamic stability analysis and control method for a linear periodic time-varying system as described in claim 4, characterized in that, The partial derivatives of the state variables and algebraic variables of the linear periodic time-varying system with respect to the parameter of interest α are calculated based on Floquet theory. and ; include: Using formula and Calculate the partial derivatives of the state variables and algebraic variables of the linear periodic time-varying system with respect to the parameter of interest α; Where τ represents the integrand of the function when convolved with respect to time t, dτ represents the infinitesimal element of τ, x represents the state variable of the system, and y represents the algebraic variable of the system. This represents the state transition matrix of the system from time 0 to t. Represents the identity matrix. This represents the state transition matrix of the system from 0 to T. Indicates when The state transition matrix of the system at time T; Indicates when -t represents the state transition matrix of the system; g represents the system of differential equations with algebraic variables; express Time corresponding ; f represents a system of differential equations for the state variables.

6. The dynamic stability analysis and control method for a linear periodic time-varying system as described in claim 3, characterized in that, The calculation of the attention parameter α In the system matrix A The display expression section A sens1 , include: Using formula Characterizing the parameters of interest α In the system matrix A The display expression section A sens1 .

7. The dynamic stability analysis and control method for a linear periodic time-varying system as described in claim 1, characterized in that, The use of the system matrix A For the parameters of interest The partial derivative of the left eigenvector l i and the right feature vector r i Calculate the dominant eigenvalue λ i For the parameters of interest The partial derivatives include: Using formula Calculate the dominant eigenvalue λ i For the parameters of interest partial derivatives ;in, , l i Represents the left eigenvector. r i This represents the right eigenvector.

8. The dynamic stability analysis and control method for a linear periodic time-varying system as described in claim 7, characterized in that, The parameters of interest The partial derivatives are calculated as follows: set up l i r i =1, then term2 is 0; If the derivative of time t does not contain a zeroth-order quantity of fft, then λ i / α The value does not include the calculation result of term3; λ i / α The value contains term1 The zeroth order quantity of the calculated FFT is then... .

9. A dynamic stability analysis and control device for a linear periodic time-varying system, characterized in that, include: The first calculation module is used to calculate the system matrix of a linear periodic time-varying system. A Dominant eigenvalues ​​of interest λ i The corresponding left feature vector l i and right eigenvectors r i ; The second calculation module is used to utilize the parameters of interest. α In the system matrix A The display expression section A sens1 and implicit expression part A sens2 Calculate the system matrix A For the parameters of interest The partial derivative; The third calculation module is used to utilize the system matrix. A For the parameters of interest The partial derivative of the left eigenvector l i and the right feature vector r i Calculate the dominant eigenvalue λ i For the parameters of interest The partial derivative; Analysis and control module, used to process the dominant feature value λ i For the parameters of interest The partial derivative of is used as the dominant eigenvalue. λ i For the parameters of interest The sensitivity of the parameter of interest is used to determine the parameter of interest. For the dominant eigenvalue λ i The magnitude and direction of the influence are used to analyze and control the dynamic stability of the linear periodic time-varying system.

10. A dynamic stability analysis and control system for a linear periodic time-varying system, comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1 to 8.

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