CN113486523A - Linear variable parameter vibration system global identification method - Google Patents

Linear variable parameter vibration system global identification method Download PDF

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CN113486523A
CN113486523A CN202110778203.3A CN202110778203A CN113486523A CN 113486523 A CN113486523 A CN 113486523A CN 202110778203 A CN202110778203 A CN 202110778203A CN 113486523 A CN113486523 A CN 113486523A
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lpv
vibration system
excitation
vibration
scheduling
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李迎光
刘旭
程英豪
蔡宇
郝小忠
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Nanjing University of Aeronautics and Astronautics
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Abstract

A linear variable parameter vibration system global identification method is characterized in that excitation is continuously applied to an LPV vibration system with continuously changing scheduling variables, an over-complete dictionary function base is constructed, an LPV-ARX type vibration system model is obtained from excitation-response data with continuously changing scheduling variables through sparse regression, the scheduling variables are taken as different values, impact excitation response or random excitation response is calculated, modal identification is carried out, and distribution of modal parameters about the scheduling variables can be obtained. Compared with the traditional local identification method of the LPV vibration system, the global identification method disclosed by the invention can obtain the vibration system model in the LPV form only by once identification, and the identification efficiency of the LPV vibration system is obviously improved on the premise of ensuring the identification precision.

Description

Linear variable parameter vibration system global identification method
Technical Field
The invention belongs to the field of vibration system identification, in particular to linear variable parameter (LPV) vibration system identification with vibration characteristics changing along with the change of a scheduling variable, and particularly relates to a method for globally identifying an LPV vibration system directly from excitation-response data with continuously changing scheduling variables.
Background
There are a number of situations in a manufacturing system where the vibration characteristics of the system change as a function of certain parameters. For example, the vibration characteristics of machine tool spindles, ball screws, and milling robot end effectors are all related to their structural pose. However, when the specific parameters influencing the vibration characteristics are fixed, the system meets the characteristics of a linear time-invariant system and can be expressed by a linear vibration differential equation. Such a system in which the system parameters change but the system structure is not changed when the specific parameters are fixed is referred to as a Linear Parameter-Varying (LPV) system. A vibration system conforming to the characteristics of the LPV system may be defined as an LPV vibration system, wherein certain parameters affecting the vibration characteristics are referred to as scheduling variables.
Vibration system identification is the basis for system vibration analysis and control. The identification of the LPV vibration system mainly adopts a local identification method at present, namely a group of discrete scheduling variables are selected, parameters of the vibration system are respectively identified and then fitting is carried out, so that an LPV model of the vibration system is established. The thesis "machine tool dynamics based on spatial statistics" adopts a spatial statistics method, and utilizes a modal experiment to measure the tool nose natural frequency under the discrete pose of an ultra-precise machine tool, thereby establishing a Kriging prediction model of the natural frequency relative to the pose. The paper Data-drive Modeling of the modular Properties of a Six-Degrees-of-free Industrial Robot and Its Application to Robotic Milling measures vibration characteristics of Milling Robot end effectors at discrete poses using Modal experiments to build a Gaussian regression model of Modal parameters with respect to pose. However, when the system characteristics are complex with respect to the scheduling variables or are related to a plurality of scheduling variables, in order to identify an accurate LPV vibration system model, the local identification method needs to perform experiments under a large number of discrete scheduling variables, which is often inefficient. Therefore, a method for accurately and efficiently identifying the LPV vibration system is needed.
Disclosure of Invention
The invention aims to accurately and efficiently identify an LPV vibration system, and discloses a global identification method of the LPV vibration system.
The technical scheme of the invention is as follows:
a global identification method of an LPV vibration system is characterized by comprising the following steps:
1. continuously applying excitation to the LPV vibration system with continuously changed scheduling variables, and synchronously acquiring scheduling variables, excitation and vibration response signals;
2. constructing an over-complete dictionary function library and obtaining an LPV-ARX-form vibration system model from the excitation-response data with continuously changed scheduling variables through sparse regression;
3. and taking the scheduling variable as different values, calculating the impact excitation response or the random excitation response of the LPV-ARX model, and performing modal identification to obtain the distribution of modal parameters about the scheduling variable.
In the LPV vibration system, when a scheduling variable p (t) influencing the vibration characteristic of the system is fixed, the system is a linear time-invariant system, and when the system modal parameter changes, the system modal parameter also changes, and the system modal parameter can be expressed by the following differential equation:
Figure BDA0003155127090000021
where x (t) and f (t) are vibration displacement and excitation force of the system, and M (p (t)), C (p (t)) and K (p (t)) respectively represent modal mass, modal damping and modal stiffness function matrices of the system with respect to the scheduling variables. The vibration differential equation and the modal parameters are both defined in a modal coordinate after the mode shape normalization.
The LPV vibration system continuously applying excitation to the continuously-changed scheduling variables requires that the continuously-changed tracks of the scheduling variables cover the whole working space as much as possible, the continuously-applied excitation needs to meet the continuous excitation condition, the excitation frequency components are as rich as possible, and random excitation or pseudo-random excitation can be adopted.
The constructing of the overcomplete dictionary function library requires: the dictionary functions in the overcomplete dictionary function library generally include constant terms, polynomial functions, trigonometric functions, exponential functions, etc., and are based on polynomials, wherein there are numerous redundant terms.
In the sparse regression, sparse constraint is added to coefficient vectors corresponding to the overcomplete dictionary function library, and L1 regularization terms of the coefficient vectors are added on the basis of an optimization target of an original regression problem when the coefficient vectors are solved.
The vibration system model of the LPV-ARX form comprises the following components: the differential equation of the n-order LPV vibration system can be dispersed in a time domain into a standard LPV-ARX form:
Figure BDA0003155127090000022
wherein p iskIs the scheduling variable of the system at the k time, and x (k) and f (k) are the vibration displacement and the exciting force of the system at the k time. a isi(pk) And bi(pk) Are coefficient functions of the LPV-ARX model, and are all functions related to scheduling variables. In order to avoid the occurrence of the fractional function form to reduce the identification difficulty, further, the coefficient function a of x (k) is used0(pk) Splitting into constant and non-constant terms, and respectively placing them at both sides of equation, dividing the equation by a0(pk) The constant term of (c) translates into:
Figure BDA0003155127090000023
wherein:
Figure BDA0003155127090000031
Figure BDA0003155127090000032
Figure BDA0003155127090000033
wherein A is0As a function of the coefficient a0(pk) Constant term in (1).
The vibration system model in the LPV-ARX form is obtained from excitation-response data with continuously changed scheduling variables by constructing an overcomplete dictionary function library and through sparse regression, and the overcomplete dictionary function library is used for the coefficient function a'i(pk) And b'i(pk) Performing representation, and solving a dictionary function library coefficient vector by using the sparse regression so as to determine a 'coefficient function'i(pk) And b'i(pk)。
The invention has the beneficial effects that:
the global identification method disclosed by the invention can obtain the vibration system model in the LPV form only by once identification, and obviously improves the identification efficiency of the LPV vibration system on the premise of ensuring the identification precision.
Drawings
Fig. 1 is a comparison graph of a local identified modal parameter fitting curve and a global identified modal parameter curve in an a-axis working space.
FIG. 2 is a graph of a fitted surface of local identified measurement points and modal parameters in an AC-axis working space.
FIG. 3 is a fitting surface diagram of globally identified modal parameters in the AC-axis working space.
Detailed Description
The invention is further described with reference to the following figures and embodiments.
The LPV vibration system global identification method provided by the invention is essentially to directly identify the vibration system model in the LPV-ARX form, firstly give a formal definition and a specific identification algorithm of the vibration system model, and give a verification effect of the vibration system model on the LPV vibration system proxy model established by modal parameter data of the tool nose structure of an actual machine tool. In specific applications, the required excitation and vibration response acquisition devices are both dependent on the prior art, and therefore are not described in detail.
The vibration differential equation of the n-order LPV vibration system under the mode state coordinates after mode shape normalization can be expressed as follows:
Figure BDA0003155127090000034
wherein, p (t) represents a scheduling variable changing along with time, and M (p (t)), C (p (t)) and K (p (t)) respectively represent a modal mass, modal damping and modal stiffness function matrix of the system relative to the scheduling variable.
Figure BDA0003155127090000035
Respectively an acceleration vector, a velocity vector and a displacement vector of the vibration,
Figure BDA0003155127090000036
is the excitation force vector.
The n-order LPV vibration system can be further represented by time domain discretization:
Figure BDA0003155127090000037
wherein p iskAnd x (k) and f (k) are respectively the vibration displacement and the excitation force at the k moment of the system. a isi(pk) And bi(pk) Is about a scheduling variable pkAll coefficient functions of (1) are all modal parameter functions M of each order of the systemi(pk)、Ci(pk) And Ki(pk) A polynomial combination of (2).
In order to avoid the occurrence of the fractional function form to reduce the identification difficulty, the coefficient function a of x (k) in the formula (2) is used0(pk) Splitting into constant and non-constant terms, and respectively placing them at both sides of equation, dividing the equation by a0(pk) The model of the vibration system in LPV-ARX form can be obtained as follows:
Figure BDA0003155127090000041
wherein:
Figure BDA0003155127090000042
Figure BDA0003155127090000043
Figure BDA0003155127090000044
in the formula, A0As a function of the coefficient a0(pk) Constant term in (1). Coefficient functions a 'in formula (3)'i(pk) And b'i(pk) Are all only formed by coefficient functions a in the formula (2)i(pk) And bi(pk) Integer divided by constant A0And obtaining the product.
The global identification of the LPV vibration system is to solve the LPV-ARX model shown in the formula (3), and has the advantages that the coefficient functions in the formula (3) are all polynomial combinations of system modal parameter functions, the complex condition that both numerator and denominator of the coefficient functions are the polynomial combinations of the system modal parameter functions is avoided, and the identification difficulty is effectively reduced.
Pairing coefficient function a 'in formula (3) by using overcomplete dictionary function library'i(pk) And b'i(pk) And performing characterization, wherein the overcomplete dictionary function library generally comprises constant terms, polynomial functions, trigonometric functions, exponential functions and the like. Defining an overcomplete dictionary function library as:
Figure BDA0003155127090000045
Figure BDA0003155127090000046
wherein the content of the first and second substances,
Figure BDA0003155127090000047
is a dictionary function. Coefficient function a'0(pk) Corresponding dictionary function library
Figure BDA0003155127090000048
Dictionary function library not containing constant terms and corresponding to other coefficient functions
Figure BDA0003155127090000049
Then both contain constant terms.
Each coefficient function may be further represented by a dictionary function library as:
Figure BDA00031551270900000410
Figure BDA00031551270900000411
Figure BDA0003155127090000051
wherein, thetai(i ═ 0,1, … 4n +1) denotes a dictionary function library coefficient vector of the corresponding coefficient function. Then, equation (3) can be further expressed as:
Figure BDA0003155127090000052
to simplify the expression, let:
Figure BDA0003155127090000053
Figure BDA0003155127090000054
wherein, theta is a dictionary function library coefficient vector of all coefficient functions to be identified, phi (k) represents that a scheduling variable is pkAnd then, combining all vibration displacements and excitation forces from k-2n to k with the corresponding dictionary function library to obtain a new state vector, and then expressing the vibration system model in the LPV-ARX form as follows:
x(k)＝Φ(k)Θ (5)
the data in actual recognition is a discrete time series composed of a scheduling variable, a state quantity, and an external input quantity. Defining:
k＝[k1 k2 … kl]
represents a discrete time series vector, where l represents the length of the time series, each kiRepresenting a discrete time instant. The discrete time instants in k need not be arranged in chronological order. From k, a vector x (k) and a matrix Ψ (k) may be constructed:
x(k)＝[x(k1) x(k2) … x(kl)]T
Ψ(k)＝[ΦT(k1) ΦT(k2) … ΦT(kl)]T
finally, the LPV vibration system identification problem can be converted into:
x(k)＝Ψ(k)Θ (6)
solving the dictionary function library coefficient vector Θ in the above equation is a standard linear regression problem. Because the dictionary function library is overcomplete, sparse regression is used to solve dictionary function library coefficient vectors in order to reduce overfitting and improve robustness to noise. Adding an L1 regularization term to the regression problem of the above equation yields:
Figure BDA0003155127090000061
where λ represents the weight of the sparsity constraint. Compared with linear regression, formula (7) increases sparse constraint on the coefficient vector Θ of the dictionary function library, thereby reducing the influence of redundant dictionary functions in the overcomplete dictionary library. Solving the sparse regression problem shown in the formula, and identifying to obtain the vibration system model in the LPV-ARX form.
When the coefficient functions are identified, equation (3) can be converted into a standard-form LPV-ARX model when applied:
Figure BDA0003155127090000062
the vibration system model in the LPV-ARX form obtained by the global identification describes the vibration characteristics of the system in the whole working space of the scheduling variables, the scheduling variables are fixed to be different values, the impact excitation response or the random excitation response of the scheduling variables are calculated, and the modal parameters under any scheduling variables can be obtained by carrying out modal identification.
The vibration characteristic of the tool nose structure of the machine tool spindle is related to the pose, and particularly when the direction of a cutter shaft is changed due to the change of a machine tool rotating shaft, the modal parameter change is obvious, so that the machine tool spindle tool belongs to a typical LPV vibration system. And respectively verifying the single-scheduling-variable LPV vibration system and the multi-scheduling-variable LPV vibration system by using the LPV vibration system proxy model established by using the modal parameter data of the tool nose structure of the actual machine tool.
In the global identification of a single-scale-variable LPV vibration system, the C axis is fixed at 0 degree, the A axis is set to move back and forth for multiple times in a working space of (-50 degrees, 50 degrees), Gaussian white noise excitation is continuously applied, and scheduling variables, excitation and vibration response sequences are synchronously acquired. The overcomplete dictionary function library is based on polynomial functions, plus a small number of other non-linear functions. The result is shown in fig. 1, where the solid line represents a modal parameter curve obtained by parameter fitting of the local identification measurement point, that is, a modal parameter curve in the proxy model, and the dotted line represents a modal parameter curve obtained by performing global identification in the proxy model. Compared with a parameter curve obtained by local identification fitting, the average error percentages of modal mass, damping and rigidity obtained by global identification are 2.63%, 2.50% and 2.65%, and the effectiveness of the method for identifying the single-gradient variable LPV vibration system is fully shown.
In the global identification of the multi-scheduling variable LPV vibration system, a rotating shaft is arranged to move in an A-axis working space [ -50 degrees, 50 degrees ] and a C-axis working space [0 degrees, 330 degrees ] according to a certain track, Gaussian white noise excitation is continuously applied, and the coordinates of an AC axis of a scheduling variable, an excitation and vibration response sequence are synchronously acquired. The dictionary function library is based on polynomial functions, plus a few other non-linear functions. Wherein the polynomial dictionary function form is:
Figure BDA0003155127090000063
Figure BDA0003155127090000064
wherein the scheduling variable paAnd pcRespectively an A-axis coordinate and a C-axis coordinate, and q is a polynomial order. The modal parameter distribution of the AC axis working space obtained by parameter fitting of the local identification measurement points is shown in fig. 2, and the modal parameter distribution obtained by global identification is shown in fig. 3, so that it can be observed that the global identification successfully identifies the overall distribution rule and most details of the modal parameters in the AC axis working space. The average error percentages of the modal mass, the modal damping and the modal stiffness are only 2.69%, 2.65% and 2.68% respectively, and the identification effectiveness of the method for the multi-scheduling variable LPV vibration system is fully shown.
The invention is not related to the part which is realized by the prior art in the same way as the prior art.

Claims (7)

1. A global identification method of a linear variable parameter vibration system is characterized by comprising the following steps:
continuously applying excitation to the LPV vibration system with continuously changed scheduling variables, and synchronously acquiring scheduling variables, excitation and vibration response signals;
constructing an over-complete dictionary function library and obtaining an LPV-ARX-form vibration system model from the excitation-response data with continuously changed scheduling variables through sparse regression;
and taking the scheduling variable as different values, calculating the impact excitation response or the random excitation response of the LPV-ARX model, and performing modal identification to obtain the distribution of modal parameters about the scheduling variable.
2. The global identification method of the linear variable-parameter vibration system according to claim 1, wherein: in the LPV vibration system, when a scheduling variable p (t) affecting vibration characteristics of the LPV vibration system is fixed, the system is a linear time-invariant system, and when the scheduling variable p (t) changes, a modal parameter of the system also changes, and the system can be expressed by the following differential equation:
Figure FDA0003155127080000011
x (t) and f (t) are the vibration displacement vector and the excitation force vector of the system,
Figure FDA0003155127080000012
the vibration differential equation and the modal parameters are defined in a modal coordinate after mode shape normalization, wherein the acceleration vector and the velocity vector of vibration are respectively, and M (p (t), C (p (t)) and K (p (t)) respectively represent a modal mass, modal damping and modal stiffness function matrix of the system relative to a scheduling variable.
3. The global identification method for the linear variable-parameter vibration system as claimed in claim 1, wherein the continuously applying the excitation to the LPV vibration system with the continuously variable scheduling variable means: the continuously variable trajectory of the scheduling variable covers the whole working space, and the continuously applied excitation needs to meet the continuous excitation condition, and random excitation or pseudo-random excitation is adopted.
4. The global identification method of the linear variable parameter vibration system according to claim 1, wherein when constructing the overcomplete dictionary function library: the dictionary functions in the overcomplete dictionary function library generally include constant terms, polynomial functions, trigonometric functions, exponential functions, and are based on polynomials, wherein there are many redundant terms.
5. The global identification method of the linear variable parameter vibration system according to claim 1, wherein the sparse regression is: and adding sparse constraint to coefficient vectors corresponding to the overcomplete dictionary function library, and adding an L1 regularization term to the coefficient vectors on the basis of an optimization target of the original regression problem when solving the coefficient vectors.
6. The global identification method for the linear variable parameter vibration system according to claim 1, wherein the differential equation of the n-order LPV vibration system in the vibration system model of LPV-ARX form can be discrete in time domain as a standard LPV-ARX form:
Figure FDA0003155127080000013
wherein p iskIs a scheduling variable of the system at the moment k, and x (k) and f (k) are vibration displacement and exciting force of the system at the moment k; a isi(pk) And bi(pk) Are coefficient functions of the LPV-ARX model, and are all functions related to scheduling variables. In order to avoid the occurrence of the fractional function form to reduce the identification difficulty, further, the coefficient function a of x (k) is used0(pk) Splitting into constant and non-constant terms, and respectively placing them at both sides of equation, dividing the equation by a0(pk) The constant term of (c) translates into:
Figure FDA0003155127080000021
wherein:
Figure FDA0003155127080000022
Figure FDA0003155127080000023
Figure FDA0003155127080000024
wherein A is0As a function of the coefficient a0(pk) Constant term in (1).
7. The global identification method for the linear variable parameter vibration system according to claim 1, wherein the construction of the overcomplete dictionary function library and the obtaining of the vibration system model in LPV-ARX form from the excitation-response data with continuously changing scheduling variables by sparse regression is to use the overcomplete dictionary function library of claim 4 to the coefficient function a 'of claim 6'i(pk) And b'i(pk) Characterizing and solving dictionary function library coefficient vectors using the sparse regression of claim 5 to determine coefficient functions a 'of claim 6'i(pk) And b'i(pk)。
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115685746A (en) * 2022-09-20 2023-02-03 四川大学 Offline and online combined system identification method for machine tool workbench

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115685746A (en) * 2022-09-20 2023-02-03 四川大学 Offline and online combined system identification method for machine tool workbench
CN115685746B (en) * 2022-09-20 2023-09-01 四川大学 Off-line and on-line combined system identification method for machine tool workbench

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