Chebyshev configuration-based periodic time-varying system trajectory obtaining method and system

Abstract

The application provides a Chebyshev configuration-based periodic time-varying system trajectory obtaining method and system, and belongs to the technical field of power electronics. The method comprises the following steps: establishing a periodic boundary value problem model of the periodic time-varying system, and determining the system period and the number of configuration points; performing time scale mapping on a single period interval, selecting Chebyshev configuration points, and constructing a spectral differential matrix; applying system dynamics constraints at each configuration point, and discretizing the differential equation into a nonlinear implicit algebraic equation set; introducing a periodic closure constraint to form a closed nonlinear algebraic equation set; solving the closed equation set by using an inner-outer iteration strategy, and outputting a steady-state periodic discrete trajectory. The application directly converts the differential equation into an algebraic equation set by configuring Chebyshev points in a single period interval and constructing a spectral differential matrix, avoids long-time domain integration, and has the advantages of high calculation efficiency, fast convergence speed, good numerical stability, and strong engineering implementability.

Description

Technical Field

[0001] This invention belongs to the field of power electronics technology, and in particular relates to a method and system for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] Periodic time-varying systems are widely used in engineering fields such as power electronic grid-connected devices, modular multilevel converters, and resonant converters. They generally exhibit nonlinear and periodic time-varying characteristics, and their steady-state operation corresponds to a periodic solution that satisfies periodic boundary conditions. Accurately obtaining the steady-state periodic trajectory is a core prerequisite for constructing linear time-varying periodic models, conducting stability analysis, harmonic calculations, and controller design, and directly affects the accuracy of system analysis and the reliability of control.

[0004] Current mainstream methods for solving steady-state trajectories mainly include long-time-domain integration to steady state followed by truncating the last period as an approximation, fixed-point iteration based on one-period mapping, and harmonic balance solutions based on Fourier series expansion. Long-time-domain integration is simple to implement, truncating a single-period waveform after long-term simulation to steady state. However, it converges extremely slowly under weakly damped and critically stable conditions, requiring integration of tens to hundreds of periods, resulting in long processing times, large accumulated errors, and inefficiencies that fail to meet engineering requirements. Fixed-point iteration using periodic mapping transforms the solution into a periodic closed-loop problem; however, each iteration still requires numerical integration of the complete period, leading to complex state transition matrix construction, high computational cost, slow convergence or divergence under strong nonlinearity, and insufficient robustness. Harmonic balance methods, based on Fourier series expansion, are suitable for harmonic analysis, but require frequent time-frequency domain transformations, rely on the harmonic truncation order, suffer from insufficient accuracy at low orders, and experience equation scaling at higher orders, making Jacobian matrix calculation difficult and resulting in poor numerical stability.

[0005] Therefore, existing methods generally suffer from drawbacks such as slow convergence, high complexity, and difficulty in balancing accuracy and scale, making them unsuitable for fast and high-precision solutions under weakly damped, critically stable, and strongly nonlinear conditions. Summary of the Invention

[0006] To overcome the shortcomings of the prior art, this invention provides a method and system for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration. The aim is to construct a framework for solving the trajectory of a periodic time-varying system that combines high accuracy and high efficiency by introducing Chebyshev configuration theory, so as to solve the technical bottlenecks of existing methods in weakly damped, critically stable and strongly nonlinear conditions, such as slow convergence, high complexity and difficulty in balancing accuracy and computational scale.

[0007] To achieve the above objectives, one or more embodiments of the present invention provide the following technical solutions: The first aspect of this invention provides a method for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration; Methods for obtaining the trajectory of periodic time-varying systems based on Chebyshev configuration include: S1. Obtain the periodic time-varying system model and parameters, determine the system period and the number of configuration points, establish the Poincaré mapping and fixed-point solution model corresponding to the steady-state periodic trajectory, and transform the problem of solving the steady-state periodic trajectory into a boundary value problem under periodic boundary conditions. S2, based on a determined period and number of configuration points, performs time-scale mapping on the single-period interval, selects Chebyshev configuration points on the standard interval and constructs a spectral differential matrix, transforming derivative operations in the continuous time domain into algebraic operations; S3. Using the constructed spectral differential matrix, system dynamic constraints are applied at each placement point, and the original differential equation is discretized into a system of nonlinear implicit algebraic equations about the state variables of the placement points. S4. Based on the aforementioned nonlinear algebraic equations, a periodic closure constraint is introduced to ensure that the discrete trajectory satisfies the steady-state periodic condition. This is achieved by forming a closed nonlinear algebraic equation system to provide a closed description of the periodic boundary conditions. S5 uses an internal and external iteration strategy to solve the closed nonlinear algebraic equation system and outputs the steady-state periodic discrete trajectory.

[0008] As a further technical solution, the model and parameters of the periodic time-varying system are obtained, and a Poincaré mapping and fixed-point solution model corresponding to the steady-state periodic trajectory are established. This transforms the problem of solving the steady-state periodic trajectory into a boundary value problem under periodic boundary conditions, including: Obtain the time-domain state equation and system parameters of a periodic time-varying system to determine the system's fundamental period; The solution of steady-state periodic trajectory is transformed into a boundary value problem that satisfies periodic boundary conditions. A Poincaré mapping and fixed-point solution model is established, and a periodic evolution operator is introduced to make the solution of periodic steady-state trajectory equivalent to the solution of fixed-point equations. Set the convergence thresholds for inner and outer layer iterations and provide a guessed value for the initial state of the period to complete the mathematical modeling of the steady-state solution problem.

[0009] As a further technical solution, based on a determined period and number of collocation points, a time-scale mapping is performed on the single-period interval. Chebyshev collocation points are selected on the standard interval, and a spectral differential matrix is ​​constructed, including: Set the time interval [0, T Mapping to standard intervals τ ∈[ The mapping relationship is: 1,1]

[0010] Select within the standard interval N+1 Chebyshev deployment point Its value is:

[0011] And based on the time mapping relationship, it is converted into the configuration time within a single period interval. ; The steady-state periodic trajectory is discretized at the selected locus points, and a spectral differential matrix is ​​established at the locus points such that the derivative at the locus points satisfies:

[0012] in, For a single-period interval [0, T The elements of the upper spectral differential matrix are used to characterize the configuration time. Discrete weighting coefficients of the derivatives of the state variables at the configuration point; System state variables x ( t During configuration The value of .

[0013] As a further technical solution, by utilizing the constructed spectral differential matrix, system dynamic constraints are applied at each placement point, discretizing the original differential equation into a system of nonlinear implicit algebraic equations concerning the state variables at the placement points, including: Substituting the obtained discrete expression into the system dynamics equations and applying dynamic constraints at each placement point, the original differential equations are transformed into a system of nonlinear algebraic equations concerning the state variables at the placement points, as shown below:

[0014] in, For parameter set.

[0015] As a further technical solution, step S3 also includes: To ensure the computability and numerical stability of the spectral differential matrix, the matrix elements are constructed according to the explicit formula for Chebyshev collocation points, and the endpoint weight coefficients are defined:

[0016] when k ≠ j When the spectral differential matrix has off-diagonal elements, it is:

[0017] when k = j To improve numerical stability, the row sum rule is used to calculate the diagonal elements:

[0018] We obtain the discrete expression of the time derivative at the intrinsic location point of a single period:

[0019] in, The time derivative is discretely expressed at the intrinsic location point of a single period; For a single-period interval [0, T Elements of the upper spectral differential matrix.

[0020] As a further technical solution, based on the aforementioned nonlinear algebraic equation system, a periodic closure constraint is introduced to ensure that the discrete trajectory satisfies the steady-state periodic condition. This achieves a closed description of the periodic boundary conditions by forming a closed nonlinear algebraic equation system, including: The state variables at each configuration point within a single period are stacked to form a discrete trajectory vector. :

[0021] And define nonlinear vector functions :

[0022] Based on the obtained dynamic constraints of the placement points, construct the discrete trajectory vector. X A system of nonlinear algebraic equations:

[0023] In the formula, D For the spectral differential matrix, F ( X () is a vector function calculated from the system's nonlinear function at each placement point; A periodic closure constraint is introduced at the endpoint of a single period to ensure that the discrete trajectory satisfies the steady-state periodic condition:

[0024] By incorporating this periodic constraint into the nonlinear equation system, a closed nonlinear algebraic equation system is obtained for solving the steady-state periodic trajectory.

[0025] As a further technical solution, an internal and external iterative strategy is employed to solve the closed nonlinear algebraic equation system, ultimately outputting a steady-state periodic discrete trajectory, including: The single-period discrete trajectory is solved by inner iteration with a given initial period value. Specifically: given the outer iteration... m The initial value of the second cycle is used to update the unknowns of the discrete trajectory by calculating the Jacobian matrix and residual function according to the Newton-Raphson iterative formula. After the inner iteration converges, the outer iteration is obtained. m The corresponding single-cycle steady-state discrete trajectory is then used to extract the single-cycle end state from its endpoint state. The initial value of the period is updated through external iteration until the periodic closure convergence accuracy is met, and finally the steady-state periodic discrete trajectory is output. Specifically, the end state of the single period obtained by the internal iteration and the initial state are used to form the external iteration residual, and the initial value of the period is corrected according to the external iteration update law. When the external iteration convergence criterion meets the external iteration convergence threshold, the steady-state periodic discrete trajectory is output.

[0026] A second aspect of the present invention provides a system for obtaining the trajectory of a periodic time-varying system based on a Chebyshev configuration.

[0027] A system for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration includes: The periodic boundary value modeling module is configured to: acquire the periodic time-varying system model and parameters, determine the system period and the number of configuration points, establish the Poincaré mapping and fixed-point solution model corresponding to the steady-state periodic trajectory, and transform the problem of solving the steady-state periodic trajectory into a boundary value problem under periodic boundary conditions; The spectral differential separation module is configured to: perform time-scale mapping on a single-period interval based on a given period and number of configuration points, select Chebyshev configuration points on the standard interval and construct a spectral differential matrix, and transform derivative operations in the continuous time domain into algebraic operations. The algebraic equation construction module is configured to: use the constructed spectral differential matrix to apply system dynamic constraints at each placement point, and discretize the original differential equation into a system of nonlinear implicit algebraic equations with respect to the state variables of the placement points; The periodic closure constraint module is configured to: introduce periodic closure constraints on the basis of the nonlinear algebraic equation system, so that the discrete trajectory satisfies the steady-state periodic condition, and realize the closed description of the periodic boundary condition by forming a closed nonlinear algebraic equation system; The iterative solution module is configured to solve the closed nonlinear algebraic equation system using internal and external iterative strategies, and output the steady-state periodic discrete trajectory.

[0028] A third aspect of the present invention provides a computer-readable storage medium having a program stored thereon, which, when executed by a processor, implements the steps of the method for obtaining the trajectory of a periodic time-varying system based on a Chebyshev configuration as described in the first aspect of the present invention.

[0029] A fourth aspect of the present invention provides an electronic device, including a memory, a processor, and a program stored in the memory and executable on the processor, wherein the processor executes the program to implement the steps in the method for obtaining the trajectory of a periodic time-varying system based on a Chebyshev configuration as described in the first aspect of the present invention.

[0030] The above one or more technical solutions have the following beneficial effects: (1) This invention uses Chebyshev placement points for discretization. The densified distribution of endpoints can improve the fitting accuracy at the periodic boundary. Combined with the calculation of the spectral differential matrix and row sum regular diagonal elements, it effectively suppresses numerical oscillations and calculation errors. The solution scale and accuracy can be flexibly adjusted by the number of placement points, avoiding the problem of insufficient accuracy or equation expansion caused by improper selection of harmonic order in the harmonic balance method. It still maintains stable convergence in strongly nonlinear and high-dimensional systems, taking into account both calculation accuracy and numerical robustness. In addition, this invention completes modeling and solving only within a single period interval, without the need for long time-domain integration, fundamentally solving the problem of slow convergence and long time consumption of traditional methods under weakly damped and critically stable conditions. Compared with long time-domain integration, this method directly discretizes and models within a single period, greatly shortening the calculation time. It is especially suitable for strong rigidity and slow convergence systems such as power electronics and high-frequency conversion, and can quickly output steady-state trajectories that meet accuracy requirements, providing efficient support for engineering simulation and online analysis.

[0031] (2) This invention constructs a unified nonlinear equation system through dynamic constraints and periodic closure constraints, and achieves fast convergence by using inner and outer double-layer Newton iterations. It eliminates the need to repeatedly construct state transition matrices or perform time-frequency domain transformations, thus reducing the complexity of the algorithm implementation. The obtained steady-state trajectory satisfies both system dynamics and periodic constraints, and can be directly used for trajectory linearization, small-signal stability analysis, and harmonic calculation. It has strong versatility and is suitable for steady-state analysis and controller design of various nonlinear periodic time-varying systems.

[0032] Advantages of additional aspects of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0033] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0034] Figure 1 This is a flowchart of the method in the first embodiment.

[0035] Figure 2 The flowchart for the single-period Chebyshev configuration of discrete and periodic closed-loop solutions in the first embodiment is shown.

[0036] Figure 3 This is a schematic diagram of the Chebyshev configuration point distribution for the first embodiment.

[0037] Figure 4 This is a system structure diagram of the second embodiment. Detailed Implementation

[0038] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0039] It should be noted that the terminology used herein is for the purpose of describing particular implementations only and is not intended to limit the exemplary implementations of the present invention.

[0040] Where there is no conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.

[0041] Example 1 This embodiment discloses a method for obtaining the trajectory of a periodic time-varying system based on Chebyshev placement. The method uses Chebyshev placement points to discretize the differential equations in a single period interval into a system of nonlinear algebraic equations, and introduces periodic closure constraints to describe the periodic boundary conditions of the steady-state trajectory. Based on this, the method of Newton is used to iteratively solve the system of algebraic equations to obtain the steady-state trajectory that satisfies both dynamic and periodic constraints.

[0042] Specifically, such as Figure 1 As shown, the method for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration includes: S1. Obtain the periodic time-varying system model and parameters, determine the system period and the number of configuration points, establish the Poincaré mapping and fixed-point solution model corresponding to the steady-state periodic trajectory, and transform the problem of solving the steady-state periodic trajectory into a boundary value problem under periodic boundary conditions.

[0043] like Figure 2 As shown, in this embodiment, by selecting the Poincaré section and calculating the one-period evolution of the system state, the Poincaré mapping relationship is established, and the solution of the steady-state periodic trajectory is transformed into a fixed-point solution problem, thereby providing a foundation for the subsequent configuration of discrete and closed equations.

[0044] In step S1, the time-domain state equation of the periodic time-varying system is first obtained, and its fundamental period is determined. T For a general form of nonlinear periodic time-varying system, it can be expressed as: (1) In the formula, For state vectors, For parameter set, The system is periodic. The steady-state trajectory of the system is a periodic solution satisfying the periodic boundary conditions. ,Right now: (2) in, For the steady-state periodic trajectory at the initial moment t =0 is the state value.

[0045] To characterize the fixed-point features of periodic steady states in phase space, a periodic... T A one-periodic evolution operator for sampling the system state at intervals Makes it possible to achieve the following in any periodic interval The above satisfies: (3) When the system reaches steady-state periodic operation, its sampling points converge to a fixed point under the Poincaré map. Therefore, solving the steady-state periodic trajectory is equivalent to solving the fixed-point equation: (4) Furthermore, in step S1, an outer-layer iteration convergence threshold is set. Inner layer iteration convergence threshold We then provide an initial guess of the initial state of the period as the initial condition for the subsequent solution process. This completes the mathematical description of the problem of finding the steady-state periodic trajectory.

[0046] S2, based on a determined period and number of configuration points, performs time-scale mapping on the single-period interval, selects Chebyshev configuration points on the standard interval and constructs a spectral differential matrix, transforming derivative operations in the continuous time domain into algebraic operations.

[0047] like Figure 3 As shown, Chebyshev collocation points are selected in the standard interval [-1,1] and mapped to a single-cycle time axis to obtain the collocation time. This point set exhibits endpoint-encrypted distribution characteristics, which can improve the discrete accuracy near the periodic endpoints and provide a basis for subsequent spectral differential matrices. D This provides a foundation for the construction and discretization of dynamic constraints.

[0048] In step S2, to perform high-precision discrete representation of the steady-state periodic trajectory within a single-period interval, the time interval [0, ... T Mapping to standard intervals τ ∈[ The mapping relationship is: 1,1] (5) Select within the standard interval N +1 Chebyshev deployment point Its value is: (6) And based on the time mapping relationship, it is converted into the configuration time within a single period interval. This is used to subsequently construct the spectral differential matrix and establish system dynamic constraints at the placement points.

[0049] The steady-state periodic trajectory is discretized at the obtained configuration points. Assume the trajectory within a single period can be represented by... N Approximate representation using an interpolation polynomial of degree: (7) In the formula, For configuration point-based The constructed Lagrange basis functions allow for the establishment of Chebyshev spectral differential operators (spectral differential matrices) at the collocation points. This ensures that the derivative at the configuration point satisfies: (8) in, For a single-period interval [0, T The elements of the upper spectral differential matrix are used to characterize the configuration time. Discrete weighting coefficients of the derivatives of the state variables at the configuration point; System state variables x ( t During configuration The value of .

[0050] S3 uses the constructed spectral differential matrix to apply system dynamic constraints at each placement point, discretizing the original differential equation into a system of nonlinear implicit algebraic equations about the state variables of the placement points.

[0051] Substituting the discrete expression obtained in step S2 into the system dynamics equations, and applying dynamic constraints at each placement point, we can obtain: (9) This transforms the original differential equation into a system of nonlinear algebraic equations concerning the state variables of the placement points, laying the foundation for the subsequent introduction of periodic closure constraints and the construction of a closed system of equations.

[0052] To ensure the spectral differential matrix D The computability and numerical stability of the matrix are ensured by the fact that its matrix elements can be constructed using the explicit formula for Chebyshev locus points, and the endpoint weight coefficients are defined. : (10) when k ≠ j When the spectral differential matrix has off-diagonal elements, it is: (11) when k = j To improve numerical stability, the row sum rule is used to calculate the diagonal elements: (12) This yields the discrete expression of the time derivative at the intrinsic location point of a single period: (13) in, System state variables x ( t During configuration The value of .

[0053] Based on the above process, Chebyshev placement discretization and spectral differential operator construction for single-period intervals were realized, so as to facilitate the subsequent establishment of placement constraints for differential equations at placement points and the formation of a system of nonlinear algebraic equations.

[0054] Furthermore, to avoid using long-term time-domain integration to calculate the single-period evolution process, the Chebyshev configuration time determined by the differential equations of the periodic time-varying system is used instead. Apply configuration constraints. For the nonlinear periodic time-varying system given in step S1: (14) Substituting the discrete expression for the time derivative obtained in step S2 into the above equation, and forcibly satisfying the system dynamics constraints at each placement point, we get: (15) Equation (15) is a nonlinear algebraic equation obtained after applying system dynamic constraints at each configuration time, and its unknowns are the state variables at each configuration point. x ( t 0), x ( t 1), …, x ( t N The purpose of this equation is to discretize the continuous-time differential equation into a finite-dimensional nonlinear algebraic equation system concerning the state variables of the location point, thereby providing a foundation for subsequently introducing periodic closure constraints and uniformly solving the steady-state periodic trajectory.

[0055] S4. Based on the nonlinear algebraic equations, a periodic closure constraint is introduced to make the discrete trajectory satisfy the steady-state periodic condition. A closed nonlinear algebraic equations system is formed to achieve a closed description of the periodic boundary conditions.

[0056] In step S4, to facilitate a unified solution, the state variables at each configuration point within a single period are stacked to form a discrete trajectory vector. : (16) And define nonlinear vector functions : (17) Based on the dynamic constraints of the placement points obtained in step S3, construct the discrete trajectory vector.X A system of nonlinear algebraic equations: (18) In the formula, D For the spectral differential matrix, F ( X The vector function is calculated from the system's nonlinear function at each placement point. To ensure that the discrete trajectory satisfies the steady-state periodic condition, a periodic closure constraint is introduced at the single-period endpoints: (19) By incorporating this periodic constraint into the aforementioned nonlinear equation system, we obtain a closed nonlinear algebraic equation system for solving the steady-state periodic trajectory. H ( X )=0.

[0057] S5. The closed nonlinear algebraic equation system is solved by internal and external iteration strategies. The single-period discrete trajectory is solved by internal iteration under a given initial period value. The initial period value is updated by external iteration until the periodic closure convergence accuracy is met. Finally, the steady-state periodic discrete trajectory is output.

[0058] In step S5, in order to solve the system of nonlinear algebraic equations constructed in step S4 H ( X When ) = 0, the Newton-Raphson method is used for iterative solution. In this embodiment, the solution process is organized into a two-layer iterative structure: For the first m The next outer iteration, given an initial value of the period. x 0 (m) The inner iteration is used to solve the single-period discrete trajectory under this initial condition, and its corresponding nonlinear algebraic equation system can be expressed as: (20) After the inner iteration converges, the th iteration is obtained. m The single-period discrete trajectory corresponding to the next outer iteration X (m) And construct a periodic closed residual from its terminal state: (twenty one) The outer iteration updates the initial value of the period based on the periodic closed residual: (twenty two) When the following conditions are met: (twenty three) When the periodic closure condition is satisfied, the corresponding steady-state periodic discrete trajectory is output.

[0059] The inner iteration is solved under a given initial period. H ( XThe single-cycle discrete trajectory is obtained by setting )=0; the external iteration updates the initial value of the cycle based on the single-cycle end state obtained from the internal iteration until the cycle closure convergence accuracy is met.

[0060] (1) Inner Iteration: Solving a system of nonlinear algebraic equations =0 Given the first m Initial value of the period corresponding to the next outer iteration x 0 (m) And take the initial guess of the inner iteration. By calculating the Jacobian matrix J ( X and residual function G ( X The unknowns of the discrete trajectory are updated according to the Newton-Raphson iterative formula. First, according to the Newton-Raphson iterative formula: (twenty four) Update the state vector. J ( X Let be a Jacobian matrix, defined as: (25) After each iteration, calculate the residual function. G ( X ), until the convergence accuracy is met: (26) In the formula, The tolerance error is set. After the inner iteration converges, the outer iteration's... m The corresponding single-period steady-state discrete trajectory And extract the single-cycle end state from its endpoint state. .

[0061] (2) External iteration: Update the initial value of the period and determine if the period is closed and converged. The final state of a single cycle obtained from the inner iteration is combined with the initial state to form the residual of the outer iteration. : (27) The initial value of the period is corrected according to the external iteration update rule, and the external iteration convergence criterion can be adopted as follows: (28) In the formula, This is the convergence threshold for the outer iteration. When the above equation is satisfied, the periodic closure is considered valid, and the steady-state periodic discrete trajectory is output. .

[0062] Example 2 This embodiment discloses a system for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration; like Figure 4 As shown, the trajectory acquisition system for a periodic time-varying system based on the Chebyshev configuration includes: The periodic boundary value modeling module is configured to: acquire the periodic time-varying system model and parameters, determine the system period and the number of configuration points, establish the Poincaré mapping and fixed-point solution model corresponding to the steady-state periodic trajectory, and transform the problem of solving the steady-state periodic trajectory into a boundary value problem under periodic boundary conditions; The spectral differential separation module is configured to: perform time-scale mapping on a single-period interval based on a given period and number of configuration points; select Chebyshev configuration points on the standard interval and construct the spectral differential matrix; and transform derivative operations in the continuous time domain into algebraic operations. The algebraic equation construction module is configured to: use the constructed spectral differential matrix to apply system dynamic constraints at each placement point, and discretize the original differential equation into a system of nonlinear implicit algebraic equations with respect to the state variables of the placement points; The periodic closure constraint module is configured to: introduce periodic closure constraints on the basis of the nonlinear algebraic equation system, so that the discrete trajectory satisfies the steady-state periodic condition, and realize the closed description of the periodic boundary condition by forming a closed nonlinear algebraic equation system; The iterative solution module is configured to: solve the closed nonlinear algebraic equation system using an internal and external iterative strategy; solve the single-period discrete trajectory under a given initial period value through internal iteration; update the initial period value through external iteration until the periodic closure convergence accuracy is met; and finally output the steady-state periodic discrete trajectory.

[0063] Example 3 The purpose of this embodiment is to provide a computer-readable storage medium.

[0064] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps in the method for obtaining the trajectory of a periodic time-varying system based on a Chebyshev configuration as described in Example 1.

[0065] Example 4 The purpose of this embodiment is to provide an electronic device.

[0066] An electronic device includes a memory, a processor, and a program stored in the memory and executable on the processor. When the processor executes the program, it implements the steps in the method for obtaining the trajectory of a periodic time-varying system based on a Chebyshev configuration as described in Embodiment 1.

[0067] The steps and methods involved in the apparatuses of Embodiments 2, 3, and 4 above correspond to those in Embodiment 1. For specific implementation details, please refer to the relevant description section of Embodiment 1. The term "computer-readable storage medium" should be understood as a single medium or multiple media including one or more instruction sets; it should also be understood as including any medium capable of storing, encoding, or carrying an instruction set for execution by a processor and enabling the processor to perform any of the methods in this invention.

[0068] Those skilled in the art will understand that the modules or steps of the present invention described above can be implemented using general-purpose computer devices. Optionally, they can be implemented using computer-executable program code, thereby allowing them to be stored in a storage device for execution by a computer device, or they can be fabricated as separate integrated circuit modules, or multiple modules or steps can be fabricated as a single integrated circuit module. The present invention is not limited to any particular combination of hardware and software.

[0069] While the specific embodiments of the present invention have been described above in conjunction with the accompanying drawings, this is not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of the present invention are still within the scope of protection of the present invention.

Claims

1. A method for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration, characterized in that, include: S1. Obtain the periodic time-varying system model and parameters, determine the system period and the number of configuration points, establish the Poincaré mapping and fixed-point solution model corresponding to the steady-state periodic trajectory, and transform the problem of solving the steady-state periodic trajectory into a boundary value problem under periodic boundary conditions. S2, based on a determined period and number of configuration points, performs time-scale mapping on the single-period interval, selects Chebyshev configuration points on the standard interval and constructs a spectral differential matrix, transforming derivative operations in the continuous time domain into algebraic operations; S3. Using the constructed spectral differential matrix, system dynamic constraints are applied at each placement point, and the original differential equation is discretized into a system of nonlinear implicit algebraic equations about the state variables of the placement points. S4, based on the nonlinear implicit algebraic equation system, introduces periodic closure constraints to make the discrete trajectory satisfy the steady-state periodic conditions, and achieves a closed description of the periodic boundary conditions by forming a closed nonlinear implicit algebraic equation system. S5 uses an internal and external iteration strategy to solve the closed nonlinear implicit algebraic equation system and outputs the steady-state periodic discrete trajectory.

2. The method for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration as described in claim 1, characterized in that, Obtain the model and parameters of the periodic time-varying system, establish the Poincaré mapping and fixed-point solution model corresponding to the steady-state periodic trajectory, and transform the problem of solving the steady-state periodic trajectory into a boundary value problem under periodic boundary conditions, including: Obtain the time-domain state equation and system parameters of a periodic time-varying system to determine the system's fundamental period; The solution of steady-state periodic trajectory is transformed into a boundary value problem that satisfies periodic boundary conditions. A Poincaré mapping and fixed-point solution model is established, and a periodic evolution operator is introduced to make the solution of periodic steady-state trajectory equivalent to the solution of fixed-point equations. Set the convergence thresholds for inner and outer layer iterations and provide a guessed value for the initial state of the period to complete the mathematical modeling of the steady-state solution problem.

3. The method for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration as described in claim 1, characterized in that, Based on a defined period and number of collocation points, a time-scale mapping is performed on the single-period interval. Chebyshev collocation points are selected on the standard interval, and a spectral differential matrix is ​​constructed, including: Set the time interval [0, T Mapping to standard intervals τ ∈[ The mapping relationship is: 1,1] Select within the standard interval N +1 Chebyshev deployment point Its value is: And based on the time mapping relationship, it is converted into the configuration time within a single period interval. t k ; The steady-state periodic trajectory is discretized at the selected placement points, and a spectral differential matrix is ​​established at each placement point such that the derivative at the placement point satisfies: in, For a single-period interval [0, T The elements of the upper spectral differential matrix are used to characterize the configuration time. Discrete weighting coefficients of the derivatives of the state variables at the configuration point; System state variables x ( t During configuration The value of .

4. The method for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration as described in claim 1, characterized in that, Using the constructed spectral differential matrix, system dynamic constraints are applied at each placement point, discretizing the original differential equation into a system of nonlinear implicit algebraic equations about the state variables at the placement points, including: Substituting the obtained discrete expression into the system dynamics equations and applying dynamic constraints at each placement point, the original differential equations are transformed into a system of nonlinear implicit algebraic equations concerning the state variables at the placement points, as shown below: in, For parameter set.

5. The method for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration as described in claim 1, characterized in that, Step S3 also includes: To ensure the computability and numerical stability of the spectral differential matrix, the matrix elements are constructed according to the explicit formula for Chebyshev collocation points, and the endpoint weight coefficients are defined: when k ≠ j When the spectral differential matrix has off-diagonal elements, it is: when k = j To improve numerical stability, the row sum rule is used to calculate the diagonal elements: We obtain the discrete expression of the time derivative at the intrinsic location point of a single period: in, The time derivative is discretely expressed at the intrinsic location point of a single period; For a single-period interval [0, T Elements of the upper spectral differential matrix.

6. The method for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration as described in claim 1, characterized in that, Based on the aforementioned nonlinear implicit algebraic equations, periodic closure constraints are introduced to ensure that the discrete trajectory satisfies the steady-state periodic condition. This is achieved by forming a closed system of nonlinear implicit algebraic equations to provide a closed description of the periodic boundary conditions, including: The state variables at each configuration point within a single period are stacked to form a discrete trajectory vector. : And define nonlinear vector functions : Based on the obtained dynamic constraints of the placement points, construct the discrete trajectory vector. X Nonlinear implicit algebraic equations: In the formula, D For the spectral differential matrix, F ( X () is a vector function calculated from the system's nonlinear function at each placement point; A periodic closure constraint is introduced at the endpoint of a single period to ensure that the discrete trajectory satisfies the steady-state periodic condition: By incorporating this periodic constraint into the nonlinear equation system, a closed nonlinear implicit algebraic equation system is obtained for solving the steady-state periodic trajectory.

7. The method for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration as described in claim 1, characterized in that, The closed nonlinear implicit algebraic equation system is solved using an internal and external iterative strategy, and the steady-state periodic discrete trajectory is finally output, including: The single-period discrete trajectory is solved by inner iteration with a given initial period value. Specifically: given the outer iteration... m The initial value of the second cycle is used to update the unknowns of the discrete trajectory by calculating the Jacobian matrix and residual function according to the Newton-Raphson iterative formula. After the inner iteration converges, the outer iteration is obtained. m The corresponding single-cycle steady-state discrete trajectory is then used to extract the single-cycle end state from its endpoint state. The initial value of the period is updated through external iteration until the periodic closure convergence accuracy is met, and finally the steady-state periodic discrete trajectory is output. Specifically, the end state of the single period obtained by the internal iteration and the initial state are used to form the external iteration residual, and the initial value of the period is corrected according to the external iteration update law. When the external iteration convergence criterion meets the external iteration convergence threshold, the steady-state periodic discrete trajectory is output.

8. A system for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration, characterized in that, include: The periodic boundary value modeling module is configured to: acquire the periodic time-varying system model and parameters, determine the system period and the number of configuration points, establish the Poincaré mapping and fixed-point solution model corresponding to the steady-state periodic trajectory, and transform the problem of solving the steady-state periodic trajectory into a boundary value problem under periodic boundary conditions; The spectral differential separation module is configured to: perform time-scale mapping on a single-period interval based on a given period and number of configuration points, select Chebyshev configuration points on the standard interval and construct a spectral differential matrix, and transform derivative operations in the continuous time domain into algebraic operations. The algebraic equation construction module is configured to: use the constructed spectral differential matrix to apply system dynamic constraints at each placement point, and discretize the original differential equation into a system of nonlinear implicit algebraic equations with respect to the state variables of the placement points; The periodic closure constraint module is configured to: introduce periodic closure constraints on the basis of the nonlinear implicit algebraic equation system, so that the discrete trajectory satisfies the steady-state periodic condition, and realize the closed description of the periodic boundary condition by forming a closed nonlinear implicit algebraic equation system; The iterative solution module is configured to solve the closed nonlinear implicit algebraic equation system using internal and external iterative strategies, and output the steady-state periodic discrete trajectory.

9. A computer-readable storage medium having a program stored thereon, characterized in that, When executed by a processor, the program implements the steps in the method for obtaining the trajectory of a periodic time-varying system based on a Chebyshev configuration as described in any one of claims 1-7.

10. An electronic device comprising a memory, a processor, and a program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the steps in the method for obtaining the trajectory of a periodic time-varying system based on Chebyshev configuration as described in any one of claims 1-7.

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