Robust coupling matrix reconfiguration optimization method based on inverse eigenvalue problem

Abstract

The invention discloses a robust coupling matrix reconfiguration optimization method based on an inverse eigenvalue problem. The method comprises the following steps of: 1, improving an optimization model based on the inverse eigenvalue problem; a second step, improvement of an optimized Levenberg-Marquardt method is carried out, and the Levenberg-Marquardt method is carried out; step 3, an optimization method based on q reduction; the method has the beneficial effects that the method can be suitable for a large-scale and more complex filter topological structure transformation problem, and the robustness of the optimization process is enhanced while the optimization efficiency is improved. Compared with the existing method in a real application example, the research provided by the invention has extremely high optimization success rate and good reliability. Therefore, an efficient comprehensive means is provided for the design of the microwave filter, the efficiency is improved for the subsequent production and debugging of the filter, and a large amount of time is saved.

Description

Technical Field

[0001] The invention relates to a coupling matrix reconfiguration optimization method, in particular to a robust coupling matrix reconfiguration optimization method based on an inverse eigenvalue problem. Background Art

[0002] At present, microwave filters are one of the most important components in communication systems. Their synthesis and design will directly affect the quality of communication. The coupling matrix used to represent the topological structure of the filter network is a basic mathematical tool for the comprehensive design and analysis of modern filters. The element values ​​in the coupling matrix represent the coupling values ​​between the resonant cavities in the corresponding filter. In practical applications, people will design and produce the corresponding filters based on the coupling matrix. For the given filter parameters in advance, such as the order (the number of resonant cavities N plus the source and load, a total of N+2), transmission zero point, return loss, etc., the N+2 type generalized Chebyshev filter function synthesis method can be used to synthesize the full specification N+2 order transverse coupling matrix T. However, since the structure of the filter represented by T is relatively complex (the source and load are connected to all the resonant cavities), it is not suitable for actual production and application. Therefore, it is necessary to transform the full specification N+2 order transverse coupling matrix T into a coupling matrix M with a specified topological structure S without changing the filter response characteristics, so as to facilitate actual production and use. This is an important engineering problem in the field of microwave filtering.

[0003] The classic method of coupling matrix topological transformation is matrix rotation elimination, that is, according to the target topology S, a series of matrix similarity transformations are used to eliminate unnecessary elements in T to obtain the final desired coupling matrix M. However, since the matrix rotation elimination method requires the prior knowledge of the rotation order and rotation angle, this method can only be applied to some classic topological structures, such as "folding type" and "wheel type". For more complex topologies such as "expanding box" structures, this method will no longer be applicable.

[0004] In order to make up for the limitations of the matrix rotation elimination method, various optimization methods for filter network topology transformation have been proposed in recent years. Among them, the most classic and effective optimization method is the optimization model proposed by Lamecki et al. based on the inverse eigenvalue problem of the coupling matrix. The model is derived from the matrix rotation elimination method. Since the four sets of eigenvalues ​​of the coupling matrix (upper and lower principal sub-matrices, core matrix, and eigenvalues ​​corresponding to the coupling matrix itself) are not changed during the rotation elimination process, Lamecki et al. use the eigenvalue difference between the current coupling matrix and the target coupling matrix to construct the optimization model. Although the model has been applied to the comprehensive design of various filters, when the optimization algorithm is used to solve the model, the optimization result is extremely sensitive to the selection of the initial value. When the initial value is not selected properly, the desired target result is often not obtained or the entire optimization process is relatively slow.

[0005] So far, although there are some initial value selection strategies (for example, the solution of the Jacobian matrix inverse eigenvalue problem), these strategies are usually heavily dependent on engineering prior knowledge, and are not always feasible for large-scale problems such as certain target topologies such as the "expanding box" topology, which limits the practicality of related methods in real application environments. Therefore, the research on the above-mentioned optimization methods based on the inverse eigenvalue problem is still not in-depth and comprehensive enough, and there is still a need for continued and sustained research.

[0006] In view of the lack of prior information about the solution of the optimization model based on the inverse eigenvalue problem, and considering the strong nonlinearity of the loss function with respect to the optimization variables, there is currently no unified and effective strategy for selecting initial values. Therefore, it is very important to reduce the sensitivity to initial values ​​and enhance the robustness of the optimization process while improving efficiency. The key to achieving this goal is to analyze the mathematical principles behind the optimization model, improve the properties of the optimization model from the bottom up, propose a general optimization model and optimization framework, and remove the restrictions on the strong dependence on initial values ​​in the optimization process. This is also a key means to improve the practicality of optimization methods in various filter topologies and different scale scenarios, truly realize the wide applicability of related optimization technologies, and enable the methods to be integrated into a variety of filter production and debugging equipment. Summary of the invention

[0007] The main purpose of the present invention is to solve the initial value sensitivity problem existing in the current optimization method of microwave filter network topology transformation, and to provide a robust coupling matrix reconfiguration optimization method based on the inverse eigenvalue problem.

[0008] The present invention provides a robust coupling matrix reconfiguration optimization method based on an inverse eigenvalue problem, and the method comprises the following steps:

[0009] The first step is to improve the optimization model based on the inverse eigenvalue problem, as follows:

[0010] The optimization model based on the inverse eigenvalue problem is as follows:

[0011]

[0012] in: is the transpose of a matrix or vector;

[0013] λ p is the eigenvalue of the current coupling matrix M; is the eigenvalue of the original coupling matrix T;

[0014] is the eigenvalue of the upper principal submatrix of the current coupling matrix M;

[0015] is the eigenvalue of the upper principal submatrix of the original coupling matrix T;

[0016] is the eigenvalue of the lower principal submatrix of the current coupling matrix M;

[0017] is the eigenvalue of the lower principal submatrix of the original coupling matrix T

[0018] λ c is the eigenvalue of the core matrix of the current coupling matrix M;

[0019] is the eigenvalue of the core matrix of the original coupling matrix T.

[0020] During the optimization process, the preset target coupling structure S is forced to be imposed on the coupling matrix M, and the optimization variables are the non-zero elements in the coupling matrix M. The coupling matrix M obtained by the optimization model satisfies the following similarity transformation relationship with the original coupling matrix T:

[0021]

[0022] Where H is an orthogonal matrix. According to this relationship, for any parameter q>0, the following matrix similarity transformation relationship still holds:

[0023]

[0024] Where I is the unit matrix. And according to the above formula, we can get the coupling matrix Topology The topological structure S of the target coupling matrix M differs only in the diagonal, that is, setting all diagonal elements in S to 1 yields This results in the following improved optimization model based on the inverse eigenvalue problem:

[0025]

[0026] During the optimization process, the pre-set target coupling structure Forced to be applied to the coupling matrix The optimization variable is the coupling matrix If the non-zero elements in are obtained by using model (1-2) Then the final target coupling matrix M can be obtained:

[0027] The second step is to optimize the improvement of the Levenberg-Marqudart method, as follows:

[0028] Levenberg-Marqudart (hereinafter referred to as LM) optimization algorithm for nonlinear least squares problems

[0029]

[0030] The main algorithm process is as follows:

[0031] x: =x0; μ: =μ0

[0032] repeat

[0033]

[0034] x:=x+αh

[0035] Updateμ

[0036] until S TOP

[0037] where I is the identity matrix, μ>0 is called the damping parameter, and J f is the Jacobi matrix corresponding to the nonlinear function f(x), F′ represents the gradient of the loss function F(x), α is used to control the step size, and h is the required step size and direction. The LM method is an optimization algorithm between the steepest descent method and the Gauss-Newton method. In the LM method, the optimization direction and step size are adjusted mainly by adjusting the damping parameter μ. When μ is large, the LM algorithm is similar to the steepest descent method, and when μ is small, it is similar to the Gauss-Newton method, and the damping parameter μ is adjusted dynamically according to the optimization process. Like other optimization algorithms, in the LM algorithm, only the optimization variables of the next step are Can reduce the loss function (that is, when ), the currently calculated step length h will be accepted; otherwise, the parameter μ will continue to be adjusted and the current step length h will be recalculated.

[0038] In the original LM algorithm, the method to determine whether to accept the current step size h is as follows:

[0039] ρ>0,

[0040]

[0041] The step size h is accepted only when the "profit factor" ρ> 0. However, this optimization process can easily fall into a local minimum and greatly increase the number of iterations, especially when the function graph corresponding to the optimization model becomes particularly narrow and long, the loss function may decrease very little in each iteration.

[0042] To solve this convergence problem, the above strategy is improved by introducing the "uphill" strategy, that is, at the i-th iteration, the step length h obtained at the i-1-th iteration is calculated. old and the step length h obtained in the i-th iteration new The cosine value θ corresponding to the angle between i =cos(h new ,h old ), and then judge (1-θ i ) b C i+1 ≥C i Is it true? If true, then regardless of whether ρ>0 or not, accept the current step h new , C i+1 and C i Respectively represent h new and h old The corresponding loss function value, parameter b is usually 1 or 2, b = 2 will accept more uphill steps than b = 1;

[0043] The improvement of the above strategy can greatly improve the probability of avoiding the LM method from falling into local extreme values ​​during the optimization process and accelerate the convergence rate of the LM method;

[0044] The third step is the optimization method based on q reduction. The specific steps are as follows:

[0045] Step 1: For a parameter q>0 and an arbitrarily selected initial value M 0 , apply the improved LM algorithm in the second step to the solution of the optimization model (1-2) based on the inverse eigenvalue problem improved in the first step. If the optimization solution M * is a global extreme point, that is is a pre-set error tolerance, then the target coupling matrix M, that is, the solution of the original optimization model (1-1) is expressed as M = M * -qI;

[0046] Step 2: If the solution M of the obtained optimization model (1-2) * It is not a global extreme point. At this time, the target coupling matrix M cannot be passed through M = M * -qI is obtained, in which case the solution is * As the initial value for the next optimization: reduce the value of parameter q, and then use M * As the initial value, the improved LM algorithm in the second step is used to re-solve the optimization model (1-2) to obtain a new optimization solution M * , and then make judgments in the above manner;

[0047] Step 3: Repeat the above steps 1 and 2 until the global extreme point M is found. *Or the parameter q is reduced to 0. For the latter, if M * It is still not a global extreme point, then M * As the initial value, the improved LM algorithm in the second step is used to directly solve the optimization model (1-1);

[0048] For the parameter q in the improved optimization model, its simplified method uses the following mathematical expression:

[0049]

[0050] Where n represents the number of reductions. represents the initial selection value of parameter q, and ρ>1 represents the reduction factor.

[0051] Beneficial effects of the present invention:

[0052] The robust coupling matrix reconfiguration optimization method based on the inverse eigenvalue problem provided by the present invention can not only solve the problem that the optimization algorithm is extremely sensitive to the initial value (for any selected initial value, the proposed algorithm can always find the desired target coupling matrix), but also the optimization method can be applied to large-scale and more complex filter topology structure transformation problems, while improving the optimization efficiency, the robustness of the optimization process is enhanced. The research of the present invention is compared with the existing methods in real application examples, and has a very high optimization success rate and good reliability. This not only provides an efficient and comprehensive means for the design of microwave filters, but also improves the efficiency of subsequent filter production and debugging, saving a lot of time. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is a schematic diagram of the flow chart of the coupling matrix reconfiguration optimization method described in the present invention. DETAILED DESCRIPTION

[0054] See also Figure 1 As shown:

[0055] The present invention provides a robust coupling matrix reconfiguration optimization method based on an inverse eigenvalue problem, and the method comprises the following steps:

[0056] Step 1: Improvement of the optimization model based on the inverse eigenvalue problem:

[0057] The optimization model based on the inverse eigenvalue problem can be expressed as:

[0058]

[0059] in: is the transpose of a matrix or vector;

[0060] λ pis the eigenvalue of the current coupling matrix M; is the eigenvalue of the original coupling matrix T;

[0061] is the eigenvalue of the upper principal submatrix of the current coupling matrix m;

[0062] is the eigenvalue of the upper principal submatrix of the original coupling matrix T;

[0063] is the eigenvalue of the lower principal submatrix of the current coupling matrix M;

[0064] is the eigenvalue of the lower principal submatrix of the original coupling matrix T

[0065] λ c is the eigenvalue of the core matrix of the current coupling matrix M;

[0066] is the eigenvalue of the core matrix of the original coupling matrix T.

[0067] During the optimization process, the preset target coupling structure S is imposed on the coupling matrix M, and the optimization variables are the non-zero elements in the coupling matrix M. In mathematical theory, we can obtain that the coupling matrix M obtained by the optimization model satisfies the following similarity transformation relationship with the original coupling matrix T:

[0068]

[0069] Where H is an orthogonal matrix. According to this relationship, for any parameter q>0, the following matrix similarity transformation relationship still holds true

[0070]

[0071] Where I is the unit matrix. At the same time, it can be seen from the above formula that the coupling matrix Topology The topological structure S of the target coupling matrix M differs only in the diagonal, that is, setting all diagonal elements in S to 1 yields Based on the above description, we obtain the following improved optimization model based on the inverse eigenvalue problem:

[0072]

[0073] Likewise, during the optimization process, the pre-set target coupling structure Forced to be applied to the coupling matrix The optimization variable is the coupling matrix Theoretically, if the target coupling matrix is ​​obtained using the above model Then the final target coupling matrix M can be obtained:

[0074] Step 2: Improvement of Levenberg-Marqudart optimization algorithm:

[0075] Levenberg-Marqudart (hereinafter referred to as LM) optimization algorithm description: For nonlinear least squares problems

[0076]

[0077] The main algorithm process of LM algorithm is:

[0078] x: =x0; μ: =μ0

[0079] repeat

[0080]

[0081] x:=x+αh

[0082] Updateμ

[0083] until S TOP

[0084] where I is the identity matrix, μ>0 is called the damping parameter, and J f is the Jacobi matrix corresponding to the nonlinear function f(x), F′ represents the gradient of the loss function F(x), α is used to control the step size, and h is the required step size and direction. In fact, the LM method is an optimization algorithm between the steepest descent method and the Gauss-Newton method. In the LM method, the optimization direction and step size are adjusted mainly by adjusting the damping parameter μ. When μ is large, the LM algorithm is similar to the steepest descent method, and when μ is small, it is similar to the Gauss-Newton method, and the damping parameter μ is dynamically adjusted according to the optimization process. However, like other optimization algorithms, in the LM method, only the optimization variables of the next step are Can reduce the loss function, that is, when , the currently calculated step length h will be accepted; otherwise, the parameter μ will continue to be adjusted and the current step length h will be recalculated. This optimization process can easily fall into a local minimum and greatly increase the number of iterations, especially when the function graph corresponding to the optimization model becomes particularly narrow and long, the loss function may decrease very little during each iteration.

[0085] In the LM algorithm, the strategy for determining whether to accept the current step length h is

[0086] ρ>0,

[0087]

[0088] When the "profit factor" ρ>0, the step length h is accepted. Here we add another strategy based on this strategy, called the "uphill" strategy, that is, at the i-th iteration, calculate the step length h that has been obtained in the i-1th iteration old and the step length h obtained in the i-th iteration new The cosine value of the angle between

[0089] θ i =cos(h new ,h old ),

[0090] Further judgement

[0091] (1-θ i ) b C i+1 ≥C i

[0092] Is it true? If true, then regardless of whether ρ>0 or not, accept the current step h new Here C i+1 and C i Respectively represent h new and h old The corresponding loss function value. The parameter b here is usually 1 or 2. Generally, b=2 will accept more uphill steps than b=1.

[0093] The improvement of the above strategies can greatly increase the probability of avoiding the LM method from falling into local extreme values ​​during the optimization process and accelerate the convergence rate of the LM method.

[0094] The third step is the optimization method based on q reduction. The specific steps are as follows:

[0095] Step 1: For a parameter q>0 and an arbitrarily selected initial value M 0 , apply the improved LM algorithm in the second step to the solution of the optimization model (1-2) based on the inverse eigenvalue problem improved in the first step. If the optimization solution M * is a global extreme point, is a pre-set error tolerance, then the target coupling matrix M, that is, the solution of the original optimization model (1-1) is expressed as M = M * -qI;

[0096] Step 2: If the solution M of the obtained optimization model (1-2) * It is not a global extreme point. At this time, the target coupling matrix M cannot be passed through M = M *-qI is obtained, in which case the solution is * As the initial value for the next optimization: reduce the value of parameter q, and then use M * As the initial value, the improved LM algorithm in the second step is used to re-solve the optimization model (1-2) to obtain a new optimization solution M * , and then make judgments in the above manner;

[0097] Step 3: Repeat the above steps 1 and 2 until the global extreme point M is found. * Or the parameter q is reduced to 0. For the latter, if M * It is still not a global extreme point, then M * As the initial value, the improved LM algorithm in the second step is used to directly solve the optimization model (1-1);

[0098] For the parameter q in the improved optimization model, its simplified mathematical expression is as follows:

[0099]

[0100] Where n represents the number of reductions. represents the initial selection value of parameter q, and ρ>1 represents the reduction factor.

[0101] The specific implementation example is as follows:

[0102] Example 1.TZs=[-j1.8,-j1.3,-j1.1,j1.25], RL=20-dB, N=10.

[0103] Example 2.TZs=[-j1.5,-j1.2,-j1.1,j1.15,j1.3], RL=20-dB, N=12.

[0104] Example 3.TZs=[-j1.4,-j1.3,-j1.1,j1.05,j1.2], RL=20-dB, N=12.

[0105] Example 4.TZs=[-j1.4,-j1.2,-j1.05,j1.1,j1.15,j1.3], RL=20-dB, N=14.

[0106] Example 5.TZs=[-j1.25,-j1.2,-j1.08,-j1.05,j1.05,j1.1,j1.15], RL=18-dB, N=16.

[0107] Where TZs represents the transmission zero point, j represents the imaginary unit, RL represents the return loss, and N represents the number of resonant cavities. Through the above filter parameters, we can use the N+2 type generalized Chebyshev filter function synthesis method to obtain the corresponding N+2 order lateral coupling matrix T. Assume that the target topology is an "expanded box" topology. In order to better demonstrate the robust coupling matrix reconfiguration optimization method based on the inverse eigenvalue problem provided by the present invention (hereinafter referred to as Algorithm1 to represent the provided optimization method, such as Figure 1 The advantages of the improved LM algorithm (shown in Figure 2) are compared with the direct optimization model (1-1) using the improved LM algorithm (hereinafter referred to as Algorithm2). We use Algorithm1 and Algorithm2 to solve the above five examples, and the initial values ​​used in Algorithm1 and Algorithm2 are randomly generated and kept the same. Each example is executed 10,000 times, and each time when the loss function C(M) or When it is less than 1e-8, the optimization is considered successful. The corresponding experimental results are given in the following table.

[0108] Table 1 Algorithm performance comparison between Algorihtm1 and Algorithm2

[0109]

[0110] In Table 1, NS represents the number of successful optimizations, SR represents the optimization success rate, AI represents the average number of iterations used by the algorithm, and AT represents the average optimization time used by the algorithm.

[0111] The above experimental results show that the method of the present invention is applicable to the problem of complex filter network topology transformation of different orders. It greatly reduces the sensitivity of the optimization algorithm to the initial value and has a very high optimization success rate. This will reduce a lot of costs for the synthesis, design and debugging of actual filters.

Claims

1. A robust coupling matrix reconfiguration optimization method based on inverse eigenvalue problem, characterized by: The method comprises the following steps: The first step is to improve the optimization model based on the inverse eigenvalue problem, as follows: The optimization model based on the inverse eigenvalue problem is as follows: in: is the transpose of a matrix or vector; λ p is the eigenvalue of the current coupling matrix M; is the eigenvalue of the original coupling matrix T; is the eigenvalue of the upper principal submatrix of the current coupling matrix M; is the eigenvalue of the upper principal submatrix of the original coupling matrix T; is the eigenvalue of the lower principal submatrix of the current coupling matrix M; is the eigenvalue of the lower principal submatrix of the original coupling matrix T λ c is the eigenvalue of the core matrix of the current coupling matrix M; is the eigenvalue of the core matrix of the original coupling matrix T. During the optimization process, the preset target coupling structure S is forced to be imposed on the coupling matrix M, and the optimization variables are the non-zero elements in the coupling matrix M. The coupling matrix M obtained by the optimization model satisfies the following similarity transformation relationship with the original coupling matrix T: Where H is an orthogonal matrix. According to this relationship, for any parameter g>0, the following matrix similarity transformation relationship still holds: Where I is the identity matrix, and according to the above formula, the coupling matrix Topology The topological structure S of the target coupling matrix M differs only in the diagonal, that is, setting all diagonal elements in S to 1 yields This results in the following improved optimization model based on the inverse eigenvalue problem: During the optimization process, the pre-set target coupling structure Forced to be applied to the coupling matrix The optimization variable is the coupling matrix If the non-zero elements in are obtained by using model (1-2) Then the final target coupling matrix M can be obtained: The second step is to improve the Levenberg-Marqudart optimization algorithm as follows: Levenberg-Marqudart (hereinafter referred to as LM) optimization algorithm for nonlinear least squares problems The main algorithm process is as follows: x: =x0; μ: =μ0 repeat x:=x+αh Update μ until STOP where I is the identity matrix, μ>0 is called the damping parameter, and J f is the Jacobi matrix corresponding to the nonlinear function f(x), F′ represents the gradient of the loss function F(x), α is used to control the step size, h is the required step size and direction, the LM method is an optimization algorithm between the steepest descent method and the Gauss-Newton method. In the LM method, the optimization direction and step size are adjusted mainly by adjusting the damping parameter μ. When μ is large, the LM algorithm is similar to the steepest descent method, and when μ is small, it is similar to the Gauss-Newton method, and the damping parameter μ is dynamically adjusted according to the optimization process. Like other optimization algorithms, in the LM algorithm, only the optimization variable of the next step is Can reduce the loss function (that is, when , the currently calculated step length h will be accepted; otherwise, the parameter μ will continue to be adjusted and the current step length h will be recalculated; In the original LM algorithm, the method to determine whether to accept the current step size h is as follows: ρ>0. The step length h is accepted only when the "profit factor" ρ>0. However, this optimization process can easily fall into a local minimum and greatly increase the number of iterations. In particular, when the function graph corresponding to the optimization model becomes particularly narrow and long, the loss function may decrease very little during each iteration. To solve this convergence problem, the above strategy is improved by introducing the "uphill" strategy, that is, at the i-th iteration, the step length h obtained at the i-1-th iteration is calculated. old and the step length h obtained in the i-th iteration new The cosine value θ corresponding to the angle between i =cos(h new ,h old ), and then judge (1-θ i ) b C i+1 ≥C i Is it true? If true, then regardless of whether ρ>0 or not, accept the current step h new , C i+1 and C i Respectively represent h new and h old The corresponding loss function value, parameter b is usually 1 or 2, b = 2 will accept more uphill steps than b = 1; The improvement of the above strategy can greatly improve the probability of avoiding the LM method from falling into local extreme values ​​during the optimization process and accelerate the convergence rate of the LM method; The third step is the optimization method based on q reduction. The specific steps are as follows: Step 1: For a parameter g>0 and an arbitrarily selected initial value M 0 , apply the improved LM algorithm in the second step to the solution of the optimization model (1-2) based on the inverse eigenvalue problem improved in the first step. If the optimization solution M * is a global extreme point, that is ∈ is a pre-set error tolerance, then the target coupling matrix M, that is, the solution of the original optimization model (1-1) is expressed as M = M * -qI; Step 2: If the solution M of the obtained optimization model (1-2) * It is not a global extreme point. At this time, the target coupling matrix M cannot be passed through M = M * -qI is obtained, in which case the solution is * As the initial value for the next optimization: reduce the value of parameter q, and then use M * As the initial value, the improved LM algorithm in the second step is used to re-solve the optimization model (1-2) to obtain a new optimization solution M * , and then make judgment according to the above method; Step 3: Repeat the above steps 1 and 2 until the global extreme point M is found. * Or the parameter q is reduced to 0. For the latter, if M * It is still not a global extreme point, then M * As the initial value, the improved LM algorithm in the second step is used to directly solve the optimization model (1-1); For the parameter q in the improved optimization model, its simplified method uses the following mathematical expression: Where n represents the number of reductions. represents the initial selection value of parameter q, and ρ>1 represents the reduction factor.

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