Vibration structure minimum norm control method based on dynamic compensator

Abstract

The invention provides a vibration structure minimum norm control method based on a dynamic compensator, and the method comprises the steps: building a vibration structure model through a finite element analysis method, and designing the dynamic compensator; processing the unchanged feature structure matrix by adopting QR decomposition, and calculating a block matrix; performing QR decomposition on the correlation matrix of the closed-loop system, and calculating an actual representation matrix of an expected characteristic value and a matrix formed by an expected characteristic structure and compensator gain; a parameterized vector is selected, and parameterized dynamic compensator gain is calculated; if the cost function exceeds a set threshold value, the optimization function is optimized through a gradient optimization method according to any given initialization parameter vector, and a dynamic compensation gain matrix and a control gain matrix of the dynamic compensator are calculated through the optimal parameterization vector. According to the method, the calculation effectiveness is improved, the calculation complexity is reduced, meanwhile, the degree of freedom of the parameterized vector is not changed, and the sufficient optimization degree of freedom is ensured.

Description

Technical Field

[0001] The present invention relates to the technical field of vibration structure control, and in particular to a minimum norm control method of a vibration structure based on a dynamic compensator. Background Art

[0002] Vibration systems, as a key research topic in engineering modeling and analysis of structural vibration, have enormous application value in vibration processes in bridges, buildings, automobiles, aircraft, power systems, aerospace, and other fields. Minimum norm of the controller means optimal control energy consumption. Researching optimal energy consumption control techniques for vibration systems plays a crucial role in promoting energy conservation and emission reduction in my country's industrial sector. Compared to general system control, vibration system control presents unique requirements. This is primarily reflected in the sensitivity of system performance indicators to the system's frequency. Generally, the control frequency of a system is neither lower nor higher; rather, it is closely related to the system's natural frequency. Examples include resonance in bridge systems and building structures, low-frequency oscillations in power systems, and vibration suppression in aircraft. Regardless of the engineering vibration problem, engineering modeling and control analysis always confront the issue of control energy consumption. Excessively weak control action results in weak control force and loss of control effectiveness, while excessive control action results in greater control effect and consumes more control energy. Achieving system control objectives with the least amount of control energy is a fundamental requirement for vibration system modeling and control, and also aligns with national energy conservation and emission reduction policies. However, due to the complexity and large-dimensional basic characteristics of the actual vibration process modeling system, it is often difficult to achieve optimal energy consumption control.

[0003] Vibration systems are widely used in vibration engineering, such as automotive suspension systems, earthquake engineering modeling, and aerospace engineering. Finite element analysis can be used to describe these practical vibration system models as typical second-order systems. The analysis and design of control problems play a vital role in improving the dynamic performance of vibration systems. Formally, vibration systems are extensions of generalized systems and state-space systems. In recent years, extensive research has been conducted on the control of vibration systems using generalized system theory and methods. As a fundamental technique for studying the stability of vibration systems, the eigenstructure assignment method has garnered widespread attention and achieved significant progress.

[0004] Similar to generalized systems, eigenvalues have a decisive influence on the dynamic performance of vibration systems. Some eigenvalues can improve the dynamic performance of vibration systems, while others may destroy it. Especially for vibration systems with singular conditions, the infinite eigenvalues of the system may cause impulsive behavior, which may completely destroy the entire system in engineering practice. In recent years, researchers have improved the dynamic performance of singular vibration systems by changing the eigenvalues of closed-loop systems. There are two types of eigenstructure configuration methods that can improve the dynamic behavior of vibration systems. When all existing eigenvalues have a negative impact on system performance, all eigenvalues need to be reconfigured. This type of method is called full eigenstructure configuration. In most cases, only a small number of eigenvalues have a negative impact on system performance, or even destroy it. In this case, only these small number of eigenvalues need to be changed. This type of method is called partial eigenstructure configuration.

[0005] In general, partial eigenvalue configuration only requires changing some unnecessary eigenvalues, resulting in relatively easy control of the system, low computational complexity, and low energy consumption. However, partial eigenvalue configuration methods based on active control do not provide sufficient degrees of freedom for the design of parametric controllers, which can complicate the optimization process. Incorporating a dynamic compensator into the active controller expands the dimensionality of the vibration system, providing additional degrees of freedom for the parametric controller, which can be used to optimize control performance.

[0006] In control engineering, optimization and controller design are often intertwined. A larger controller norm implies higher control energy consumption. Furthermore, the increased dimensionality of the dynamic compensator increases energy consumption compared to conventional controllers. At the same time, the increased degrees of freedom gained by the dynamic compensator allow for optimization of minimum norm gain and eigenvalue robustness. Therefore, the minimum norm controller design problem has attracted considerable research interest due to its potential economic and practical significance.

[0007] Existing methods for studying the eigenstructure configuration of vibration systems mostly focus on designing complex controllers to precisely control the system's eigenvalues to the desired position, focusing on aspects such as the robustness of the eigenvalues, the rapidity of system stability, and the complexity of the controller. These methods often overlook the cost of control. This is because the higher the control requirements for a system, the greater the difficulty and cost of control, and the significantly increased energy consumption. Summary of the Invention

[0008] Existing methods for studying the characteristic structure configuration of vibration systems mostly focus on designing complex controllers so that the system's eigenvalues can be accurately controlled at the desired position, and often ignore the technical issue of control cost. The present invention proposes a minimum norm control method for vibration structures based on dynamic compensators. Different from traditional research ideas, the research goal is to focus on how to use less control cost to improve the control performance of the system, that is, not only to improve the operating performance of the system through characteristic structure configuration, but also to optimize the performance of the applied controller to achieve optimal energy consumption control.

[0009] In order to achieve the above object, the technical solution of the present invention is implemented as follows: a minimum norm control method for a vibration structure based on a dynamic compensator, the steps of which are as follows:

[0010] Step 1: Based on the vibration structure, use the finite element analysis method to establish a vibration structure model, design a dynamic compensator, and calculate the eigenvalues and eigenvectors of the compensated augmented open-loop system;

[0011] Step 2: Determine the desired eigenvalues to be changed, perform QR decomposition on the unchanged eigenstructure matrix, and calculate the block matrix corresponding to the orthogonal matrix; perform QR decomposition on the closed-loop system correlation matrix, and calculate the real representation matrix of the eigenvector corresponding to the desired eigenvalues and the matrix consisting of the desired eigenstructure and compensator gain;

[0012] Step 3: Select a parameterized vector and calculate the parameterized dynamic compensator gain based on the block matrix, the real representation matrix of the eigenvector corresponding to the expected eigenvalue, and the matrix composed of the expected eigenstructure and the compensator gain;

[0013] Step 4: If the optimization function exceeds the set threshold, the optimization function is optimized by the gradient optimization method using the given arbitrary initialization parameter vector to obtain the optimal parameterization vector, and the characteristic matrix composed of the eigenvectors and the transition matrix are calculated using the optimal parameterization vector and the block matrix, thereby calculating the dynamic compensation gain matrix and the control gain matrix of the dynamic compensator.

[0014] Preferably, the vibration structure model is

[0015]

[0016] y(t)=C d x(t)

[0017]

[0018] in, are the inertia matrix, damping matrix and stiffness matrix of the vibrating structure respectively. The inertia matrix M is a non-singular matrix, and n is the dimension of the vibrating structure. is the input control matrix, whose rank satisfies rank(B)=m, where m is the dimension of the control input; and are the displacement gain matrix and the velocity gain matrix respectively, and are the output vectors related to displacement and velocity at time t, r is the dimension of the output related to velocity, and q is the dimension of the output related to displacement; and are the displacement vector and the input control vector respectively; and represent velocity and acceleration respectively;

[0019] The dynamic compensator is:

[0020]

[0021] in, is the compensation vector of the vibrating structure, and s is the dimension of the compensation; and denote the derivative and second-order derivative of the compensation vector w(t) respectively; are the dynamic compensation gain matrices related to the compensation vector w(t) and the control output vectors y(t), z(t), respectively. is the control gain matrix to be determined.

[0022] Preferably, the vibration structure model after compensation obtained from the vibration structure model and the dynamic compensator is:

[0023]

[0024] After considering the compensation vector w(t), the augmented open-loop system of the vibration structure model is:

[0025]

[0026] The closed-loop system of the vibration structure model is:

[0027]

[0028] Among them, the augmented inertia matrix Augmented Damping Matrix Augmented stiffness matrix Augmented Input Control Matrix Displacement compensation vector

[0029] Preferably, the eigenvalues and corresponding eigenvectors of the compensated augmented open-loop system are denoted as There are p eigenvalues Need to reconfigure, p<2(n+s); design the dynamic compensation gain matrix of the parameterized dynamic compensator and the control gain matrix The eigenvalue of the closed-loop system is Reconfigured eigenvalues The corresponding eigenvector is

[0030] Preferably, under the parameterized dynamic compensator, appropriate parameters are selected so that the F norm of the compensation gain correlation matrix Ψ is Minimum, where the compensation gain correlation matrix is

[0031]

[0032] Preferably, the unchanged characteristic structure matrix is in, The real representation matrix representing the invariant eigenvalues, The real representation matrix representing the eigenvectors corresponding to the invariant eigenvalues;

[0033] The closed-loop system correlation matrix is is the i-th expected eigenvalue,

[0034] i=1,…,p.

[0035] Preferably, the method of using QR decomposition to process the unchanged eigenstructure matrix is: in, is an orthogonal matrix, is a non-singular upper triangular matrix; denoted as orthogonal matrix

[0036]

[0037] in, It is a block matrix obtained by dividing the orthogonal matrix W into blocks; a new matrix is constructed by the block matrices W1 and W2 and and

[0038]

[0039] in, is the real representation matrix of the expected eigenvalues, is the real representation matrix of the eigenvector corresponding to the expected eigenvalue;

[0040] The method for QR decomposition of the closed-loop system correlation matrix is:

[0041]

[0042] in, is an orthogonal matrix, is a nonsingular upper triangular matrix, a zero matrix (·) H Denotes conjugate transpose; let the orthogonal matrix S i =(S i1 ,S i2 ), where the orthogonal submatrix Orthogonal submatrix

[0043] Left multiply the orthogonal submatrix Available

[0044]

[0045] S i is the orthogonal matrix rank(S i2 )=m+s;

[0046] Orthogonal submatrix The row vectors of A set of basis vectors,

[0047]

[0048] but

[0049]

[0050] in

[0051]

[0052] Intermediate Matrix is the orthogonal submatrix S i2 The block matrix obtained by block division, D i Represents the expected eigenvalue of the i-th Eigenvector and the compensator gain; then the real representation matrix of the eigenvector corresponding to the expected eigenvalue and the matrix composed of the expected characteristic structure and the compensator gain are obtained respectively

[0053]

[0054] Preferably, the vibration structure satisfies rank(γ 2 M+γD+K,B)=n, If a dynamic compensator is used to reconfigure part of the characteristic structure of the augmented open-loop system to improve the dynamic performance of the system, the gain matrix of the dynamic compensator can be calculated as follows:

[0055]

[0056] in,

[0057]

[0058] N=(S 212 ξ1,…,S p22 ξ p )

[0059]

[0060] in, S 112 、S p12 、S 212 、S p22 Both represent intermediate matrices, I s represents the identity matrix, (·) + Represents the Moore-Penrose inverse;

[0061] And the block matrix W2=(W 21 ,W 22 ) is a constant matrix, the intermediate matrix represents the block submatrix obtained by partitioning the block matrix W2, is a selectable parameter vector that satisfies:

[0062] (a) There are parameters l,k, when the eigenvalue When , the corresponding eigenvector * indicates conjugation;

[0063]

[0064] Preferably, the optimization function Among them, the compensation gain correlation matrix

[0065] For any nonzero parameterized matrix The column vector satisfies hour, Then the gradient of the optimization function J with respect to the non-zero parameterized matrix Ξ is

[0066]

[0067] Among them, the three correlation matrices are

[0068]

[0069] Intermediate Matrix

[0070]

[0071] Preferably, for the unchanged 2(n+s)-p constant eigenvalues and corresponding eigenvectors, assuming that 2τ are pairwise conjugates and the remaining 2(n+s)-p-2τ are real eigenvalues, then

[0072]

[0073] Among them, δ represents the imaginary unit, λ 2ι-1 、 represents the eigenvalue of the ιth conjugate pair, λ ιR ,λ ιI The eigenvalue λ represents the conjugation 2ι-1 、 The real and imaginary parts, v 2ι-1 、 represents the eigenvalue λ of the ιth conjugate 2ι-1 、 The corresponding eigenvectors, v ιR 、v ιI Represents the feature vector v 2ι-1 、 The real and imaginary parts of the matrix; then the real representation of the unchanged eigenvalue

[0074]

[0075] Real representation matrix of eigenvectors corresponding to invariant eigenvalues

[0076]

[0077] Among them, diag represents the diagonal matrix composed of all elements;

[0078] The expected p eigenvalues and corresponding eigenvectors are represented by real numbers. Suppose 2ε are pairwise conjugates and the remaining p-2ε are real eigenvalues. Then

[0079]

[0080] in, represents the eigenvalue of the ηth conjugate pair, The eigenvalue representing the conjugation The real and imaginary parts of Denotes the eigenvalue of the ηth conjugate pair The corresponding eigenvector, Represents the feature vector The real and imaginary parts of the expected eigenvalues are represented by the real matrix

[0081]

[0082] The real representation matrix of the expected eigenvalue corresponding to the eigenvector

[0083] Reconfigured feature pairs and unchanged feature pairs If both satisfy the closed-loop system, then

[0084]

[0085] in, is the eigenvalue The corresponding eigenvector, v j is the eigenvalue λ j The corresponding eigenvector;

[0086] Unchanged feature pairs To satisfy the augmented open-loop system, there is

[0087]

[0088] but

[0089]

[0090] definition

[0091]

[0092] It can be seen that

[0093]

[0094] but

[0095]

[0096] in, is a zero matrix, and the matrix N is composed of the desired characteristic structure and the compensator gain. p ),and represents the complex field, is the eigenvector corresponding to the expected eigenvalue;

[0097] By rank(γ 2 M+γD+K,B)=n Left Null Space The dimension of is m+s.

[0098] Compared with the prior art, the present invention has the following beneficial effects:

[0099] A minimum-norm control method for non-singular vibration structures based on a dynamic compensation approach is studied, aiming to achieve optimal performance and optimal operating conditions for closed-loop systems using optimal control energy consumption. The eigenstructure configuration method is one of the most classic and effective control techniques commonly used to study vibration system control problems. To address the limited parameterization freedom provided by active control, which limits system optimization, the present invention innovatively proposes using a dynamic compensator to expand the dimensionality of the original vibration system, significantly increasing the parameterization freedom of the controller and providing sufficient freedom for optimal energy consumption control optimization. Expanding the system dimensionality also introduces another problem: increased computational complexity during the system control process. To address this computational complexity, the present invention cleverly employs an orthogonal triangular (QR) decomposition method to effectively reduce the dimensionality of the relevant control process (matrix), improving computational efficiency and reducing computational complexity while maintaining the degrees of freedom of the parameterization vector, ensuring sufficient optimization freedom. To better and more effectively reduce the dynamic compensator gain norm, the present invention employs a gradient-based steepest descent method to optimize the compensator parameterization vector, minimizing controller energy consumption and achieving optimal utilization of control energy. Simulation results also show that the gain norm of the dynamic compensator after optimization is significantly reduced compared to the unoptimized case. The research method of the present invention plays an important role in solving the problem of excessive energy consumption in the vibration field. Its research results and technology can be used to optimize bridge vibration design systems, power system low-frequency oscillation optimization, aircraft vibration suppression and other optimization problems related to vibration process energy consumption control, playing an extremely important role in promoting energy conservation and emission reduction in the industrial field. BRIEF DESCRIPTION OF THE DRAWINGS

[0100] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0101] Figure 1 Flowchart of the present invention. DETAILED DESCRIPTION

[0102] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without creative work are within the scope of protection of the present invention.

[0103] like Figure 1As shown in FIG, a minimum norm control method for a vibration structure based on a dynamic compensator, the steps of which are as follows:

[0104] Step 1: Based on the vibration structure, a vibration structure model is established using the finite element analysis method, and a corresponding dynamic compensator is designed to calculate the eigenvalues and eigenvectors of the compensated augmented open-loop system.

[0105] Consider the following vibration model of the vibration structure established by the finite element analysis method:

[0106]

[0107] in, are the system matrices whose model is known, respectively referred to as the inertia matrix, damping matrix, and stiffness matrix of the vibration system. Here, the inertia matrix is a nonsingular matrix. n is the dimension of the system, i.e., the dimensions of the system matrices M, D, and K. The inertia matrix M is also called the mass matrix. When the inertia matrix M is nonsingular, the corresponding system is called a nonsingular vibration system. This invention studies nonsingular vibration system models. It is the input control matrix, whose rank satisfies rank(B)=m, where m is the dimension of the control input, that is, the number of columns of the input control matrix B. and are the displacement gain matrix and the velocity gain matrix respectively, and is the output vector or control output vector related to displacement and velocity at time t, r is the output dimension related to velocity, and q is the output dimension related to displacement. and are the displacement vector and input control vector respectively. and They represent the derivative and second-order derivative of the displacement vector x(t), namely velocity and acceleration, respectively.

[0108] The characteristic of the non-singular vibration system model is that the system's degrees of freedom are determined. The control and optimization of the system are often limited by the degrees of freedom. More degrees of freedom means better optimization effects. The present invention considers taking the method of designing a dynamic compensator to expand the dimension of the system in order to increase the system's optimization degrees of freedom and improve the system's optimization effect. For the non-singular vibration model (1), the designed dynamic compensator is as follows:

[0109]

[0110] in, is the compensation vector of the vibration model, and s is the dimension of the compensation (or the dimension of the compensator). and represent the derivative and second-order derivative of the compensation vector w(t) respectively. are the dynamic compensation gain matrices related to the compensation vector w(t) and the control output vectors y(t), z(t), respectively. is the control gain matrix to be determined. The following relationship can be obtained from the vibration model of formula (1) and the dynamic compensator of formula (2):

[0111]

[0112] Note that the vibration model, i.e., formula (1), is an open-loop system. After considering the compensation vector w(t), its augmented open-loop system can be expressed as follows:

[0113]

[0114] Combining formula (3) and formula (4) we can get the closed-loop system:

[0115]

[0116] Among them, the augmented inertia matrix Augmented Damping Matrix Augmented stiffness matrix Augmented Input Control Matrix Displacement compensation vector

[0117] For the non-singular vibration model (1) with a known model matrix, the eigenvalue of the system is determined by the inertia matrix M, damping matrix D, and stiffness matrix K of the system. The eigenvalue of the system plays a decisive role in the various performances of the system, so changing the eigenvalue of the system is an effective way to enhance and improve the performance of the vibration system. For the original non-singular vibration system model obtained by finite element method modeling, the study adopts the method of designing a dynamic compensator to change the characteristic structure of the vibration model (1) so that the system has the desired performance indicators. Inevitably, since the dynamic compensator is used to expand the system dimension, the control intensity will increase, that is, the gain norm of the dynamic compensator increases. Therefore, changing the system characteristic structure so that the closed-loop system after adding the dynamic compensator has the desired performance while reducing the gain norm of the controller as much as possible can effectively reduce the control intensity, thereby reducing the consumption of control energy, that is, the optimal energy consumption control problem. Therefore, the core issues to be studied in this invention can be summarized into two aspects: on the one hand, the characteristic structure of the non-singular vibration system is arbitrarily configured to improve the dynamic performance of the system, among which stability is the performance that must be guaranteed first. On the other hand, while arbitrarily configuring the system characteristic structure, the gain norm of the dynamic compensator is ensured to be minimized, that is, the energy consumption control index is minimized. To facilitate the derivation and implementation of the algorithm, the above problem is described in mathematical language as follows:

[0118] For a certain non-singular vibration system model, that is, formula (1), the matrix and For the compensated augmented open-loop system (4), its characteristic structure (i.e., eigenvalues and corresponding eigenvectors) can be easily calculated and is denoted as In order to make the vibration system (structure) have a certain desired performance index, some eigenvalues of the system need to be replaced by eigenvalues That is, there are p eigenvalues that need to be changed (reconfigured), p<2(n+s). The purpose of this invention is: (1) to design the dynamic compensation gain matrix and control gain matrix of the parameterized dynamic compensator The characteristic value of the closed-loop system (5) is Its reconfigured eigenvalue The corresponding eigenvector is recorded as (2) Under the parameterized dynamic compensator, select appropriate parameters so that the F norm of the compensation gain correlation matrix Minimum, where the compensation gain correlation matrix

[0119]

[0120] For ease of calculation, the present invention uses real representation to describe the eigenvalues and eigenvectors of the system. That is, the original complex eigenvalues are extracted and their real parts are used to form new vectors or matrices. The following is the specific operation process. The real representation methods mentioned in the present invention refer to the following methods or are obtained by operating according to the following process:

[0121] The first step is to represent the unchanged 2(n+s)-p eigenvalues and their corresponding eigenvectors. Let's assume that 2τ are pairwise conjugates and the remaining 2(n+s)-p-2τ are real eigenvalues. Then

[0122]

[0123] Among them, δ represents the imaginary unit, λ 2ι-1 、 represents the eigenvalue of the ιth conjugate pair, λ ιR ,λ ιI The eigenvalue λ represents the conjugation 2ι-1 、 The real and imaginary parts, v 2ι-1 、 represents the eigenvalue λ of the lth conjugate 2ι-1 、 The corresponding eigenvectors, v ιR 、v ιI Represents its real and imaginary parts. These eigenvalues and eigenvectors form matrices with the remaining real eigenvalues and real eigenvectors, which are recorded as

[0124]

[0125] in, The real representation matrix representing the invariant eigenvalues, The real matrix represents the eigenvectors corresponding to the invariant eigenvalues. diag represents the diagonal matrix consisting of all elements (blocks) (in parentheses). The subscripts above indicate numbering starting from p+1 up to p+τ, and starting from p+2τ+1 up to 2(n+s).

[0126] The second step is to use real representation for the expected p eigenvalues and corresponding eigenvectors. The process is similar to the first step. Let's assume that 2ε are pairwise conjugate and the remaining p-2ε are real eigenvalues. Then

[0127]

[0128] in, represents the eigenvalue of the ηth conjugate pair, The eigenvalue representing the conjugation The real and imaginary parts of Denotes the eigenvalue of the ηth conjugate pair The corresponding eigenvector, Represents the feature vector The real and imaginary parts of these eigenvalues and eigenvectors form matrices with the remaining real eigenvalues and real eigenvectors, which are recorded as

[0129]

[0130] in, The real representation matrix representing the expected eigenvalues, The real matrix representing the eigenvectors corresponding to the expected eigenvalues. diag represents the diagonal matrix consisting of all elements (blocks) (in parentheses). The subscripts above indicate numbering from 1 up to ε and from 2ε+1 up to p.

[0131] Step 2: Determine the desired eigenvalues to be changed, perform QR decomposition on the unchanged eigenstructure matrix, and calculate the block matrix corresponding to the orthogonal matrix; perform QR decomposition on the closed-loop system correlation matrix, and calculate the real representation matrix of the eigenvector corresponding to the desired eigenvalue and the matrix composed of the desired eigenstructure and compensator gain.

[0132] To design a parameterized dynamic compensator, several important preparatory steps are required. The characteristic structures of a vibration system must satisfy the characteristic equation. Since the present invention only reconfigures some of the characteristic structures of the vibration system, the remaining characteristic structures remain unchanged. Therefore, there are p characteristic pairs that satisfy the closed-loop system (5), and 2n+2s-p characteristic pairs that satisfy both the open-loop system (4) and the closed-loop system (5). Therefore, the following conclusions can be drawn.

[0133] Reconfigured feature pairs and unchanged feature pairs All satisfy the closed loop system (5)

[0134]

[0135] in, is the eigenvalue The corresponding eigenvector, v j is the eigenvalue λ j The corresponding eigenvector.

[0136] Notice is an unchanged characteristic pair, which also satisfies the open-loop system (4), so we have

[0137]

[0138] Furthermore, from formulas (7) and (8), we can know that

[0139]

[0140] definition

[0141]

[0142] Among them, N i represents the matrix consisting of the i-th desired eigenvalue, eigenvector, and compensator gain.

[0143] From formulas (9) and (10), we can see that

[0144]

[0145] In formula (11), we can easily obtain

[0146]

[0147] in, is a zero matrix, matrix N=(N1,…,N p ),and Represents a complex field. is the eigenvector corresponding to the desired eigenvalue, and N is the matrix consisting of the desired eigenstructure and compensator gain.

[0148] In order to reduce the complexity of calculation, we further use QR decomposition on formula (12) and let

[0149]

[0150] in, is an orthogonal matrix, is a nonsingular upper triangular matrix.

[0151] remember

[0152]

[0153] Among them, W1 and w2 are block matrices obtained by dividing the orthogonal matrix w into blocks according to the corresponding dimensions. Construct new matrices from the block matrices W1 and W2, denoted as Z1 and Z2, and

[0154]

[0155] Then there is a matrix and The above matrices W1, W2, Z1, and Z2 are all defined for the convenience of calculation. Matrix Z2 is used as an intermediate transition matrix and is replaced later only to calculate the compensator gain.

[0156] Theorem 1 The vibration system satisfies rank(γ 2 M+γD+K,B)=n, If a dynamic compensator such as formula (2) is used to reconfigure part of the characteristic structure of the augmented vibration system to improve the dynamic performance of the system, the gain matrix of the dynamic compensator can be calculated as follows:

[0157]

[0158] in,

[0159]

[0160] N=(S 212 ξ1,…,S p22 ξ p )(20)

[0161]

[0162] in, S 112 、S p12 、S G12 、S pGGBoth represent intermediate matrices, and a complex combination matrix is represented by a new matrix to facilitate subsequent calculations. s represents the identity matrix. (·) + represents the Moore-Penrose inverse (or generalized inverse).

[0163] And, the block matrix W2=(W 21 ,W 22 ) is a constant matrix, is a selectable parameter vector that satisfies:

[0164] (a) There are parameters l,k, when the eigenvalue When , the corresponding eigenvector also has * indicates conjugation.

[0165] (b)

[0166] Proof: From formula (12), we can see

[0167]

[0168] Substituting formula (14) into formula (23), we can get

[0169]

[0170] Substituting formulas (13), (15) and (16) into formula (24), we can obtain

[0171]

[0172] Define the block matrix

[0173] W2=(W 21 ,W 22 )(26)

[0174] Among them, the intermediate matrix represents the block submatrix obtained by partitioning the block matrix W2 accordingly. From formulas (16) and (26), we can see that:

[0175]

[0176] Thus there is

[0177]

[0178] Combining formula (6) and (10) into a compact form, we have

[0179]

[0180] Taking the conjugate transpose of formula (29) we can get

[0181]

[0182] make

[0183]

[0184] From formula (30), we can immediately know

[0185]

[0186] Among them, Ω i represents the correlation matrix of the closed-loop system, i=1,…,p. Represents the left null space. By rank(γ 2 M+γD+K,B)=n

[0187]

[0188] Thus, we know that the left null space The dimension of is m+s. For the closed-loop system correlation matrix Ω i Taking the orthogonal triangular (QR) decomposition, we can get

[0189]

[0190] in, is an orthogonal matrix, is a nonsingular upper triangular matrix, (·) H Denotes the conjugate transpose. Let

[0191] S i =(S i1 ,S i2 ),

[0192] Among them, the orthogonal submatrix Orthogonal submatrix Multiply formula (33) by the orthogonal submatrix on the left Available

[0193]

[0194] Note that S i is an orthogonal matrix, and

[0195] rank(S i2 )=m+s

[0196] This shows that the orthogonal submatrices The row vectors of A set of basis vectors. Then we have

[0197]

[0198] It is easy to know from formula (35)

[0199]

[0200] in

[0201]

[0202] Here, the intermediate matrix is the orthogonal submatrix S i2 The two block matrices are obtained by the corresponding blocks. In formula (36), let i = 1,…,p, then we can get

[0203]

[0204] It is easy to find that the following formula is always true

[0205]

[0206] here

[0207]

[0208] (·) + It is easy to get from formulas (28) and (38) that

[0209]

[0210] Proof completed.

[0211] Theorem 1 and its proof provide a general approach for designing a parametric dynamic compensator. This dynamic compensator can partially reconfigure the eigenstructure of a vibration system to improve its performance. Based on the above analysis, the steps for calculating the gain of this parametric dynamic compensator can be summarized as the following algorithm.

[0212] Algorithm 1

[0213] Step 1. Use the root-finding method of the characteristic polynomial to calculate the eigenvalues and eigenvectors of the augmented open-loop system, and determine the eigenvalues that are expected to be changed according to the requirements of the system performance indicators.

[0214] Step 2. Use the first step in the above real representation method to obtain the real representation matrix of the invariant eigenvalue And the real representation matrix of the eigenvector corresponding to the invariant eigenvalue And the matrix According to formula (13), QR decomposition is performed and the corresponding orthogonal matrix W and non-singular upper triangular matrix R are obtained.

[0215] Step 3. Select the desired eigenvalue of the closed-loop system and use the second step of the real representation method to obtain the real representation matrix of the desired eigenvalue According to formula (33), the matrix Take QR decomposition and calculate the block matrix S i21 ,S i22 .

[0216] Step 4. Choose the parameter vector ξ i , i=1,…,p, satisfying when the eigenvalue When , there is a eigenvector l,k∈1,…,p.

[0217] Step 5. Calculate the real representation matrix of the eigenvector corresponding to the expected eigenvalue according to formula (37) based on the block matrix and parameter vector and the matrix N consisting of the desired eigenstructure and compensator gains.

[0218] Step 6. Use equations (14), (16) and (26) to calculate the block matrices W2, Z2, W 21 ,W 22 Formulas (14) and (26) are formulas for direct block division according to the dimension, and formula (16) is a formula for direct substitution.

[0219] Step 7. Calculate the gain matrix P of the dynamic compensator using equations (17) and (18) respectively v ,F v ,Q v ,G v ,P d ,G d ,Q2,G d .

[0220] Step 3: Select a parameterized vector and calculate the parameterized dynamic compensator gain based on the block matrix, the real representation matrix of the eigenvector corresponding to the expected eigenvalue, and the matrix composed of the expected eigenstructure and the compensator gain.

[0221] Step 4: If the cost function exceeds the set threshold, the F norm of the compensation gain correlation matrix is optimized by the gradient optimization method using the given arbitrary initialization parameter vector to obtain the optimal parameterization vector, and the characteristic matrix composed of the eigenvectors and the transition matrix are calculated using the optimal parameterization vector and the block matrix, thereby calculating the dynamic compensation gain matrix and the control gain matrix of the dynamic compensator.

[0222] The previous section discussed the partial characteristic structure configuration method for improving system performance using a dynamic compensator and gave a design algorithm for the dynamic compensator. Considering that the present invention designs a controller for an augmented system after the original vibration system is expanded, the increase in dimension makes the control of the system more difficult, which is mainly reflected in the significant increase in the control gain norm, thereby requiring more control energy consumption. The present invention will consider how to design a dynamic compensator to consume less control energy while keeping the control target unchanged, that is, to use as little energy as possible to achieve the goal of improving system performance. The energy consumption of the controller is mainly determined by the gain norm of the controller. For a dynamic compensator such as Equation (2), in order to obtain the minimum controller energy to achieve the goal of improving the performance of the vibration system, that is, to complete the task of partial characteristic structure configuration, it is necessary to optimize the relevant gain norm of the dynamic compensator and design a suitable optimization method to minimize the norm of the compensator.

[0223] The energy consumption of the controller is mainly determined by the controller gain. Therefore, the norm of the dynamic compensator gain can indirectly characterize the energy consumption index. In order to obtain the minimum energy consumption, the present invention selects the following optimization function

[0224]

[0225] Among them, the compensation gain correlation matrix

[0226]

[0227] J is the optimization function that reflects the energy consumption index of the dynamic compensator. To achieve better optimization results, a gradient optimization method is used. The gradient calculation method is given by the following theorem.

[0228] Theorem 2 For any nonzero parameterized matrix Its column vector satisfies hour, Then the gradient of the optimization function J with respect to the non-zero parameterized matrix Ξ can be calculated as follows

[0229]

[0230] Among them, the three correlation matrices of the optimization function J with respect to the non-zero parameterization matrix Ξ gradient are

[0231]

[0232] Among them, (α 11 ,…,α 1p )、(α 21 ,…,α 2p ) and (α 31 ,…,α 3p ) are the columns of the corresponding matrix on the right side of the equation.

[0233] Proof: From the definition of the dynamic compensator gain norm, we know that

[0234]

[0235] Differentiating both sides of equation (47) yields

[0236]

[0237] Where Δ represents the differential operator and tr represents the trace of the matrix.

[0238] Memoization optimization function

[0239] J 11 =tr(Ψ H ΔΨ),J 12 =tr(ΨΔΨ H ). (49)

[0240] From formula (25), we can know

[0241]

[0242] as well as

[0243]

[0244] Substituting equations (50) and (51) into equation (49) yields the suboptimal function

[0245]

[0246] Note that differentiating the inverse matrix yields:

[0247]

[0248] So there is a sub-optimization function

[0249]

[0250] Now we need to further simplify the second equation of equation (53), and to do this we denote the optimization function

[0251]

[0252] Differentiating both sides of equation (27) yields

[0253]

[0254] Substituting equation (55) into equation (54) yields the suboptimal function

[0255]

[0256] make

[0257]

[0258] Among them, the parameterized correlation matrix From equations (53), (56) and (57), we can immediately see that

[0259]

[0260] Thus there is

[0261]

[0262] Where vec represents the matrix straightening operation, that is, the operation of expanding the matrix into column vectors by column. The matrix N composed of the desired characteristic structure and the compensator gain is (S 122 ξ1,…,S p22 ξ p ).

[0263] remember

[0264] Ξ=(ξ1,…,ξ p ),

[0265]

[0266] Then there is

[0267]

[0268] Calculate using a similar method and available

[0269]

[0270] here

[0271]

[0272] Combining equations (59), (60) and (61), it is easy to know

[0273] J 11 =tr(Γ1ΔΞ)-tr(Γ2ΔΞ)-tr(Γ3ΔΞ)

[0274] =tr((Γ1-Γ2-Γ3)ΔΞ) (62)

[0275] Similarly, we can get

[0276] J 12 =tr((Γ1-Γ2-Γ3) H ΔΞ H ) (63)

[0277] From equations (62) and (63), according to the properties of matrix trace and differential operation rules, we can get

[0278]

[0279] Algorithm 1 presents the calculation process for the dynamic compensator under arbitrary parameterization conditions. The controller gains derived from this algorithm may not necessarily have the minimum norm, meaning that energy control may not be optimal. Based on Algorithm 1, Theorem 2 provides a gradient calculation method. Given any given initialization parameter vector, the optimal parameterization vector can be obtained through gradient optimization. This allows us to calculate the optimal dynamic compensator gain norm and obtain a dynamic compensator with minimum energy control. Based on Algorithm 1 and the results of Theorem 2, the calculation process for the optimal energy dynamic compensator can be summarized as follows: Algorithm 2.

[0280] Algorithm 2

[0281] Step 1. Calculate the eigenvalues and eigenvectors of the augmented vibration system, and determine the eigenvalues that are expected to be changed based on the requirements of the system performance indicators.

[0282] Step 2. Matrix According to formula (13), QR decomposition is performed and the corresponding matrix W,R is obtained.

[0283] Step 3. Select the desired eigenvalues of the closed-loop system and use real representation to obtain the corresponding matrix According to formula (33), the matrix Take QR decomposition and calculate the block matrix S i21 ,S i22 .

[0284] Step 4. Select the initialization parameter vector ξ i0 , i=1,…,p, when Sometimes l,k∈1,…,p.

[0285] Step 5. Based on the set initialization parameter vector, the gradient optimization method is used to calculate the optimal parameterization vector and record it as

[0286] Step 6. Calculate the optimal parameterized vector obtained by calculation according to formula (37) N.

[0287] Step 7. Calculate W2, Z2, W using equations (14), (16), and (26) respectively. 21 ,W 22 .

[0288] Step 8. Use equations (17) and (18) to calculate the gain matrix P of the dynamic compensator for optimal energy consumption control. v ,F v ,Q v ,G v ,P d ,F d ,Q2,G d .

[0289] The present invention provides relevant numerical examples to verify the proposed relevant optimal energy consumption control algorithm.

[0290] Example 1. Consider the following vibration system

[0291]

[0292] The characteristic polynomial of the system (65) is solved by using the solve function of MATLAB. It is easy to calculate and the eigenvalue set of the vibration system (65) is 2.1478, 0.3656, -1.1628±0.6208, -2.9588, -2.3389.

[0293] Design a dynamic compensator as shown in formula (2). Assume that the compensator dimension s = 2. Then, according to formula (4), expand the dimension of the vibration system (65) and obtain the eigenvalue set of the augmented open-loop system as 2.1478, 0.3656, -1.1628 ± 0.6208, -2.9588, -2.3389, 0, 0, 0, 0. In order to verify the effectiveness of the algorithm, the eigenvalues 2.1478, 0.3656, 0, 0, 0, 0 are replaced by -4, -5, -6, -7, -8, -9. The parameter vector generally needs to satisfy (a) and (b) in Theorem 1. Here, since the expected eigenvalues are different from each other, it is sufficient as long as the eigenvectors are not equal. The parameter vector selected for initialization is

[0294]

[0295] The relevant gain matrix P of the dynamic compensator can be calculated by Algorithm 1: v ,F v ,Q v ,G v ,P d ,F d ,Q2,G d They are

[0296]

[0297] Example 1 is only used to verify Algorithm 1.

[0298] Example 2. Consider the following vibration system model

[0299]

[0300] It is easy to calculate. The eigenvalue set of the vibration system (66) is: 1.4018, -1.4980, -3.4464 ± 4.9412i. Design a dynamic compensator as shown in formula (2). Assume that the compensator dimension s = 1. Then, according to formula (4), the vibration system (66) is expanded and the eigenvalue set of the augmented open-loop system is obtained as: 1.4018, -1.4980, -3.4464 ± 4.9412i, 0, 0. In order to verify the effectiveness of the algorithm, the eigenvalues 1.4018, 0, 0 are replaced by -5, -6, -7. The parameterized vector is selected as

[0301]

[0302] (1) The correlation gain matrix P of the dynamic compensator can be calculated by Algorithm 1 v ,F v ,Q v ,G v ,P d ,F d ,Q2,G d They are:

[0303] P v =[193.6517 285.0687],F v =[0.7755],

[0304] Q v =[2.3086 3.3984],G v =[-12.0638],

[0305] P d =[114.3410 56.0794],F d =[3.8775],

[0306] Q d =[1.3631 0.6685],G d =[-35.3189].

[0307] (2) Considering the optimal energy consumption control problem, select the parameterized initial vector:

[0308]

[0309] According to the optimal energy consumption control algorithm, the optimal parameter solution vector of the energy consumption control problem is

[0310]

[0311] According to the optimal energy consumption control algorithm 2, the relevant gain matrix P of the dynamic compensator v ,F v ,Q v ,G v ,P d ,F d ,Q2,G d The calculation result is

[0312] P b =[98.0474 144.3326],F v =[20.3250],

[0313] Q v =[22.6219 10.6791],G v =[-15.6865],

[0314] P d =[145.2818 56.0794],F d =[101.1674],

[0315] Q d =[33.5200 13.1021],G d =[-53.2866].

[0316] Table 1: Comparison of calculation results

[0317]

[0318] The data in the table shows a significant decrease in the value of the cost function J. Note that the cost function J is a function of the norm of the dynamic compensator gain. This indicates that after optimization, the controller gain has been significantly reduced. Since the controller gain norm is positively correlated with control energy consumption, the optimization algorithm based on this invention significantly reduces the norm of the dynamic compensator, thereby reducing the control energy consumption of the vibration system. This plays a significant role in optimizing energy consumption and reducing energy consumption in the operation of vibrating structures in practical engineering applications.

[0319] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A minimum norm control method for a vibration structure based on a dynamic compensator, characterized in that: The steps are as follows: Step 1: Based on the vibration structure, use the finite element analysis method to establish a vibration structure model, design a dynamic compensator, and calculate the eigenvalues and eigenvectors of the compensated augmented open-loop system; Step 2: Determine the desired eigenvalues to be changed, perform QR decomposition on the unchanged eigenstructure matrix, and calculate the block matrix corresponding to the orthogonal matrix; perform QR decomposition on the closed-loop system correlation matrix, and calculate the real representation matrix of the eigenvector corresponding to the desired eigenvalues and the matrix consisting of the desired eigenstructure and compensator gain; Step 3: Select a parameterized vector and calculate the parameterized dynamic compensator gain based on the block matrix, the real representation matrix of the eigenvector corresponding to the expected eigenvalue, and the matrix composed of the expected eigenstructure and the compensator gain; Step 4: If the optimization function exceeds the set threshold, the optimization function is optimized by the gradient optimization method using the given arbitrary initialization parameter vector to obtain the optimal parameterization vector, and the characteristic matrix composed of the eigenvectors and the transition matrix are calculated using the optimal parameterization vector and the block matrix, thereby calculating the dynamic compensation gain matrix and the control gain matrix of the dynamic compensator.

2. The minimum norm control method of a vibration structure based on a dynamic compensator according to claim 1, characterized in that: The vibration structure model is y(t)=C d x(t) in, are the inertia matrix, damping matrix and stiffness matrix of the vibrating structure respectively. The inertia matrix M is a non-singular matrix, and n is the dimension of the vibrating structure. is the input control matrix, whose rank satisfies rank(B)=m, where m is the dimension of the control input; and are the displacement gain matrix and the velocity gain matrix respectively, and are the output vectors related to displacement and velocity at time t, r is the dimension of the output related to velocity, and q is the dimension of the output related to displacement; and are the displacement vector and the input control vector respectively; and represent velocity and acceleration respectively; The dynamic compensator is: in, is the compensation vector of the vibrating structure, and s is the dimension of the compensation; and denote the derivative and second-order derivative of the compensation vector w(t) respectively; are the dynamic compensation gain matrices related to the compensation vector w(t) and the control output vectors y(t), z(t), respectively. is the control gain matrix to be determined.

3. The minimum norm control method of a vibration structure based on a dynamic compensator according to claim 2, characterized in that: The vibration structure model after compensation can be obtained from the vibration structure model and the dynamic compensator: After considering the compensation vector w(t), the augmented open-loop system of the vibration structure model is: The closed-loop system of the vibration structure model is: Among them, the augmented inertia matrix Augmented Damping Matrix Augmented stiffness matrix Augmented Input Control Matrix Displacement compensation vector 4. The minimum norm control method of a vibration structure based on a dynamic compensator according to claim 3, characterized in that: The eigenvalues and corresponding eigenvectors of the compensated augmented open-loop system are denoted as There are p eigenvalues Need to reconfigure, p<2(n+s); design the dynamic compensation gain matrix of the parameterized dynamic compensator and the control gain matrix The eigenvalue of the closed-loop system is Reconfigured eigenvalues The corresponding eigenvector is 5. The minimum norm control method for a vibration structure based on a dynamic compensator according to any one of claims 2 to 4, characterized in that: Under the parameterized dynamic compensator, appropriate parameters are selected so that the F norm of the compensation gain correlation matrix Ψ Minimum, where the compensation gain correlation matrix is 6. The minimum norm control method of a vibration structure based on a dynamic compensator according to claim 4, characterized in that: The unchanged eigenstructure matrix is in, The real representation matrix representing the invariant eigenvalues, The real representation matrix representing the eigenvectors corresponding to the invariant eigenvalues; The closed-loop system correlation matrix is is the i-th expected eigenvalue, i=1,…,p.

7. The minimum norm control method of a vibration structure based on a dynamic compensator according to claim 6, characterized in that: The method of using QR decomposition to process the unchanged eigenstructure matrix is: in, is an orthogonal matrix, is a non-singular upper triangular matrix; denoted as orthogonal matrix in, It is a block matrix obtained by dividing the orthogonal matrix W into blocks; a new matrix is constructed by the block matrices W1 and W2 and and in, is the real representation matrix of the expected eigenvalues, is the real representation matrix of the eigenvector corresponding to the expected eigenvalue; The method for QR decomposition of the closed-loop system correlation matrix is: in, is an orthogonal matrix, is a nonsingular upper triangular matrix, a zero matrix (·) H Denotes conjugate transpose; let the orthogonal matrix S i =(S i1 ,S i2 ), where the orthogonal submatrix Orthogonal submatrix Left multiply the orthogonal submatrix Available S i is the orthogonal matrix rank(S i2 )=m+s; Orthogonal submatrix The row vectors of A set of basis vectors, but in Intermediate Matrix is the orthogonal submatrix S i2 The block matrix obtained by block division, N i Represents the expected eigenvalue of the i-th Eigenvector and the compensator gain; then the real representation matrix of the eigenvector corresponding to the expected eigenvalue and the matrix composed of the expected characteristic structure and the compensator gain are obtained respectively 8. The minimum norm control method of a vibration structure based on a dynamic compensator according to claim 7, characterized in that: Vibration structure meets If a dynamic compensator is used to reconfigure part of the characteristic structure of the augmented open-loop system to improve the dynamic performance of the system, the gain matrix of the dynamic compensator can be calculated as follows: in, N=(S 212 ξ1,…,S p22 x p ) in, S 112 、S p12 、S 212 、S p22 Both represent intermediate matrices, I s represents the identity matrix, (·) + Represents the Moore-Penrose inverse; And the block matrix W2=(W 21 ,W 22 ) is a constant matrix, the intermediate matrix represents the block submatrix obtained by partitioning the block matrix W2, is a selectable parameter vector that satisfies: (a) There are parameters l,k, when the eigenvalue When , the corresponding eigenvector * indicates conjugation; (b) 9. The minimum norm control method for vibration structure based on dynamic compensator according to claim 7 or 8, characterized in that: The optimization function Among them, the compensation gain correlation matrix For any nonzero parameterized matrix The column vector satisfies hour, Then the gradient of the optimization function J with respect to the non-zero parameterized matrix Ξ is Among them, the three correlation matrices are Intermediate Matrix 10. The minimum norm control method of a vibration structure based on a dynamic compensator according to claim 9, characterized in that: For the unchanged 2(n+s)-p constant eigenvalues and corresponding eigenvectors, let 2τ be pairwise conjugates and the remaining 2(n+s)-p-2τ be real eigenvalues. Among them, δ represents the imaginary unit, λ 2ι-1 、 represents the eigenvalue of the lth conjugate pair, λ ιR ,λ ιI The eigenvalue λ represents the conjugation 2ι-1 、 The real and imaginary parts, v 2ι-1 、 represents the eigenvalue λ of the ιth conjugate 2ι-1 、 The corresponding eigenvectors, v ιR 、v ιI Represents the feature vector v 2ι-1 、 The real and imaginary parts of the matrix; then the real representation of the unchanged eigenvalue Real representation matrix of eigenvectors corresponding to invariant eigenvalues Among them, diag represents the diagonal matrix composed of all elements; The expected p eigenvalues and corresponding eigenvectors are represented by real numbers. Suppose 2ε are pairwise conjugates and the remaining p-2ε are real eigenvalues. Then in, represents the eigenvalue of the ηth conjugate pair, The eigenvalue representing the conjugation The real and imaginary parts of represents the eigenvalue of the ηth conjugate pair The corresponding eigenvector, Represents the feature vector The real and imaginary parts of the expected eigenvalues are represented by the real matrix The real representation matrix of the expected eigenvalue corresponding to the eigenvector Reconfigured feature pairs and unchanged feature pairs If both satisfy the closed-loop system, then in, is the eigenvalue The corresponding eigenvector, v j is the eigenvalue λ j The corresponding eigenvector; Unchanged feature pairs To satisfy the augmented open-loop system, there is but definition It can be seen that but in, is a zero matrix, and the matrix N is composed of the desired characteristic structure and the compensator gain. p ),and represents the complex field, is the eigenvector corresponding to the expected eigenvalue; By rank(γ 2 M+γD+K,B)=n Left Null Space The dimension of is m+s.

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