CN113742936A - Complex manufacturing process modeling and predicting method based on functional state space model - Google Patents

Abstract

The invention discloses a complex manufacturing process modeling and predicting method based on a functional state space model, which comprises the following steps: expressing the observed quantity of the function time sequence as a basis function with an implicit variable by using basis function extension; establishing a state space model for the function time sequence; and estimating unknown parameters in the state space equation by using an EM algorithm and a Kalman filtering algorithm, establishing a complete function state space equation based on FSSM estimation, and calculating the advanced prediction error of the previous step. The method does not need to perform dimensionality reduction processing on the data, so that the loss of information is reduced; completely describing a system containing function data by using a past and present minimum information form; the parameters needing to be estimated are fewer, the analytic solution of the problem can be obtained, and the estimation accuracy is higher; the FSSM not only depicts the internal state of the system, but also can reveal the direct connection between the internal state of the system and the external output variables, which is reflected more comprehensively.

Description

Complex manufacturing process modeling and predicting method based on functional state space model

Technical Field

The invention belongs to the technical field of complex manufacturing process modeling and prediction, and relates to a complex manufacturing process modeling and prediction method based on a functional state space model.

Background

In recent years, multi-channel profile data has become increasingly popular and important in a variety of applications, particularly in complex manufacturing processes, where it is widely used to assess the temporal or spatial performance of a system. One of the challenges of multi-channel profile data is the need to consider the interrelationship between channels, especially for profile data streams in complex manufacturing processes. This correlation exists in more and more application scenarios, and is very different from typical data, and therefore cannot be handled well with traditional methods. More appropriate modeling of multi-channel contour data to extract valuable information has become an increasingly prominent problem in manufacturing. For this reason, it is crucial to develop a more efficient multi-channel contour modeling method.

Conventional modeling schemes are based on the basic assumption that process observations are statistically independent of each other. In real-world production, the multi-channel profiles of complex manufacturing processes do not always satisfy assumptions that are statistically independent of each other. Due to some uncontrollable or unknown factors, the observations of the previous epoch tend to have an effect on the observations of the subsequent epoch, which makes the observations appear to be interrelated. Such a process is called an autocorrelation process, or a sequential correlation process. Many scholars have studied multichannel autocorrelation data in the manufacturing process. Bhowmik (Bahrami, H., Niaki, S.T.A., & Khedmati, M. (2019). Monitoring multivariable profiles in multistage processes.communication in Statistics-Simulation and calculation, 1-29, translation Bhowmik et al, 2019. by using recursive singular spectroscopy to perform real-time unified single-channel and multi-channel structural damage detection.structural health Monitoring, 18(2) & 563-589. https:// doi.org/10.1177/1475921718760483) using recursive singular spectroscopy in combination with a time varying autoregressive model for continuous online damage detection of multi-degree-of-freedom vibrating structures, when applied to streaming data without any baseline data, facilitates online damage detection. Wang (Wang, T., Lu, G., & Yan, P. (2019); Multi-sensors based conditioning monitoring of robot mechanics: An advance of Multi-dimensional time-series analysis. measurement,134,326-33) proposes a novel Multi-sensor based modeling strategy, which can be used to detect changes in machine operating conditions during continuous operation. The method is based on automatic change detection, and is realized by combining multidimensional time series analysis (MultiDTSA) with an extended autoregressive integrated moving average (ARIMA) regression process. Wang (Wang, X., Zhu, Z., & Lu, G. (2020), Multiple regression analysis for change detection in Multiple-sensor monitoring data with application to induction motor speed condition monitoring, measurement Science and Technology,31(9) and 95103, translation X.Wang., 2020, application of Multiple regression analysis for Multiple sensory monitoring data change detection and induction motor speed condition monitoring, measurement Science and Technology,31(9) and 95103) propose a framework of an asynchronous motor on-line monitoring system based on Multiple Regression Analysis (MRA) that is capable of detecting speed changes in motor operation. An MRA-based predictive model is established using the collaborative information of the multi-sensor state data. The model defines the residual between the model output and the sensor observation as a dynamic stability indicator characterizing the motor operating state. These articles above all consider the autocorrelation of multi-channel data, but they are only for vectors or scalars, while the modeling of complex manufacturing process autoregressive multi-channel profiles is rarely discussed. Except for the correlation between each channel data. In addition, in these manufacturing systems, there are also an unequal number of hidden variables that affect the profile data, but are often difficult to find directly.

Disclosure of Invention

The technical problem to be solved by the invention is as follows: a complex manufacturing process modeling and prediction method based on a functional state space model is provided to solve the technical problems in the prior art.

The technical scheme adopted by the invention is as follows: the complex manufacturing process modeling and predicting method based on the functional state space model comprises the following steps:

(1) expressing the observed quantity of the function time sequence as a basis function with an implicit variable by using basis function extension;

(2) establishing a state space model for the function time sequence;

(3) estimating unknown parameters in the state space equation by using an EM algorithm and a Kalman filtering algorithm, and establishing a complete function state space equation;

(4) based on the estimation of the FSSM, a step (the step is that the process is iterative, the step is equivalent to a time section, and the prediction step is that the error of the actual data and the predicted data of the next time section is predicted) is calculated to lead the prediction error, so that the prediction of the autoregressive profile data of the complex manufacturing process is realized.

The method for setting the basis function with the hidden variable in the step (1) comprises the following steps:

assuming that N identical cycles or identical types of equipment are monitored within a pre-specified time frame, Y is set for each equipment_iThere are readings from P different or partially identical sensors at T time points; the reading of the sensors in the same time period forms profile data, and the observation times and the observation time are different due to different equipment or different periods, so the length of the profile can be different; the goal of the prediction problem is to learn the mathematical mapping from the observed variable Y to the implied variable X;

basis function expansion of multi-channel profile data: the main method for converting the original discrete data into the smooth function is to use the linear combination of the known basis functions to represent each function, and assume that N independent and identically distributed multi-channel contour samples are provided; each sample containing the P-channel profile is a square integrable random process, denoted as Y_i(t),{i＝1,2,…,N}

Definition of X ═ X₁,...,x_K]′∈R^K×T,{x_kK is the decomposition matrix (non observable state variables) for the ith sample, where x is 1_kIs a T-dimensional row vector;

then let there be a set of K basis functions C ═ C₁,c₂,...,c_K]1, …, K, where c is a P-dimensional column vector;

then the basis function for the ith sample is expanded as:

the matrix is expressed in the form of

The method for establishing the state space model in the step (2) comprises the following steps:

constructing a new Functional State Space Model (FSSM) based on the extension of the basis function obtained in the step (1):

wherein

C∈^P×K,X∈R^K×T,ε∈R^p×T,A∈R^K×K,ω∈R^K×T；C＝[c₁,c₂,...,c_K]K1, …, K is a combination of K basis functions, i 1,2, N is the sample number, j 1, P is the profile of the jth channel, T1, T represents the data at the time of the tth sample, e_i～N(0,R),R∈R^P×TIs the error matrix of the observed quantity, omega_i～N(0,Q),Q∈R^K×TIs the error matrix of the system;

y in the equation_i(t) is the profile data of the ith sample, X_i(t) is the hidden state matrix of the sample that is not observable, and the parameter set to be estimated is Θ ═ a, Q, R, epsilon_i,ω_i]。

The state space equation evaluation method in the step (3) comprises the following steps: for a function state space model, an EM algorithm and Kalman filtering are combined, two steps in an evaluation algorithm are respectively realized, and the method comprises an E step and an M step, wherein the E step is to realize the estimation of the expectation of a state X by utilizing the Kalman filtering and the Kalman smoothing algorithm, and the M step is to realize the estimation of parameters and noise covariance by maximizing the expectation of a log-likelihood function;

a state space set is established for each single channel as follows:

wherein, y_i,j＝{y_i,1,y_i,2,…,y_i,PThe j channel profile data of the ith sample is obtained, and the total number of the j channel profile data is P;

x_i,j(t)＝{x₁,x₂,…,x_Nthe j contour of the ith sample is an observable state vector, and the total number of the observable state vectors is N;

p is an observed variable y_i,jK is the state variable x_i,jDimension (d);

q is the system noise omega_i,jIs the observation noise epsilon_i,jThe covariance matrix of (2).

The step E is as follows: assume that the initial state is θ₀＝{A₀,R,Q,x_0,j,V_0,jThe state space parameter to be estimated is Θ ═ a, Q, R, epsilon_i,ω_i]；

The assumed model is shown as formula (5), ε_i,jAnd ω_i,jAll obey a Gaussian distribution, thenThe following relationships exist:

y_i,jthe j-th channel profile data for the i-th sample has a length of N, i.e. { y }₁,y₂,…,y_N}；

x_i,j(t) is the non-observable state vector for the jth profile of the ith sample, which is also N in length, i.e. { x }₁,x₂,…,x_N}；

Q is system noise omega_i,jCovariance matrix, R is the observation noise ε_i,jA covariance matrix;

according to the characteristics of the Markov process, there is the following relationship:

setting the probability density function of the initial state as:

the corresponding log-likelihood function is then:

in step E, the expectation of the log-likelihood function needs to be calculated, namely:

Q＝E[lnP({x},{y})|{y}] (11)

this expectation is iteratively maximized as a function of θ until convergence;

this process can be decomposed into the calculation of E [ x ]_i,j|{y}],E[x_i,jx_i,j|{y}]E[x_i,jx_i-1,j|{y}]；

Is defined as follows:

P_i,j＝E[x_i,jx′_i,j|{y}]，P_(i,i-1),j＝E[x_i,jx′_i-1,j|{y}]；

the above expectation is calculated by the kalman filter method, resulting in:

wherein,

a is a state transition matrix, and C is an observation matrix;

as can be obtained by the above calculation,

and P_i,jI.e. by

P_i,jObtained by means of a kalman smoothing method:

substituting the 3 expected estimated values obtained by calculation, and maximizing a log-likelihood function to obtain parameter estimation and noise covariance estimation;

observation noise covariance matrix R:

to obtain

For state transition matrix A:

to obtain

For the process noise covariance matrix Q:

to obtain

Initial state estimate pi₁：

Then the

Initial state covariance V_1,j

Namely:

the M steps are as follows: the steps for obtaining the maximum likelihood estimate using the EM algorithm are as follows:

first, setting an initial value θ₀＝{A₀,R,Q,x_0,j,V_0,jAnd x is obtained using formulae (20) to (24)_k,jAnd V_k,jAn estimated value of (d); then three conditional expectations were obtained

P_i,j＝E[x_i,jx′_i,j|{y}]And P_(i,i-1),j＝E[x_i,jx′_i-1,j|{y}]；

Secondly, calculating a condition expected value shown in a formula;

thirdly, maximizing the conditional expectation value of the log-likelihood function and obtaining A, R, Q and x by using formulas (20) to (24)_0.j,V_0,jAn estimated value of (d);

and step four, substituting the parameters obtained in the step three into the step one, and repeating the process until the log-likelihood function converges or reaches the set times.

The invention has the beneficial effects that: compared with the prior art, the invention has the following effects:

(1) compared with FPCA, the invention does not need to carry out dimensionality reduction processing on the data, thus reducing the loss of information;

(2) the method of B-spline smoothing and basis function expansion is combined with a state space model based on contour data, and a system containing function data can be completely described by using the past and present minimum information forms;

(3) compared with SSM, the method has the advantages that the number of parameters needing to be estimated is small, the analytic solution of the problem can be obtained, and the estimation accuracy is higher;

(4) the FSSM not only depicts the internal state of the system, but also can reveal the direct connection between the internal state of the system and the external output variables, which is reflected more comprehensively.

Drawings

FIG. 1 is a flow chart of a complex manufacturing process modeling and prediction based on a functional state space model;

FIG. 2 is a graph of the look-ahead one-step prediction results for FSSM and SSM of three models;

FIG. 3 is a MSE plot of FSSM and SSM for three models;

FIG. 4 is a graph of observed variations of milling machine data;

FIG. 5 is a diagram of a one step advance prediction of FSSM and SSM for milling machine data;

FIG. 6 is a diagram of predictions 1-100 for FSSM and SSM of milling machine data one step ahead.

FIG. 7 is a diagram of a one-step-ahead prediction 4401-4500 of FSSM and SSM of milling machine data;

FIG. 8 is a diagram of a one step-ahead prediction 8901-9000 of FSSM and SSM of milling machine data.

Detailed Description

The invention is further described with reference to the accompanying drawings and specific embodiments.

Example 1: 1-8, a method for modeling and predicting a complex manufacturing process based on a functional state space model, the method comprising the steps of:

(1) expressing the observed quantity of the function time sequence as a basis function with an implicit variable by using basis function extension;

(2) establishing a state space model for the function time sequence;

(3) estimating unknown parameters in the state space equation by using an EM algorithm and a Kalman filtering algorithm, and establishing a complete function state space equation;

(4) based on the estimation of FSSM, the advanced prediction error of one step (the error between the actual data and the predicted data in the next period) is calculated, and the prediction of the autoregressive profile data in the complex manufacturing process is realized.

The method for setting the basis function with the hidden variable in the step (1) comprises the following steps:

assuming that N identical cycles or identical types of equipment are monitored within a pre-specified time frame, Y is set for each equipment_iThere are readings from P different or partially identical sensors at T time points; the reading of the sensors in the same time period forms profile data, and the observation times and the observation time are different due to different equipment or different periods, so the length of the profile can be different; the goal of the prediction problem is to learn the mathematical mapping from the observed variable Y to the implied variable X;

basis function expansion of multi-channel profile data: the main method for converting the original discrete data into the smooth function is to use the linear combination of the known basis functions to represent each function, and assume that N independent and identically distributed multi-channel contour samples are provided; each sample containing the P-channel profile is a square integrable random process, denoted as Y_i(t),{i＝1,2,…,N}

Definition of X ═ X₁,...,x_K]′∈R^K×T,{x_kK is the ith oneDecomposition matrix (non observable state variable) of the sample, where x_kIs a T-dimensional row vector;

then let there be a set of K basis functions C ═ C₁,c₂,...,c_K]1, …, K, where c is a P-dimensional column vector;

then the basis function for the ith sample is expanded as:

the matrix is expressed in the form of

The method for establishing the state space model in the step (2) comprises the following steps:

constructing a new Functional State Space Model (FSSM) based on the extension of the basis function obtained in the step (1):

wherein

C∈^P×K,X∈R^K×T,ε∈R^p×T,A∈R^K×K,ω∈R^K×T；C＝[c₁,c₂,...,c_K]K1, …, K is a combination of K basis functions, i 1,2, N is the sample number, j 1, P is the profile of the jth channel, T1, T represents the data at the time of the tth sample, e_i～N(0,R),R∈R^P×TIs the error matrix of the observed quantity, omega_i～N(0,Q),Q∈R^K×TIs the error matrix of the system;

y in the equation_i(t) is the profile data of the ith sample, X_i(t) is the non-observable hidden state matrix of the samples, and the parameter set to be estimated is Θ＝[A,Q,R,ε_i,ω_i]。

The state space equation evaluation method in the step (3) comprises the following steps: for a function state space model, an EM algorithm and Kalman filtering are combined, two steps in an evaluation algorithm are respectively realized, and the method comprises an E step and an M step, wherein the E step is to realize the estimation of the expectation of a state X by utilizing the Kalman filtering and the Kalman smoothing algorithm, and the M step is to realize the estimation of parameters and noise covariance by maximizing the expectation of a log-likelihood function;

a state space set is established for each single channel as follows:

wherein, y_i,j＝{y_i,1,y_i,2,…,y_i,PThe j channel profile data of the ith sample is obtained, and the total number of the j channel profile data is P;

x_i,j(t)＝{x₁,x₂,…,x_Nthe j contour of the ith sample is an observable state vector, and the total number of the observable state vectors is N;

p is an observed variable y_i,jK is the state variable x_i,jDimension (d);

q is the system noise omega_i,jIs the observation noise epsilon_i,jThe covariance matrix of (2).

The step E is as follows: assume that the initial state is θ₀＝{A₀,R,Q,x_0,j,V_0,jThe state space parameter to be estimated is Θ ═ a, Q, R, epsilon_i,ω_i]；

The assumed model is shown as formula (5), ε_i,jAnd ω_i,jObey the gaussian distribution, the following relationships exist:

y_i,jthe j-th channel profile data for the i-th sample has a length of N, i.e. { y }₁,y₂,…,y_N}；

x_i,j(t) is the non-observable state vector for the jth profile of the ith sample, which is also N in length, i.e. { x }₁,x₂,…,x_N}；

Q is system noise omega_i,jCovariance matrix, R is the observation noise ε_i,jA covariance matrix;

according to the characteristics of the Markov process, there is the following relationship:

setting the probability density function of the initial state as:

the corresponding log-likelihood function is then:

in step E, the expectation of the log-likelihood function needs to be calculated, namely:

Q＝E[lnP({x},{y})|{y}] (11)

this expectation is iteratively maximized as a function of θ until convergence;

this process can be decomposed into the calculation of E [ x ]_i,j|{y}],E[x_i,jx_i,j|{y}]E[x_i,jx_i-1,j|{y}]；

Is defined as follows:

P_i,j＝E[x_i,jx′_i,j|{y}]，P_(i,i-1),j＝E[x_i,jx′_i-1,j|{y}]；

the above expectation is calculated by the kalman filter method, resulting in:

wherein,

a is a state transition matrix, and C is an observation matrix;

as can be obtained by the above calculation,

and P_i,jI.e. by

P_i,jObtained by means of a kalman smoothing method:

substituting the 3 expected estimated values obtained by calculation, and maximizing a log-likelihood function to obtain parameter estimation and noise covariance estimation;

observation noise covariance matrix R:

to obtain

For state transition matrix A:

to obtain

For the process noise covariance matrix Q:

to obtain

Initial state estimate pi₁：

Then the

Initial state covariance V_1,j

Namely:

the M steps are as follows: the steps for obtaining the maximum likelihood estimate using the EM algorithm are as follows:

first, setting an initial value θ₀＝{A₀,R,Q,x_0,j,V_0,jAnd x is obtained using formulae (20) to (24)_k,jAnd V_k,jAn estimated value of (d); then three conditional expectations were obtained

P_i,j＝E[x_i,jx′_i,j|{y}]And P_(i,i-1),j＝E[x_i,jx′_i-1,j|{y}]；

Secondly, calculating a condition expected value shown in a formula;

thirdly, maximizing the conditional expectation value of the log-likelihood function and obtaining A, R, Q and x by using formulas (20) to (24)_0.j,V_0,jAn estimated value of (d);

and step four, substituting the parameters obtained in the step three into the step one, and repeating the process until the log-likelihood function converges or reaches the set times.

The invention adopts a function state space model of an autoregressive curve, which can not only flexibly describe the cross correlation and autocorrelation structure of function data through the combination of basis functions, but also simplify the calculation of parameter estimation, and make the modeling easier.

To illustrate the effects of the present invention, the following simulation studies were conducted: the performance of FSSM methods was thoroughly evaluated by several numerical studies and compared with some of the most advanced methods. In particular, the following three different models are considered.

i) Model (I): exist of

Satisfy the requirement of

Wherein v is_k(k 1.., 8) are 1, 4, 7, 10, 13, 16, 19, and 22B-spline basis functions of order 3 at [0,1 ]]With a grid of 10 equidistant nodes. t is t_l(l ═ 1.. times, n) (where n ═ 10) is at the same position as the junction, ξ_ik(k 1.. 8) is set as a sparse vector whose ith component is represented by ξ_kl＝β_klL(β_klI > 1.5), where L (·) is an index function that equals L when its condition is true, and 0 otherwise. Here beta_k＝[β_k1,...,β_k50]Follow a 50-dimensional multivariate normal distribution with mean b _k0, covariance matrix of B_k. Setting (B)_k)_lh＝(0.5)^l-h，l,h＝1,...,50，

For noise, each component follows a mean of 0 and a variance of σ²0.04, wherein mu0 is [1,0.75,0.5, -1,1, -0.5, -0.75,1]'；P0＝eye(M)*1-4；Q＝eye(M)*1-4；R＝eye(N)*1-5。

II) model (II): exist of

Satisfy the requirement of

Wherein v is_k(k ═ 1.., 8.) is the first 8 Vaidyanathan wavelet bases at [0,1 ·,1 @]Grid with 16 equidistant nodes in it, ξ_ik(k 1.. 8) is a sparse vector whose ith component is represented by ξ_kl＝β_klL(|β_klI > 1.5) is generated. Here beta_k＝[β_k1,...,β_k60]Follow a 60-dimensional multivariate normal distribution, mean b _k0 and covariance matrix B_k. Setting (B)_k)_lh＝(0.5)^l-h，l,h＝1,...,120，

Is generated in the same manner as model (I).

III) model (III): exist of

Satisfy the requirement of

Wherein v is_k(k 1.., 6) is a B-spline basis function of 1 st, 4 th, 7 th, 10 th, 13 th, 16 th order with a grid of [0,1 th]6 equidistant nodes. t is t_l(l ═ 1.., n) (where n ═ 6) is at the same position as the junction.

Wherein v is_k(k 7.., 12) is the first 6 non-constant fourier bases, e.g., ν_k＝cos(kt+kπ)，ν_kCos (kt + k pi), at [0,2 pi]Grid with 50 equidistant nodes in it, ξ_ik(k ═ 1.., 6) and e_iGenerated in the same manner as model (I).

In the present invention, to better evaluate the performance of the FSSM method, the maximum likelihood estimates Q and R from the models (20) and (22) are used for Y_iGiven Y_i-1Is a reaction of Y_iNormalized by the one-step prediction error (OSFEs) of (1), expressed as e_i～N(0,S_j) The definition is as follows.

S_i＝var(e_i)＝CV_i ^i-1C′+R (26)

Table I MSE for different methods of the three models

Fig. 3 shows modeling predictions for these three models by FSSM. It can be seen that in general the FSSM predicted contours are very close (almost overlapping) to the real data, demonstrating the accuracy of the proposed estimation procedure. FSSM has relatively accurate prediction results regardless of whether the model consists of B-spline or wavelet basis alone, or mixed basis functions. Meanwhile, as can be seen from the comparison between MES of the two methods based on the state space model in fig. 5, the prediction effect of FSSM is significantly better than that of the conventional SSM method.

Table 1 shows the MSE of the different methods for the smoothed autoregressive curve. The prediction gap for the S1 channel was slightly larger relative to the other channels due to the larger variation between each sample of the raw S1 data. Taken together, the MSE for FSSM and SSM methods is significantly smaller than other methods. This shows that the SSM-based prediction method shows certain advantages compared to the PCA-based method.

Actual case study: in many real-world manufacturing systems, various sensors are installed in equipment to monitor key process variables in the production process, such as motion, vibration, pressure, temperature, voltage, etc. For each sensor, a large amount of high dimensional data is collected, which can be viewed as function data for each manufacturing cycle. Based on historical function data, changes of the process variables are effectively predicted, so that a more accurate reference basis can be provided for state monitoring, and early warning detection can be performed on abnormal conditions in a more timely manner.

In this example, the FSSM method was applied to a real case of a manufacturing system for milling machine process state monitoring to demonstrate the effectiveness of FSSM based autoregressive models. The performance of FSSM was compared to various state-of-the-art methods of function analysis, including SSM, SMFPCA, mfcca, VPCA, and others. Experimental results show that the proposed FSSM method is clearly superior to all these alternative methods.

This set of data represents experiments run on a milling machine under different operating conditions. In particular, tool wear was investigated in conventional cutting as well as in-cut and out-cut. Data are collected at multiple locations by three different types of sensors (acoustic emission sensors, vibration sensors, current sensors).

In this study, there were six sets of data from different sensors representing the six process variables monitored during the manufacturing process for each product sample. These sensor variables are referred to as S1 through S6. Fig. 4 shows an overview of these six sensor variables. It should be noted that under different operating conditions, the profile variation for each process variable is different due to the complex relationships of the variables within the manufacturing system. In the present invention, only profile variations under a single working condition are considered for the study. Figures 5-8 show the predicted results of FSSM compared to the actual profile.

The overall data is first estimated and predicted. Then, in order to fully explain the prediction effect of the method on different stages of the production process, three parts of data are respectively extracted for prediction, and the data come from the early stage, the middle stage and the later stage of the monitoring process. The results are shown in FIG. 6.

Table II illustrates the predicted results of the FSSM model at different stages of the milling process. It can be seen that the predicted MSE at the later stage is quite small, since the state values at the later stage generally tend to decrease, but the periodic fluctuations are still large, and FSSM is sensitive in capturing peak-to-valley variations. The prediction results of 6 different channels are comprehensively analyzed, and the fact that the predicted MSE of S1 is larger in any period is found, because the operation mode of S1 is obviously different from that of other channels, the upper and lower changes are larger, and accurate prediction is more difficult. However, combining several methods, FSSM remains the best method to predict the operating state of S1. Fig. 6-8 show the fit of the two best performing methods FSSM and SSM to the raw data. It can be seen that FSSM is, in general, more accurate in predicting the trend changes of the operating state.

Table II RMSE of different methods in different datasets

In connection with this real example it can be seen that the predictive power of other methods is in most cases not comparable to FSSM. Notably, the overall performance of the SSM method is the second best. Since its model structure is the same as FSSM. Despite the superiority and flexibility of the non-basis function extension of SSM, the state space model still captures the characteristics of some sensors to some extent. Therefore, the predictive ability of this method is not poor. As for the other four PCA methods, they are poorly predictive and unstable because their models cannot describe the variable changes implied by the data. For channels with little fluctuation in state values, the performance of FSSM is similar to the SSM method. This is because for channels whose operating conditions are relatively stable, the features between them are very similar, so the implicit variables learned by SSM separately for each channel are very close to the implicit variables learned by FSSM jointly for all three channels. In other words, the channels may be collectively regarded as one channel. In this case, the predictive power of the base spread monitoring will be similar to the detection power of the monitoring alone. Furthermore, it should be noted that UMPCA performs the worst, as it has the inherent disadvantage of being able to extract only a limited number of features. In summary, these baseline methods are somewhat unable to extract different features from different contours, and therefore are unable to detect changes in features.

And (4) simulation conclusion: functional Data Analysis (FDA) has originated in the field of statistics and is under development. More and more manufacturing systems and industrial processes are generating this type of structured data. Functional regression, consisting of regression methods involving functional responses or predictions, is one of the most interesting areas in functional data analysis in industrial applications and process development. The key to FDA is the need to combine information across functions and internally, and is also the key to the functional regression method. Despite extensive research in the literature on prediction and regression of functional data, the challenge of how to design a reasonable prediction method to estimate a multi-channel autoregressive curve remains to be solved. In the invention, a monitoring scheme based on FSSM is provided to fill the research gap. In particular, each contour is treated as a state and the FSSM is proposed by adding different basis function extensions to the state space equation. Basis functions are key components of the method and determine how the model draws intensity from the observation curves when performing regression. In this way, FSSM allows for the combination of basis functions for each hidden variable, thereby enabling the modeling of multi-channel autoregressive curves with different characteristics. Numerical studies and case studies demonstrate the effectiveness and applicability of the proposed method. By researching cases of different manufacturing systems and selecting and establishing different basis functions according to different conditions, a suitable FSSM is constructed, the purposes of prediction and monitoring are achieved, a reliable basis is provided for decision making of the manufacturing industry, and superiority of a function data analysis method in the research of the manufacturing industry is truly embodied.

There are several benefits to using the complex manufacturing process modeling and prediction method of the present invention. Firstly, compared with FPCA, the method has the advantages that the data does not need to be subjected to dimensionality reduction, and the loss of information is reduced; second, combining spline smoothing and radix expansion methods with a state space model based on contour data, a complete description of the system containing functional data can be made using past and present forms of minimal information. Secondly, compared with SSM, the parameters needing to be estimated are fewer, the analytic solution of the problem can be obtained, and the estimation accuracy is higher; finally, the FSSM not only depicts the internal state of the system, but also can reveal the direct connection between the internal state of the system and the external output variables, which is reflected more comprehensively. However, a limitation of this approach is that SSM is generally based on markov properties, i.e. given the current state of the system, the future of the system is independent of its past. If the system does not satisfy the Markov property, it is not appropriate to use the state space model.

With this research approach, there are several potentially valuable extensions. First, it is contemplated that the model of the FSSM can be included in the monitoring to make it more accurate in predicting abnormal situations. Second, a more efficient optimization algorithm can be explored, taking into account the selection and setting of basis functions in the joint estimation FSSM. Finally, the proposed modeling technique can also be combined with neural networks for exploring the impact of different characteristics on the desired index. In summary, there are still many areas to be investigated for the application of FSSM methods in the field of manufacturing.

The above description is only an embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can easily conceive of changes or substitutions within the technical scope of the present invention, and therefore, the scope of the present invention should be determined by the scope of the claims.

Claims (6)

1. The complex manufacturing process modeling and predicting method based on the functional state space model is characterized in that: the method comprises the following steps:

(1) expressing the observed quantity of the function time sequence as a basis function with an implicit variable by using basis function extension;

(2) establishing a state space model for the function time sequence;

(3) estimating unknown parameters in the state space equation by using an EM algorithm and a Kalman filtering algorithm, and establishing a complete function state space equation;

(4) and calculating a one-step advanced prediction error based on the estimation of the FSSM to realize the prediction of the autoregressive profile data in the complex manufacturing process.

2. The method of claim 1, wherein the method comprises: the method for setting the basis function with the hidden variable in the step (1) comprises the following steps:

assuming that N identical cycles or identical types of equipment are monitored within a pre-specified time frame, Y is set for each equipment_iThere are readings from P different or partially identical sensors at T time points; the sensor readings in the same time period form a profile data; the goal of the prediction problem is to learn the mathematical mapping from the observed variable Y to the implied variable X;

basis function expansion of multi-channel profile data: the main method for converting the original discrete data into the smooth function is to use the linear combination of the known basis functions to represent each function, and assume that N independent and identically distributed multi-channel contour samples are provided; each sample containing the P-channel profile is a square integrable random process, denoted as Y_i(t),{i＝1,2,…,N}

Definition of X ═ X₁,...,x_K]′∈R^K×T,{x_kK is the decomposition matrix (non observable state variables) for the ith sample, where x is 1_kIs a T-dimensional row vector;

then let there be a set of K basis functions C ═ C₁,c₂,...,c_K]1, …, K, where c is a P-dimensional column vector;

then the basis function for the ith sample is expanded as:

the matrix is expressed in the form of

3. The method of claim 2, wherein the method comprises: the method for establishing the state space model in the step (2) comprises the following steps:

constructing a new Functional State Space Model (FSSM) based on the extension of the basis function obtained in the step (1):

wherein

ω∈R^K×T；C＝[c₁,c₂,...,c_K]K1, …, K is a combination of K basis functions, i 1,2, N is the sample number, j 1, P is the profile of the jth channel, T1, T represents the data at the time of the tth sample, e_i～N(0,R),R∈R^P×TIs the error matrix of the observed quantity, omega_i～N(0,Q),Q∈R^K×TIs the error matrix of the system;

y in the equation_i(t) is the profile data of the ith sample, X_i(t) is the hidden state matrix of the sample that is not observable, and the parameter set to be estimated is Θ ═ a, Q, R, epsilon_i,ω_i]。

4. The method of claim 3, wherein the method comprises: the state space equation evaluation method in the step (3) comprises the following steps: for a function state space model, an EM algorithm and Kalman filtering are combined, two steps in an evaluation algorithm are respectively realized, and the method comprises an E step and an M step, wherein the E step is to realize the estimation of the expectation of a state X by utilizing the Kalman filtering and the Kalman smoothing algorithm, and the M step is to realize the estimation of parameters and noise covariance by maximizing the expectation of a log-likelihood function;

a state space set is established for each single channel as follows:

wherein, y_i,j＝{y_i,1,y_i,2,…,y_i,PThe j channel profile data of the ith sample is obtained, and the total number of the j channel profile data is P;

x_i,j(t)＝{x₁,x₂,…,x_Nthe j contour of the ith sample is an observable state vector, and the total number of the observable state vectors is N;

p is an observed variable y_i,jK is the state variable x_i,jDimension (d);

q is the system noise omega_i,jIs the observation noise epsilon_i,jThe covariance matrix of (2).

5. The method of claim 4, wherein the method comprises: the step E is as follows: assume that the initial state is θ₀＝{A₀,R,Q,x_0,j,V_0,jThe state space parameter to be estimated is Θ ═ a, Q, R, epsilon_i,ω_i]；

The assumed model is shown as formula (5), ε_i,jAnd ω_i,jObey the gaussian distribution, the following relationships exist:

y_i,jthe j-th channel profile data for the i-th sample has a length of N, i.e. { y }₁,y₂,…,y_N}；

x_i,j(t) is the non-observable state vector for the jth profile of the ith sample, which is also N in length, i.e. { x }₁,x₂,…,x_N}；

Q is system noise omega_i,jCovariance matrix, R is the observation noise ε_i,jA covariance matrix;

according to the characteristics of the Markov process, there is the following relationship:

setting the probability density function of the initial state as:

the corresponding log-likelihood function is then:

in step E, the expectation of the log-likelihood function needs to be calculated, namely:

Q＝E[lnP({x},{y})|{y}] (11)

continuously iterating and maximizing the expectation of the log-likelihood function as a function of theta until convergence;

this process decomposes into computing E [ x ]_i,j|{y}],E[x_i,jx_i,j|{y}]E[x_i,jx_i-1,j|{y}]；

Is defined as follows:

P_i,j＝E[x_i,jx′_i,j|{y}]，P_(i,i-1),j＝E[x_i,jx′_i-1,j|{y}]；

calculating the above expectation by Kalman Filter method

P_i，j＝E[x_i，jx′_i，j|{y}]，P_{(i，i-1)，j}＝E[x_i，jx′_i-1，j|{y}]Obtaining:

wherein,

a is a state transition matrix, and C is an observation matrix;

as can be obtained by the above calculation,

and P_i,jI.e. by

P_i,jObtained by means of a kalman smoothing method:

substituting the 3 expected estimated values obtained by calculation, and maximizing a log-likelihood function to obtain parameter estimation and noise covariance estimation;

observation noise covariance matrix R:

to obtain

For state transition matrix A:

to obtain

For the process noise covariance matrix Q:

to obtain

Initial state estimate pi₁：

Then the

Initial state covariance V_1,j

Namely:

6. the method of claim 6, wherein the method comprises: the M steps are as follows: the steps for obtaining the maximum likelihood estimate using the EM algorithm are as follows:

first, setting an initial value θ₀＝{A₀,R,Q,x_0,j,V_0,jAnd x is obtained using formulae (20) to (24)_k,jAnd V_k,jAn estimated value of (d); then three conditional expectations were obtained

P_i,j＝E[x_i,jx′_i,j|{y}]And P_(i,i-1),j＝E[x_{i, j}x′_i-1,j|{y}]；

Secondly, calculating a condition expected value shown in a formula;

thirdly, maximizing the conditional expectation value of the log-likelihood function and obtaining A, R, Q and x by using formulas (20) to (24)_0.j,V_0,jAn estimated value of (d);

and step four, substituting the parameters obtained in the step three into the step one, and repeating the process until the log-likelihood function converges or reaches the set times.

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