CN113486523B - Global identification method for linear variable parameter vibration system - Google Patents
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Abstract
A global identification method of a linear variable parameter vibration system is characterized in that an overcomplete dictionary function library is constructed by continuously applying excitation to an LPV vibration system with continuously changed scheduling variables, a vibration system model in the form of LPV-ARX is obtained from excitation-response data with continuously changed scheduling variables through sparse regression, impact excitation responses or random excitation responses of the scheduling variables are calculated by taking the scheduling variables as different values, and modal identification is carried out, so that 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 LPV vibration system model only by one-time identification, and the identification efficiency of the LPV vibration system is obviously improved on the premise of ensuring the identification precision.
Description
Technical Field
The invention belongs to the field of vibration system identification, in particular to a method for identifying a linear variable parameter (LPV) vibration system with vibration characteristics changing along with the change of a dispatching variable, and particularly relates to a method for globally identifying the LPV vibration system directly from excitation-response data with continuous change of the dispatching variable.
Background
There are a number of situations in manufacturing systems where the vibration characteristics of the system vary with particular 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 affecting the vibration characteristics are fixed, the system satisfies the characteristics of a linear time-invariant system, and can be represented by a linear vibration differential equation. Such a system in which a specific parameter is fixed and constant and is represented as a linear time-constant system, and in which a system parameter is changed and a system structure is constant when a specific parameter is changed is called a linear variable parameter (LPV) system. A vibration system that meets the characteristics of an LPV system may be defined as an LPV vibration system, where certain parameters affecting vibration characteristics are referred to as schedule variables.
Vibration system identification is the basis for system vibration analysis and control. At present, a local identification method is mainly adopted for identifying the LPV vibration system, namely, a group of discrete scheduling variables are selected, and parameters of the vibration system are respectively identified and then fitted so as to establish an LPV model of the vibration system. The paper "machine tool dynamics characteristic based on space statistics" adopts a space statistics method, and utilizes a modal experiment to measure the natural frequency of the tool nose under the discrete pose of the ultra-precise machine tool, so as to establish a Kriging prediction model of the natural frequency on the pose. The paper "Data-Driven Modeling of the Modal Properties of a Six-Degrees-of-Freedom Industrial Robot and Its Application to Robotic Milling" utilizes a modal experiment to measure vibration characteristics of a milling robot end effector of discrete pose and establishes a Gaussian regression model of modal parameters with respect to the pose. However, when the system characteristics are complex with respect to the variation of 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 tends to be 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, comprising the steps of:
1. Continuously applying excitation to the LPV vibration system with continuously-changed scheduling variable, and synchronously collecting the scheduling variable, excitation and vibration response signals;
2. Constructing an overcomplete dictionary function library and obtaining an LPV-ARX vibration system model from excitation-response data of continuous change of a scheduling variable through sparse regression;
3. And taking the scheduling variable as different values, calculating the impulse excitation response or the random excitation response of the LPV-ARX model, and carrying out mode identification to obtain the distribution of the mode parameters about the scheduling variable.
In the LPV vibration system, the system is a linear time-invariant system when a scheduling variable p (t) affecting the vibration characteristics of the system is fixed, and the modal parameters of the system also change when the system changes, and can be expressed by the following differential equation:
where x (t) and f (t) are the vibration displacement and excitation force of the system, M (p (t)), C (p (t)) and K (p (t)) represent the modal mass, modal damping and modal stiffness function matrices of the system with respect to the schedule variables, respectively. The vibration differential equation and the modal parameters are defined under the modal coordinates after the normalization of the vibration modes.
The continuous excitation of the LPV vibration system with continuously variable scheduling variable requires that the track of the continuously variable scheduling variable cover the whole working space as much as possible, the continuously applied excitation meets the continuous excitation condition, the excitation frequency components are as rich as possible, and random excitation or pseudo random excitation can be adopted.
The construction of the overcomplete dictionary function library requires: dictionary functions in an overcomplete dictionary function library typically include constant terms, polynomial functions, trigonometric functions, exponential functions, etc., and have polynomials as the main basis, with numerous redundant terms.
And the sparse regression is used for adding sparse constraint to the coefficient vector corresponding to the overcomplete dictionary function library, and adding an L1 regularization term to the coefficient vector on the basis of the optimization target of the original regression problem when solving the coefficient vector.
The vibration system model of the LPV-ARX form: the differential equation of the n-order LPV vibration system can be time-domain discretized into a standard LPV-ARX form:
Where p k is the schedule variable of the system at time k, and x (k) and f (k) are the vibration displacement and excitation force at time k of the system. a i(pk) and b i(pk) are coefficient functions of the LPV-ARX model, both as functions of the scheduling variables. To avoid the occurrence of the fractional function form to reduce the difficulty of recognition, further, the coefficient function a 0(pk) of x (k) is split into a constant term and a non-constant term, and placed on both sides of the equation, and the constant term of the whole equation divided by a 0(pk) is converted into:
wherein:
Where a 0 is a constant term in the coefficient function a 0(pk).
The vibration system model of the LPV-ARX form is obtained from excitation-response data of continuous change of scheduling variables by constructing an overcomplete dictionary function library, wherein coefficient functions a 'i(pk) and b' i(pk) are represented by the overcomplete dictionary function library, and coefficient vectors of the dictionary function library are solved by using the sparse regression, so that coefficient functions a 'i(pk) and b' i(pk) are determined.
The beneficial effects of the invention are as follows:
The global identification method disclosed by the invention can obtain the vibration system model in the LPV form by only one-time 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 graph comparing local and global identification modal parameter fitting curves in an A-axis workspace.
FIG. 2 is a graph of a locally identified measurement point and modal parameter fit curve in an AC axis workspace.
FIG. 3 is a graph of a global recognition modality parameter fit surface within an AC axis workspace.
Detailed Description
The invention will be further described with reference to the drawings and the specific embodiments.
The global identification method of the LPV vibration system is essentially to directly identify the vibration system model in the LPV-ARX form, firstly, the formal definition and specific identification algorithm of the vibration system model are given, and the verification effect of the vibration system model on the LPV vibration system agent model established by the modal parameter data of the actual machine tool nose structure is given. In specific application, the required excitation and vibration response acquisition equipment all depend on the prior art, so that no further description is given.
The vibration differential equation of the n-order LPV vibration system under the mode coordinates after the vibration mode normalization can be expressed as:
Where p (t) represents a time-varying schedule variable, and M (p (t)), C (p (t)), and K (p (t)) represent the modal mass, modal damping, and modal stiffness function matrices of the system with respect to the schedule variable, respectively. Acceleration vector, velocity vector and displacement vector of vibration respectively,/>Is the excitation force vector.
The above-described n-order LPV vibration system may be further expressed in terms of time domain dispersion as:
Wherein p k is a scheduling variable at the time of k of the system, and x (k) and f (k) are vibration displacement and excitation force at the time of k respectively. a i(pk) and b i(pk) are coefficient functions for the schedule variable p k, which are polynomial combinations of the system's respective order modal parameter functions M i(pk)、Ci(pk) and K i(pk).
In order to avoid the occurrence of a fractional function form to reduce the identification difficulty, the coefficient function a 0(pk) of x (k) in the formula (2) is split into a constant term and a non-constant term, and the constant term is respectively arranged on two sides of an equation, and the whole equation is divided by the constant term of a 0(pk), so that the vibration system model in the LPV-ARX form can be obtained as follows:
wherein:
Where A 0 is a constant term in the coefficient function a 0(pk). Each of the coefficient functions a 'i(pk) and b' i(pk) in the formula (3) is obtained by dividing the whole of each of the coefficient functions a i(pk) and b i(pk) in the formula (2) by the constant a 0.
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 polynomial combinations of system modal parameter functions, so that the complex situation that the coefficient functions are combined by the polynomial of the system modal parameter functions in the molecular and denominator mode is avoided, and the identification difficulty is effectively reduced.
The coefficient functions a 'i(pk) and b' i(pk) in equation (3) are characterized by an overcomplete dictionary function library, which typically contains constant terms, polynomial functions, trigonometric functions, exponential functions, and the like. Defining an overcomplete dictionary function library as:
Wherein, Is a dictionary function. Coefficient function a' 0(pk) corresponding dictionary function library/>Dictionary function library/>, corresponding to the rest coefficient functions, without constant itemsThen both contain constant terms.
The coefficient functions may be further expressed as:
where θ i (i=0, 1, … 4n+1) represents a dictionary function library coefficient vector of the corresponding coefficient function. Then, equation (3) may be further expressed as:
To simplify the expression, let:
Wherein Θ is a dictionary function library coefficient vector of all coefficient functions to be identified, Φ (k) represents a new state vector formed by combining all vibration displacement and excitation force from k-2n to k moment and the corresponding dictionary function library when the scheduling variable is p k, and then the vibration system model in the form of LPV-ARX can be expressed as follows:
x(k)＝Φ(k)Θ (5)
The data at the time of actual recognition is a discrete time series composed of schedule variables, state quantities, and external input quantities. Definition:
k＝[k1 k2 … kl]
Represents a discrete time series vector, where l represents the length of the time series and each k i represents a discrete time instant. The discrete moments in k need not be arranged in time order. From k, a vector x (k) and a matrix ψ (k) can 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 translated 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 for dictionary function library coefficient vectors in order to reduce the overcomplete and improve the robustness to noise. Adding an L1 regularization term to the regression problem of the above formula to obtain:
Where λ represents the weight of the sparse constraint. Compared with linear regression, the formula (7) increases the sparse constraint on the dictionary function library coefficient vector theta, thereby reducing the influence of redundant dictionary functions in the overcomplete dictionary library. And solving the sparse regression problem shown in the formula, and identifying to obtain the vibration system model in the LPV-ARX form.
After identifying each coefficient function, the equation (3) can be converted into a standard LPV-ARX model when in use:
The vibration system model in the LPV-ARX form obtained through global identification describes the vibration characteristics of the system in the whole working space of the scheduling variable, the scheduling variable is fixed to be different values, the impact excitation response or the random excitation response of the scheduling variable is calculated, the modal identification is carried out, and the modal parameters under any scheduling variable can be obtained.
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 rotating shaft of the machine tool, the change of modal parameters is obvious, and the machine tool spindle nose structure belongs to a typical LPV vibration system. And verifying the single-dispatching variable and multi-dispatching variable LPV vibration system respectively by using the LPV vibration system proxy model established by the modal parameter data of the actual machine tool nose structure.
In global identification of the single-modulation variable LPV vibration system, the C axis is fixed at 0 degrees, the A axis is set to reciprocate for a plurality of times in a working space of minus 50 degrees and 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 nonlinear functions. As a result, as shown in fig. 1, the solid line represents a modal parameter curve obtained by locally identifying the measurement point parameter fitting, 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 the 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 invention on the identification of the single-tone variable LPV vibration system is fully shown.
In global identification of the multi-scheduling variable LPV vibration system, a rotating shaft is arranged to move in a certain track in an A-axis working space of minus 50 degrees, 50 degrees and a C-axis working space of 0 degrees and 330 degrees, gaussian white noise excitation is continuously applied, and scheduling variable AC axis coordinates, excitation and vibration response sequences are synchronously acquired. The dictionary function library is based on polynomial functions, plus a small number of other nonlinear functions. The polynomial dictionary function form is as follows:
Wherein the scheduling variables p a and p c are an A-axis coordinate and a C-axis coordinate, respectively, and q is a polynomial order. The modal parameter distribution of the AC axis working space is obtained by fitting the local identification measuring point parameters is shown in fig. 2, the modal parameter distribution obtained by global identification is shown in fig. 3, and the overall distribution rule and most of details of the modal parameters in the AC axis working space can be observed when the global identification is successfully identified. 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 effectiveness of the invention on the identification of the multi-scheduling variable LPV vibration system is fully shown.
The invention is not related to the implementation of the prior art, which is partially the same as the prior art.
Claims (4)
1. The global identification method of the linear variable parameter vibration system is characterized by comprising the following steps of:
Continuously applying excitation to the LPV vibration system with continuously-changed scheduling variable, and synchronously collecting the scheduling variable, excitation and vibration response signals;
constructing an overcomplete dictionary function library and obtaining an LPV-ARX vibration system model from excitation-response data of continuous change of a scheduling variable through sparse regression;
taking the dispatching variable as different values, calculating the impulse excitation response or the random excitation response of the vibration system model in the LPV-ARX form, and carrying out modal identification to obtain the distribution of modal parameters about the dispatching variable;
in the vibration system in the LPV-ARX form, the system is a linear unchanged system when a scheduling variable p (t) affecting the vibration characteristics of the model is fixed, and the modal parameters of the system are changed along with the change of the scheduling variable p (t), and the system is represented by the following differential equation:
c (t) and f (t) are vibration displacement vectors and excitation force vectors of the system, Respectively representing the modal mass, modal damping and modal stiffness function matrix of the system about the scheduling variable by using an acceleration vector, a speed vector, M (p (t)), C (p (t)) and K (p (t)) of vibration, wherein the vibration differential equation and the modal parameters are defined under the modal coordinates after the vibration mode normalization; the differential equation of the n-order LPV vibration system in the LPV-ARX form vibration system model may be time domain discretized into a standard LPV-ARX form:
Wherein, p k is the scheduling variable of the system at the moment k, and x (k) and f (k) are the vibration displacement and excitation force at the moment k of the system; a i(pk) and b i(pk) are coefficient functions of the LPV-ARX model, both functions of the scheduling variables; to avoid the occurrence of the fractional function form to reduce the difficulty of recognition, further, the coefficient function a 0(pk) of x (k) is split into a constant term and a non-constant term, and placed on both sides of the equation, and the constant term of the whole equation divided by a 0(pk) is converted into:
wherein:
Where a 0 is a constant term in the coefficient function a 0(pk).
2. The global recognition method of a linear variable parameter vibration system according to claim 1, wherein continuously applying excitation to the LPV vibration system in which the tuning variable is continuously changed means: the track of the continuous change of the scheduling variable is required to cover the whole working space, and the continuously applied excitation is required to meet the continuous excitation condition, and random excitation or pseudo random excitation is adopted.
3. The global recognition method of a linear variable parameter vibration system according to claim 1, wherein when constructing an overcomplete dictionary function library: dictionary functions in the overcomplete dictionary function library comprise constant terms, polynomial functions, trigonometric functions, exponential functions.
4. The global identification method of a linear variable parameter vibration system according to claim 1, wherein the sparse regression means: and adding sparse constraint to the coefficient vector corresponding to the overcomplete dictionary function library, and adding an L1 regularization term to the coefficient vector on the basis of an optimization target of the original regression problem when solving the coefficient vector.
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| CN102667755A (en) * | 2009-09-03 | 2012-09-12 | ***·E.·拉里莫尔 | Method and system for empirical modeling of time-varying, parameter-varying, and nonlinear systems via iterative linear subspace computation |
| CN108762284A (en) * | 2018-05-17 | 2018-11-06 | 北京航空航天大学 | A kind of spacecraft attitude tracking and controlling method and device based on LPV technologies |
| CN112100894A (en) * | 2020-10-14 | 2020-12-18 | 南京航空航天大学 | Flexible antenna structure impact load identification method and system |
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