# Regularized Least Squares Recursive Algorithm with Forgetting Factor for Identifying Parameters in the Grinding Process

Authors: Yang Yu; Rui Deng; Gang Yu; Yu Wang; Guodong Yang; DaYong Zhao
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper investigates a parameter identification problem in the grinding process. Due to the data saturated phenomenon and the ill-posed of parameter identification inverse problem, this paper presents a regularized least squares recursive algorithm with a forgetting factor (RLSRAFF), the basic idea of which is to combine the forgetting factor with regularization parameters. Moreover, based on RLSRAFF, this paper verifies the recursive calculation of criterion function, analyzes the effect of calculation error from the gain matrix and proves the convergence of the proposed algorithm. Finally, effectiveness of RLSRAFF is verified by simulation experiments and grinding data. Compared with other algorithms, RLSRAFF can give a more convergence rate to the real data and reduce the error from the true value.


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## Body

## 1. Introduction

Parameters identification is one of the most important areas in system modeling and signal processing [1], and the related identification methods are attracted by many scholars. As one of most important parameters identification methods, the least square method (LSM) has been applied in various fields. In [2], to identify a ship’s linear sway-yaw manoeuvring coefficients and drag-area parameters in current and wind, Wayne and Gash developed a simple least squares technique. In [3], the loss in localization accuracy induced by time difference of arrival noises and velocity errors is reduced by the constrained total least squares method. In [4], to deal with the problem of overenhancement results, an image enhancement scheme based on weighted least squares are proposed. In [5], due to the complex biochemical characteristics of the wastewater treatment process, an adaptive dynamic nonlinear partial least squares model is proposed to improve the prediction performance and stability of effluent quality indexes. In [6], Ramezani applied the collocated discrete least squares meshless method to improve the node moving technique. In order to enhance the performance of LSM, some improved algorithms are developed. Reference [7] presents an iteratively reweighted LSM to improve the antioutlier performance of the least squares support vector machine. Reference [8] combines the partial least square with the attention mechanism in a neural network named the attention-PLS. Zhang, et al. [9] proposed the Lagrange energy-least squares similitude method to deduce output scaling laws. Furthermore, some scholars study the recursive least squares (RLS) method with the forgetting factor. In [10], Paleologu et al. pointed out that the performance of the recursive least squares algorithm is governed by the forgetting factor and proposed a variable forgetting factor RLS (VFF-RLS) algorithm for system identification. Reference [11] based on the framework of recursive least squares-temporal difference proposed a new reinforcement learning method by using the forgetting factor. Sun et al. [12] presented an adaptive forgetting factor RLS method for online identification of the second order resistor-capacitance equivalent circuit model parameters. Meanwhile, the regularization method is used to improve the LSM. Wang et al. [13] introduced the least squares regularization method to solve the ill-posed problem of the multiplicative error model. Zhou et al. [14] investigated the method of the anti-ill-conditioned population-weighted median based on the least squares regularization method. Jin et al. [15] used the kurtosis regularization algorithm for circuit joint optimization in neural network training to increase the information entropy of neural network weight data. Bai et al. [16] proposed a generic model for least squares nonnegative matrix factorizations with Tikhonov regularization. However, in the process of parameter identification, the data saturated phenomenon and the ill-posed problem often occur simultaneously. Therefore, this paper presents a regularized least squares recursive algorithm with forgetting factor (RLSRAFF).At the same time, grinding process [17–19], as one of the most important procedures in the mineral processing, extracts the valuable minerals from the discernible gangue after physical grinding and classification. The role of this process can dissociate different useful minerals from each other and avail to the subsequent sorting process. The grinding and classification process has complex characteristics, such as large inertia, time-varying parameters, and nonlinearity. Recently, to realize the automatic production, some control technology strategies [20–24] for the grinding process have been attracted by majority of scientific researchers. As the core of the control system, the accuracy of the mathematical model of the grinding process plays a vital role. Reference [25] identified the parameters of the prediction model of steel ball wear law in the grinding process, [26] used a nonlinear parameter identification method based on the improved differential evolution algorithm to solve the parameter of the nonlinear model in the bauxite grinding classification process. Furthermore, Chen [27] investigated the least square recursive algorithm with forgetting factor to systematically identify the unknown parameters of the grinding process model.Due to characteristics of complexity and nonlinearity, parameter identification in the mathematical model of the grinding process can lead to the ill-posed. The above literature does not consider the effectiveness of ill-posed on the identification results. Regularization [28] is proved to be an efficient approach for the inverse problem. Therefore, this paper develops RLSRAFF for identifying the parameters in the mathematical model of the grinding process. The main contributions can be summed as follows:(i)
The ill-posed problem is considered for identifying the parameters in the mathematical model of the grinding process.(ii)
RLSRAFF, which combines the forgetting factor with the regularization parameter, is presented.(iii)
The recursive calculation of criterion function, the effect of calculation error from the gain matrix, and the convergence of the proposed algorithm are analyzed.The rest of the paper is organized as follows. In Section2, the grinding process is described and then the parameter identification model is introduced. In Section 3, RLSRAFF is proposed and the calculation error of the gain matrix and convergence of the algorithm are proved. In Section 4, two experiments are carried out to verify the proposed method. In Section 5, the conclusion is given.


## 2. Grinding Process and Parameter Identification Model

### 2.1. Grinding Process

The grinding process (see Figure1) can be briefly described as follows. The fresh ores are firstly sent into the ball mill by the conveyor belt and then crushed ceaselessly by the steel balls to produce the pulp within a certain concentration. Meanwhile, a certain amount of water is added into the mill to adjust the concentration of ore pulp within limits. After grinding, the mixed ore pulp is continuously discharged from mill into the spiral classifier for classification, and a certain amount of water is added into the spiral classifier to adjust the concentration. The substandard ore pulp returns to the first stage of ball mill for regrinding, and the standard ore, extracting from the classifier, enters into the pump sump for the next process [27].Figure 1
The grinding process of the primary ball mill.


### 2.2. Mathematical Model of the Grinding Process Parameter Identification

Mathematical model of the grinding process mainly includes the ball mill model and spiral classifier model. The ball mill model [29] has been detailed in a variety of studies. It describes the relationship between the ore feeding quantity, the water feeding quantity, the return sand quantity, and the mill pulp concentration. The spiral classifier model gives the relationship between the classifier overflow concentration, the pulp fineness, the ore discharge quantity, and the return sand quantity.However, these models only give internal mechanism of each part in the grinding process, and it has not reflected the multi-variable coupling and time delay between the primary ball mill and the classifier.Thus, the mathematical model [27] between the ball mill and the classifier can be described as follows:(1)CcQD=β1α1s+1e−τ1sβ2α2s+1e−τ2sβ3α3s+1α5s+1e−τ3sβ4α4s+1e−τ4sU3U1,where Cc is the actual value of the classifier overflow concentration, (%); QD is the actual value of the return sand quantity,m3/h; U1 is the actual value of the ore feeding quantity, t/h; U3 is the actual value of the classifier adding water quantity, m3/h.By analyzing the model (1), we know that theU1 and U3 are the input variables, and Cc and QD are the output variables. The model identifies between the ball mill and the classifier is described as follows:(2)CcQD=G11′SG12′SG21′SG22′SU3U1,where G11′S,G12′S,G21′S,G22′S are consisted by the inertia link and the delay link, and G11′S=β1/α1S+1e−τ1s,G12′S=β2/α2S+1e−τ2s,G21′S=β3/α3S+1α5S+1e−τ3s,G22′S=β4/α4S+1α4S+1e−τ4s. Different identification methods are used to identify the parameters in these two links, respectively. The inertia link is identified by the least square algorithm, and the delay link is identified by the cross-correlation function algorithm [27]. The major study in this paper focuses on identification of the parameters for the inertia link a1,b1,c1,d1;a2,a3,b2,b3,c2,d2 to deal with the data saturated phenomenon and the ill-posed problem. For the inertia link, we have the following equations:(3)Cc=G11SU3+G12SU1=β1α1S+1U3+β2α2S+1U1,(4)QD=G21SU3+G22SU1=β3α3S+1α5S+1U3+β4α4S+1U1,where G11S=β1/α1S+1,G12S=β2/α2S+1,G21S=β3/α3S+1α5S+1,G22S=β4/α4S+1. According to equations (3) and (4), we have(5)Ccz=b1z−11+a1z−1U3z+c1z−11+d1z−1U1z,(6)QDz=b2z−1+b3z−11+a2z−1+a3z−1U3z+c2z−11+d2z−1U1z,Thus, the input and output observation data are used to estimate the unknown parametersa1,b1,c1,d1;a2,a3,b2,b3,c2,d2 in equations (5) and (6).Remark 1.
For the delay link, cross-correlation function [27] can be used to identify the delay time τ1,τ2,τ3,τ4, because the main work of this paper is to identify the unknown parameters, the details of identifying the delay time are not given.


## 2.1. Grinding Process

The grinding process (see Figure1) can be briefly described as follows. The fresh ores are firstly sent into the ball mill by the conveyor belt and then crushed ceaselessly by the steel balls to produce the pulp within a certain concentration. Meanwhile, a certain amount of water is added into the mill to adjust the concentration of ore pulp within limits. After grinding, the mixed ore pulp is continuously discharged from mill into the spiral classifier for classification, and a certain amount of water is added into the spiral classifier to adjust the concentration. The substandard ore pulp returns to the first stage of ball mill for regrinding, and the standard ore, extracting from the classifier, enters into the pump sump for the next process [27].Figure 1
The grinding process of the primary ball mill.


## 2.2. Mathematical Model of the Grinding Process Parameter Identification

Mathematical model of the grinding process mainly includes the ball mill model and spiral classifier model. The ball mill model [29] has been detailed in a variety of studies. It describes the relationship between the ore feeding quantity, the water feeding quantity, the return sand quantity, and the mill pulp concentration. The spiral classifier model gives the relationship between the classifier overflow concentration, the pulp fineness, the ore discharge quantity, and the return sand quantity.However, these models only give internal mechanism of each part in the grinding process, and it has not reflected the multi-variable coupling and time delay between the primary ball mill and the classifier.Thus, the mathematical model [27] between the ball mill and the classifier can be described as follows:(1)CcQD=β1α1s+1e−τ1sβ2α2s+1e−τ2sβ3α3s+1α5s+1e−τ3sβ4α4s+1e−τ4sU3U1,where Cc is the actual value of the classifier overflow concentration, (%); QD is the actual value of the return sand quantity,m3/h; U1 is the actual value of the ore feeding quantity, t/h; U3 is the actual value of the classifier adding water quantity, m3/h.By analyzing the model (1), we know that theU1 and U3 are the input variables, and Cc and QD are the output variables. The model identifies between the ball mill and the classifier is described as follows:(2)CcQD=G11′SG12′SG21′SG22′SU3U1,where G11′S,G12′S,G21′S,G22′S are consisted by the inertia link and the delay link, and G11′S=β1/α1S+1e−τ1s,G12′S=β2/α2S+1e−τ2s,G21′S=β3/α3S+1α5S+1e−τ3s,G22′S=β4/α4S+1α4S+1e−τ4s. Different identification methods are used to identify the parameters in these two links, respectively. The inertia link is identified by the least square algorithm, and the delay link is identified by the cross-correlation function algorithm [27]. The major study in this paper focuses on identification of the parameters for the inertia link a1,b1,c1,d1;a2,a3,b2,b3,c2,d2 to deal with the data saturated phenomenon and the ill-posed problem. For the inertia link, we have the following equations:(3)Cc=G11SU3+G12SU1=β1α1S+1U3+β2α2S+1U1,(4)QD=G21SU3+G22SU1=β3α3S+1α5S+1U3+β4α4S+1U1,where G11S=β1/α1S+1,G12S=β2/α2S+1,G21S=β3/α3S+1α5S+1,G22S=β4/α4S+1. According to equations (3) and (4), we have(5)Ccz=b1z−11+a1z−1U3z+c1z−11+d1z−1U1z,(6)QDz=b2z−1+b3z−11+a2z−1+a3z−1U3z+c2z−11+d2z−1U1z,Thus, the input and output observation data are used to estimate the unknown parametersa1,b1,c1,d1;a2,a3,b2,b3,c2,d2 in equations (5) and (6).Remark 1.
For the delay link, cross-correlation function [27] can be used to identify the delay time τ1,τ2,τ3,τ4, because the main work of this paper is to identify the unknown parameters, the details of identifying the delay time are not given.


## 3. Regularized Least Squares Recursive Algorithm with Forgetting Factor and Property Analysis

### 3.1. Regularized Least Squares Recursive Algorithm with Forgetting Factor

In order to identify the parameters in model (5),U3z and U1z are defined as the input variable, Ccz is defined as the output variable. The difference equation is used to discrete the linear system as(7)2Cck+a1Cck−1+d1Cck−1=b1U3k−1+c1U1k−1+εk,where k=2,3,…,1+N, εkis uncorrelated random variable that subjects to N0,1 distribution, U3k−1 and U1k−1 are the actual input signal of recorded data, Cck is the actual output signal of recorded data, and Cck−1 is the function value of the output signal Cck at the previous sampling period. So we have the following equation from the following equation:(8)2Cck=−a1Cck−1−d1Cck−1+b1U3k−1+c1U1k−1+εk.It can be seen from (8) that the above parameters can be estimated by N sets of the observation equations:(9)Cc1+1=−0.5a1Cc1−0.5d1Cc1+0.5b1U31+0.5c1U11+0.5ε1+1Cc1+2=−0.5a1Cc2−0.5d1Cc2+0.5b1U32+0.5c1U12+0.5ε1+2⋮Cc1+N=−0.5a1CcN−0.5d1CcN+0.5b1U3N+0.5c1U1N+0.5ε1+N.Equation (9) is defined as an observation equation. We define CcN=Cc1+1,Cc1+2,…,Cc1+NT,θN=0.5a1,0.5d1,0.5b1,0.5c1T and εN=ε1+1,ε1+2,…,ε1+NT. Equation (9) can be written into the following form:(10)CCN=ΦNθN+εN,We define(11)ΦN=ϕT1,ϕT2,…,ϕTNT,(12)ϕTi=−Cci,−Cci,U3i,U1i,where i=1,2,…,N. So the observation equation (9) can be expressed as(13)Cc1+i=ϕTiθN+ε1+i.The least square method is to minimize the sum of residuals by the observation equation (9). Thus, we assume the sum of residuals squares as(14)F=∑i=1Nε2n+i=εTNεN,we use equation (10) into equation (14), take the derivative of θ and set this derivative as zero, we have(15)−2ΦCC−Φθ=0,so we can obtain(16)θ=ΦTΦ−1ΦTCC,due to the ill-posed of itemΦTΦ−1, the regularization term is introduced as(17)θ=ΦΤΦ+λI−1ΦTCC,where λ is the regularization parameter.Because the phenomenon of data saturation, which makes the parameters to be identified and not improve with the increase of new sampling values, occurs in the least square method, this paper presents the regularized least squares recursive algorithm with forgetting factor (RLSRAFF). RLSRAFF combines the regularization parameter with forgetting factor, which is shown as follows:(18)θN+1=θN+KNCcN+2−ϕTθN,KN=PNϕμ+ϕTPNϕ−1,PN+1=1μI−PNϕμ+ϕTPNϕ−1ϕTPN,where PN=ϕTϕ+ϕI−1. N is the number of data group, μ is the forgetting factor 0<μ≤1. If μ is chosen as a smaller one, the forgetting rate of old data is larger. Therefore, the forgetting factor μ is adjusted according to the process characteristics.Remark 2.
The innovation of this paper mainly focuses on the combination of regularization parameterλ and the forgetting factor μ. The regularization parameter λ, added in the itemPN=ϕTϕ+λI−1, can eliminate the ill-posed caused byϕTϕ−1 and reduce error.


### 3.2. Property Analysis of RLSRAFF

The property of recursive least square includes the following points: the recursive calculation of criterion function, the effect of calculation error from the gain matrix, and the convergence of the proposed algorithm. In this section, the property of RLSRAFF is analyzed.Theorem 1.
We assume thatJN+1 and JN are described as follows:(19)JN=CCTNCCN,(20)JN+1=CCTN+1CCN+1.
On the basis of RLSRAFF, the recurrence equation of the criterion function can be given as follows:(21)JN+1=μJN+μ+ϕTPNϕ−1Cc˜2N+2,where C˜cN+2=CcN+2−ϕTθN, JN, and JN+1 are the criterion function of parameter estimation at N and N+1, respectively.Proof.
By analyzing Section3.1, we have(22)ΦN+1=ρΦNϕTN+1,(23)CCN+1=ρCCNCcN+2,where ρ0<ρ≤1 is a factor, and we have(24)CC1N+1=CCN+1−ΦN+1θN+1=ρCc˜NCc˜N+2−ρΦNϕTN+1KNCc˜N+2.
So we have(25)JN+1=CC1TN+1CC1N+1=ρ2JN−2ρ2KTNΦTNCC1NCc˜N+2+ρ2KTNΦTNΦNKNCc˜2N+2+I−ΦTKNCc˜2N+2.
Using equation (19), the third and fourth terms of equation (26) can be transformed into(26)ρ2KTNΦTNΦNKNCc˜2N+2=ρ2ϕTPNϕμ+ϕTPNϕ−2Cc˜2N+2,and(27)I−ϕTKN2Cc˜2N+2=μ2μ+ϕTPNϕ−2Cc˜2N+2.
Substituting equation (26) into equations (27) and (28), and using ϕTNCC1N+1=0, we have(28)JN+1=μJN+μμ+ϕTPNϕ−1C˜c2N+2=μJN+μ+ϕTPNϕ−1C˜c2N+2.
Theorem1 has been proved.Theorem 2.
In the process of recursive calculation, if the gain matrixKN contains calculated error ΔKN, which is affected by the rounding off and other factors, it can take the calculated error ΔPN into the matrix PN. So we can obtain the following equation:(29)ΔPN+1=−ΔKNϕTPN,where ΔPN can transmit the calculation error to the gain matrix again by equation (19). If this cycle continues, the final identified result will be affected.Proof.
According to equation (19), equation PN+1 can be described as follows:(30)PN+1=I−KNϕTPN−PNϕKTN+PNϕKTN=I−KNϕTPN−PNϕKTN+KNμ+ϕTPNϕKTN,so we have(31)PN+1=I−KNϕTPNI−KNϕTT+μKNKTN.
By equation (27), we know that ΔPN+1 is only related to the quadratic form of ΔKN, which can effectively reduce the transmission of the calculation error and ensure the identification accuracy. However, if the number of parameters is less than 10, it is not necessary to avoid increasing the amount of calculation.
Meanwhile, we bringKN=KN+ΔKN in to equation (27) and subtract from equation (27), we can obtain equation (25).Theorem2 has been proved.Theorem 3.
Ifεk is known to be an uncorrelated random variable with zero mean, the parameter estimation θN given in (19) is uniformly convergent, and we have(32)limN⟶∞θN=θ0,where θ0 is true value of model parameters.Proof.
We defineθ1N=θ0−θN, and suppose(33)limN⟶∞θ1N=θ1′,Then according to equation (12), we have(34)CC=ϕTθ0+ε.
Based on equation (19), we have(35)θ1N+1=θ1N−KNCcN+2−ϕTθN=θ1N−KNϕTθ0+ε−ϕTθN=I−KNϕTθ1N−KNε=μPN+1P−1Nθ1N−KNε.We define(36)AN+1=μPN+1P−1N=μμP−1N+ϕϕT−1P−1N.
Using (32), θ1N+1 can be obtained as(37)θ1N+1=AN+1θ1N−KNε.
Further, we define the eigenvalue of matrixAN is η, and then the following equation holds(38)AN+1x=ηx,where x is nonzero eigenvector. Substituting (32) into (34), we have(39)μ1−ηP−1Nx=ηϕϕTx.
Then, we get(40)μ1−ηxTP−1Nx=ηxTϕϕTx,
According toμ>0,P−1N and ϕϕT are all positive definite matrix. Thus, for all nonzero vector x,μ1−η and η are the same number on both sides of (36), we have(41)μ1−ηη=μη−μ>0,we can gain that the eigenvalue of matrix AN satisfies 0<η<1, so (33) is stable at the balance point θ′1=0, and then according to (29), we have(42)limN⟶∞θ1N=0.
Becauseθ1N=θ0−θN, (28) is obtained.Theorem3 has been proved.


## 3.1. Regularized Least Squares Recursive Algorithm with Forgetting Factor

In order to identify the parameters in model (5),U3z and U1z are defined as the input variable, Ccz is defined as the output variable. The difference equation is used to discrete the linear system as(7)2Cck+a1Cck−1+d1Cck−1=b1U3k−1+c1U1k−1+εk,where k=2,3,…,1+N, εkis uncorrelated random variable that subjects to N0,1 distribution, U3k−1 and U1k−1 are the actual input signal of recorded data, Cck is the actual output signal of recorded data, and Cck−1 is the function value of the output signal Cck at the previous sampling period. So we have the following equation from the following equation:(8)2Cck=−a1Cck−1−d1Cck−1+b1U3k−1+c1U1k−1+εk.It can be seen from (8) that the above parameters can be estimated by N sets of the observation equations:(9)Cc1+1=−0.5a1Cc1−0.5d1Cc1+0.5b1U31+0.5c1U11+0.5ε1+1Cc1+2=−0.5a1Cc2−0.5d1Cc2+0.5b1U32+0.5c1U12+0.5ε1+2⋮Cc1+N=−0.5a1CcN−0.5d1CcN+0.5b1U3N+0.5c1U1N+0.5ε1+N.Equation (9) is defined as an observation equation. We define CcN=Cc1+1,Cc1+2,…,Cc1+NT,θN=0.5a1,0.5d1,0.5b1,0.5c1T and εN=ε1+1,ε1+2,…,ε1+NT. Equation (9) can be written into the following form:(10)CCN=ΦNθN+εN,We define(11)ΦN=ϕT1,ϕT2,…,ϕTNT,(12)ϕTi=−Cci,−Cci,U3i,U1i,where i=1,2,…,N. So the observation equation (9) can be expressed as(13)Cc1+i=ϕTiθN+ε1+i.The least square method is to minimize the sum of residuals by the observation equation (9). Thus, we assume the sum of residuals squares as(14)F=∑i=1Nε2n+i=εTNεN,we use equation (10) into equation (14), take the derivative of θ and set this derivative as zero, we have(15)−2ΦCC−Φθ=0,so we can obtain(16)θ=ΦTΦ−1ΦTCC,due to the ill-posed of itemΦTΦ−1, the regularization term is introduced as(17)θ=ΦΤΦ+λI−1ΦTCC,where λ is the regularization parameter.Because the phenomenon of data saturation, which makes the parameters to be identified and not improve with the increase of new sampling values, occurs in the least square method, this paper presents the regularized least squares recursive algorithm with forgetting factor (RLSRAFF). RLSRAFF combines the regularization parameter with forgetting factor, which is shown as follows:(18)θN+1=θN+KNCcN+2−ϕTθN,KN=PNϕμ+ϕTPNϕ−1,PN+1=1μI−PNϕμ+ϕTPNϕ−1ϕTPN,where PN=ϕTϕ+ϕI−1. N is the number of data group, μ is the forgetting factor 0<μ≤1. If μ is chosen as a smaller one, the forgetting rate of old data is larger. Therefore, the forgetting factor μ is adjusted according to the process characteristics.Remark 2.
The innovation of this paper mainly focuses on the combination of regularization parameterλ and the forgetting factor μ. The regularization parameter λ, added in the itemPN=ϕTϕ+λI−1, can eliminate the ill-posed caused byϕTϕ−1 and reduce error.


## 3.2. Property Analysis of RLSRAFF

The property of recursive least square includes the following points: the recursive calculation of criterion function, the effect of calculation error from the gain matrix, and the convergence of the proposed algorithm. In this section, the property of RLSRAFF is analyzed.Theorem 1.
We assume thatJN+1 and JN are described as follows:(19)JN=CCTNCCN,(20)JN+1=CCTN+1CCN+1.
On the basis of RLSRAFF, the recurrence equation of the criterion function can be given as follows:(21)JN+1=μJN+μ+ϕTPNϕ−1Cc˜2N+2,where C˜cN+2=CcN+2−ϕTθN, JN, and JN+1 are the criterion function of parameter estimation at N and N+1, respectively.Proof.
By analyzing Section3.1, we have(22)ΦN+1=ρΦNϕTN+1,(23)CCN+1=ρCCNCcN+2,where ρ0<ρ≤1 is a factor, and we have(24)CC1N+1=CCN+1−ΦN+1θN+1=ρCc˜NCc˜N+2−ρΦNϕTN+1KNCc˜N+2.
So we have(25)JN+1=CC1TN+1CC1N+1=ρ2JN−2ρ2KTNΦTNCC1NCc˜N+2+ρ2KTNΦTNΦNKNCc˜2N+2+I−ΦTKNCc˜2N+2.
Using equation (19), the third and fourth terms of equation (26) can be transformed into(26)ρ2KTNΦTNΦNKNCc˜2N+2=ρ2ϕTPNϕμ+ϕTPNϕ−2Cc˜2N+2,and(27)I−ϕTKN2Cc˜2N+2=μ2μ+ϕTPNϕ−2Cc˜2N+2.
Substituting equation (26) into equations (27) and (28), and using ϕTNCC1N+1=0, we have(28)JN+1=μJN+μμ+ϕTPNϕ−1C˜c2N+2=μJN+μ+ϕTPNϕ−1C˜c2N+2.
Theorem1 has been proved.Theorem 2.
In the process of recursive calculation, if the gain matrixKN contains calculated error ΔKN, which is affected by the rounding off and other factors, it can take the calculated error ΔPN into the matrix PN. So we can obtain the following equation:(29)ΔPN+1=−ΔKNϕTPN,where ΔPN can transmit the calculation error to the gain matrix again by equation (19). If this cycle continues, the final identified result will be affected.Proof.
According to equation (19), equation PN+1 can be described as follows:(30)PN+1=I−KNϕTPN−PNϕKTN+PNϕKTN=I−KNϕTPN−PNϕKTN+KNμ+ϕTPNϕKTN,so we have(31)PN+1=I−KNϕTPNI−KNϕTT+μKNKTN.
By equation (27), we know that ΔPN+1 is only related to the quadratic form of ΔKN, which can effectively reduce the transmission of the calculation error and ensure the identification accuracy. However, if the number of parameters is less than 10, it is not necessary to avoid increasing the amount of calculation.
Meanwhile, we bringKN=KN+ΔKN in to equation (27) and subtract from equation (27), we can obtain equation (25).Theorem2 has been proved.Theorem 3.
Ifεk is known to be an uncorrelated random variable with zero mean, the parameter estimation θN given in (19) is uniformly convergent, and we have(32)limN⟶∞θN=θ0,where θ0 is true value of model parameters.Proof.
We defineθ1N=θ0−θN, and suppose(33)limN⟶∞θ1N=θ1′,Then according to equation (12), we have(34)CC=ϕTθ0+ε.
Based on equation (19), we have(35)θ1N+1=θ1N−KNCcN+2−ϕTθN=θ1N−KNϕTθ0+ε−ϕTθN=I−KNϕTθ1N−KNε=μPN+1P−1Nθ1N−KNε.We define(36)AN+1=μPN+1P−1N=μμP−1N+ϕϕT−1P−1N.
Using (32), θ1N+1 can be obtained as(37)θ1N+1=AN+1θ1N−KNε.
Further, we define the eigenvalue of matrixAN is η, and then the following equation holds(38)AN+1x=ηx,where x is nonzero eigenvector. Substituting (32) into (34), we have(39)μ1−ηP−1Nx=ηϕϕTx.
Then, we get(40)μ1−ηxTP−1Nx=ηxTϕϕTx,
According toμ>0,P−1N and ϕϕT are all positive definite matrix. Thus, for all nonzero vector x,μ1−η and η are the same number on both sides of (36), we have(41)μ1−ηη=μη−μ>0,we can gain that the eigenvalue of matrix AN satisfies 0<η<1, so (33) is stable at the balance point θ′1=0, and then according to (29), we have(42)limN⟶∞θ1N=0.
Becauseθ1N=θ0−θN, (28) is obtained.Theorem3 has been proved.


## 4. Simulation Experiment

In this section, the proposed RLSRAFF is compared with other methods by two experiments. Firstly, the algorithm is compared with LSRAFF [30], the least squares method (LSM) [30], and the regularized least squares method (RLSM) [31] in Section 4.1 and Section 4.2. Then, the paper tests the performance of RLSRAFF in the parameter identification of the grinding process in Section 4.3.


### 4.1. Comparison of the Algorithm Performance

In this simulation, we choose the following equation to identify the model parameter as(43)zk+2=−a1′zk+1+a2′zk+b1′uk+1+b2′uk+εk,where a′1=−1.5,a′2=0.7,b′1=1.0,b′2=0.5;εk is a random variable that subjects to N0,1 distribution; uk is the system input, which is generated by fourth order M sequences (the amplitude is 1); zk is the system output, and the length of observation data L = 400. Meanwhile, the forgetting factor is chosen as μ=0.98 in this simulation.In this section, we choose 13 regularization parametersλ of the proposed RLSRAFF method to find the better identification results. The parameter identification results of different parameters λ are listed in Table 1. We can see from this table that the best one is given when λ=9. Next, the proposed RLSRAFF, LSM [30], RLSM [30], and LSRAFF [31] methods are used to identify the identification results, which are given in Table 2. From this table, we can find that the proposed RLSRAFF method has better performance than others.Table 1
The parameter identification results of differentλ.
The value ofλIdentification resultsa′1Absolute errora′2Absolute errorb′1Absolute errorb′2Absolute errorλ=0.01−1.54340.04340.76940.06941.05430.05430.51640.0164λ=0.1−1.50590.00590.70760.00761.10590.10590.45320.0468λ=0.5−1.45270.04730.69510.00491.08170.08170.51510.0151λ=1−1.53000.03000.74470.04471.01460.01460.74090.2409λ=2−1.47970.02030.66420.03580.98260.01740.38030.0297λ=3−1.46310.03690.68480.01520.96660.03640.59160.0916λ=4−1.52270.02270.72300.02300.92140.07860.72390.2239λ=5−1.55570.05570.70100.00100.77030.22970.40510.0949λ=6−1.58050.03050.76680.06680.85590.14410.55730.0573λ=7−1.60590.10590.79830.09830.91480.08520.40760.0924λ=8−1.49070.00930.75460.05461.15870.15870.61390.1139λ=9−1.49760.00220.69870.00131.01500.01500.51670.0167λ=10−1.44070.00830.68250.01751.17350.02490.68390.1839Table 2
Parameter identification results.
AlgorithmIdentification resultsa1′a2′b1′b2′LSM [30]−1.47130.66290.80650.5918RLSM [30]−1.49420.66740.87370.4274LSRAFF [31]1.49110.68830.96710.5446RLSRAFF−1.49760.69271.04500.5167The parameter estimation process is shown in Figure2. By observing the parameter estimation process from Figure 2, we can find that the curve (LSM and RLSM) changes very little in the later stage. However, the curve of RLSRAFF and LSRAFF [30] fluctuate all the time, and the error between the RLSRAFF curve and asymptotic property is smaller than the LSRAFF curve.Figure 2
Parameter estimation variation process.
(a)(b)(c)(d)The output results of the system of these four algorithms are given in Figure3, and the relative errors between the estimated value and the true value of system output are shown in Figure 4. From these figures, it can be seen that estimated values of system output obtained by the proposed RLSRAFF match with the true value very well. The average and maximum relative errors of system output of these four algorithms are described in Table 3, and the average and maximum relative error of RLSRAFF is 5.93% and 74.49%, respectively, which are smaller than others.Figure 3
The output results of the system.
(a)(b)(c)(d)Figure 4
The relative error between the estimated value and the true value of system output.
(a)(b)(c)(d)Table 3
The average and maximum relative error between the estimated value and the true value of system output.
AlgorithmAverage relative error (%)Maximum relative error (%)LSM [30]14.72201.5LSRAFF [30]11.86180.0RLSM [31]9.14129.3RLSRAFF5.9374.79


### 4.2. Statistical Results and Analysis

In this section, the performance of the proposed RLSRAFF method is tested by the statistical way. In this experiment, we select 10 groups of different parametersa1′,a2′,b1′,b2′, the true values of which are given in Table 3. The proposed RLSRAFF, LSM [30], RLSM [31], and LSRAFF [30] methods are used to identify these parameters, the results of which are shown in Table 4. From the identification results of Table 4, we can see that, the proposed RLSRAFF method gives a better result.Table 4
Identification results of 10 experiments.
Parametersa1′a2′b1′b2′Truth value−1.511.51RLSRAFF−1.50090.99891.49850.9958RLSM [31]−1.49650.99841.48741.0582LSM [30]−1.50391.00441.58520.8855LSRAFF [30]−1.49800.99381.48120.9872Truth value−1.00.83.04.0RLSRAFF-0.99630.79962.99213.9940RLSM [31]−1.01910.80232.93653.8294LSM [30]−1.10400.81433.10933.7783LSRAFF [30]−1.00480.80543.04744.0382Truth value−0.90.32.03.0RLSRAFF−0.90370.30172.01702.9730RLSM [31]−0.90580.31752.02992.9587LSM [30]−0.93060.32331.94543.1175LSRAFF [30]−0.92570.30961.97083.0338Truth value−0.50.14.010.0RLSRAFF−0.49890.10083.99819.9925RLSM [31]−0.51280.09593.93819.9606LSM [30]−0.47900.09453.85419.9446LSRAFF [30]−0.48850.09484.109010.0459Truth value−0.10.410.07.0RLSRAFF−0.10210.40209.92496.9556RLSM [31]−0.08570.36949.88606.9263LSM [30]−0.07980.407710.02637.0824LSRAFF [30]−0.10770.39799.89646.8515Truth value−0.81.0−2.06.0RLSRAFF−0.80150.0010−2.01045.9948RLSM [31]−0.80130.9965−1.81125.9769LSM [30]−0.80221.0106−1.97786.1568LSRAFF [30]−0.80201.0003−1.98515.9752Truth value−0.70.42.0−3.0RLSRAFF−0.70110.40162.0014−2.9984RLSM [31]−0.71570.38971.9152−2.9785LSM [30]−0.67050.36681.8951−3.0987LSRAFF [30]−0.73810.38992.1158−2.9857Truth value−0.6−0.2−4.0−3.0RLSRAFF−0.6060−0.1988−4.0408−3.0135RLSM [31]−0.6247−0.1962−3.9870−2.9425LSM [30]-0.6309-0.1755-4.0537−2.8846LSRAFF [30]−0.5882−0.2156−3.9819−2.9502Truth value0.30.48.05.5RLSRAFF0.30040.40067.99875.4976RLSM [31]0.29210.39827.97625.5475LSM [30]0.28340.38777.89145.3927LSRAFF [30]0.28950.39898.02395.4837Truth value0.3−0.5−3.01.0RLSRAFF0.2935−0.4956−3.01480.9929RLSM [31]0.3283−0.4890−2.97720.9802LSM [30]0.2553−0.5161−2.89331.0424LSRAFF [30]0.2621−0.5128−3.01511.0201By analyzing the RLSRAFF algorithm, we can see that forgetting factors and regularization parameters are the key elements for this method. When forgetting factors are the same, the identification results of the two algorithms (RLSRAFF and LSRAFF [30]) with regularization parameters are better than those without regularization parameters. Furthermore, the forgetting factor of the proposed RLSRAFF method can eliminate the data saturated phenomenon.Remark 3.
According to equation (19), we can see that the proposed RLSRAFF method is equal to LSRAFF [30] when λ=0. By analyzing the results of statistical results, it can be seen that the regularization parameters not only can solve ill-posed problems but also improve the identification results by adjusting the appropriate regularization parameters λ.


### 4.3. Application of Parameter Identification in the Grinding Process

In this section, the data in the grinding process are given to identify the parameters in the model by the proposed RLSRAFF. We set the sampling time is 10 seconds and collect 70 groups of data. The sampling data are shown in Table5 [27]. In this table, the time is from 11 : 45 : 14 to 11 : 56 : 54; the process data of the ore feeding quantity is set from 25 t/h to 233 t/h; the process data of the classifier adding water quantity is set from 5.5 m3/h to 35 m3/h; the process data of the sand return quantity is set from 22 m3/h to 105 m3/h; the process data of the classifier overflow concentration is set from 42% to 92%, the forgetting factor is 0.3, and 0.5∗randn is added in the sampling data [27].Table 5
Sampling data [27].
SequenceTimeOre feeding quantityt/hClassifier adding water quantitym3/hSand return quantitym3/hClassifier overflow concentrationm3/h111 : 45 : 1424.8845.45722.11442.05211 : 45 : 2427.1625.49630.50658.306311 : 45 : 3434.7825.76231.11862.482………………7011 : 56 : 54233.09135.059105.13192.101The observation data of input variables and output variables are selected for identifying the parameters in the grinding process model and RLSRAFF is used. The average relative error between the calculated value and the actual value with different regularization parametersλ are used to test the performance of RLSRAFF, and the results are given in Table 6. The best result is given when λ=8.Table 6
The average relative error between the calculated value and the actual value.
Regularization parametersAverage relative error (%)Regularization parametersAverage relative error (%)λ=13.87λ=73.85λ=23.94λ=83.83λ=34.02λ=93.88λ=44.12λ=103.89λ=54.00λ=113.94λ=63.93λ=124.03The experiment compares RLSRAFF and LSRAFF [27] (We select the obtained parameters θT=−0.303−0.4090.2360.136 as the parameter identification results of LSRAFF in [27].), and the output results are given in Figure 5, which compares the output contrasting curves obtained by RLSRAFF and LSRAFF. It is observed from this figure that RLSRAFF can achieve a better performance than LSRAFF.Figure 5
Output of the system.Further, the relative errors of the RLSRAFF method and the LSRAFF method are given in Figure6, and the relative error of RLSRAFF is smaller than LSRAFF. Meanwhile, the average relative error of RLSRAFF is 3.83%, and the maximum relative error is 8.02%, but for LSRAFF is 8.79% and 18.32%, respectively. Therefore, RLSRAFF gives a better performance.Figure 6
The relative error of the RLSRAFF method and the LSRAFF method [27].


## 4.1. Comparison of the Algorithm Performance

In this simulation, we choose the following equation to identify the model parameter as(43)zk+2=−a1′zk+1+a2′zk+b1′uk+1+b2′uk+εk,where a′1=−1.5,a′2=0.7,b′1=1.0,b′2=0.5;εk is a random variable that subjects to N0,1 distribution; uk is the system input, which is generated by fourth order M sequences (the amplitude is 1); zk is the system output, and the length of observation data L = 400. Meanwhile, the forgetting factor is chosen as μ=0.98 in this simulation.In this section, we choose 13 regularization parametersλ of the proposed RLSRAFF method to find the better identification results. The parameter identification results of different parameters λ are listed in Table 1. We can see from this table that the best one is given when λ=9. Next, the proposed RLSRAFF, LSM [30], RLSM [30], and LSRAFF [31] methods are used to identify the identification results, which are given in Table 2. From this table, we can find that the proposed RLSRAFF method has better performance than others.Table 1
The parameter identification results of differentλ.
The value ofλIdentification resultsa′1Absolute errora′2Absolute errorb′1Absolute errorb′2Absolute errorλ=0.01−1.54340.04340.76940.06941.05430.05430.51640.0164λ=0.1−1.50590.00590.70760.00761.10590.10590.45320.0468λ=0.5−1.45270.04730.69510.00491.08170.08170.51510.0151λ=1−1.53000.03000.74470.04471.01460.01460.74090.2409λ=2−1.47970.02030.66420.03580.98260.01740.38030.0297λ=3−1.46310.03690.68480.01520.96660.03640.59160.0916λ=4−1.52270.02270.72300.02300.92140.07860.72390.2239λ=5−1.55570.05570.70100.00100.77030.22970.40510.0949λ=6−1.58050.03050.76680.06680.85590.14410.55730.0573λ=7−1.60590.10590.79830.09830.91480.08520.40760.0924λ=8−1.49070.00930.75460.05461.15870.15870.61390.1139λ=9−1.49760.00220.69870.00131.01500.01500.51670.0167λ=10−1.44070.00830.68250.01751.17350.02490.68390.1839Table 2
Parameter identification results.
AlgorithmIdentification resultsa1′a2′b1′b2′LSM [30]−1.47130.66290.80650.5918RLSM [30]−1.49420.66740.87370.4274LSRAFF [31]1.49110.68830.96710.5446RLSRAFF−1.49760.69271.04500.5167The parameter estimation process is shown in Figure2. By observing the parameter estimation process from Figure 2, we can find that the curve (LSM and RLSM) changes very little in the later stage. However, the curve of RLSRAFF and LSRAFF [30] fluctuate all the time, and the error between the RLSRAFF curve and asymptotic property is smaller than the LSRAFF curve.Figure 2
Parameter estimation variation process.
(a)(b)(c)(d)The output results of the system of these four algorithms are given in Figure3, and the relative errors between the estimated value and the true value of system output are shown in Figure 4. From these figures, it can be seen that estimated values of system output obtained by the proposed RLSRAFF match with the true value very well. The average and maximum relative errors of system output of these four algorithms are described in Table 3, and the average and maximum relative error of RLSRAFF is 5.93% and 74.49%, respectively, which are smaller than others.Figure 3
The output results of the system.
(a)(b)(c)(d)Figure 4
The relative error between the estimated value and the true value of system output.
(a)(b)(c)(d)Table 3
The average and maximum relative error between the estimated value and the true value of system output.
AlgorithmAverage relative error (%)Maximum relative error (%)LSM [30]14.72201.5LSRAFF [30]11.86180.0RLSM [31]9.14129.3RLSRAFF5.9374.79


## 4.2. Statistical Results and Analysis

In this section, the performance of the proposed RLSRAFF method is tested by the statistical way. In this experiment, we select 10 groups of different parametersa1′,a2′,b1′,b2′, the true values of which are given in Table 3. The proposed RLSRAFF, LSM [30], RLSM [31], and LSRAFF [30] methods are used to identify these parameters, the results of which are shown in Table 4. From the identification results of Table 4, we can see that, the proposed RLSRAFF method gives a better result.Table 4
Identification results of 10 experiments.
Parametersa1′a2′b1′b2′Truth value−1.511.51RLSRAFF−1.50090.99891.49850.9958RLSM [31]−1.49650.99841.48741.0582LSM [30]−1.50391.00441.58520.8855LSRAFF [30]−1.49800.99381.48120.9872Truth value−1.00.83.04.0RLSRAFF-0.99630.79962.99213.9940RLSM [31]−1.01910.80232.93653.8294LSM [30]−1.10400.81433.10933.7783LSRAFF [30]−1.00480.80543.04744.0382Truth value−0.90.32.03.0RLSRAFF−0.90370.30172.01702.9730RLSM [31]−0.90580.31752.02992.9587LSM [30]−0.93060.32331.94543.1175LSRAFF [30]−0.92570.30961.97083.0338Truth value−0.50.14.010.0RLSRAFF−0.49890.10083.99819.9925RLSM [31]−0.51280.09593.93819.9606LSM [30]−0.47900.09453.85419.9446LSRAFF [30]−0.48850.09484.109010.0459Truth value−0.10.410.07.0RLSRAFF−0.10210.40209.92496.9556RLSM [31]−0.08570.36949.88606.9263LSM [30]−0.07980.407710.02637.0824LSRAFF [30]−0.10770.39799.89646.8515Truth value−0.81.0−2.06.0RLSRAFF−0.80150.0010−2.01045.9948RLSM [31]−0.80130.9965−1.81125.9769LSM [30]−0.80221.0106−1.97786.1568LSRAFF [30]−0.80201.0003−1.98515.9752Truth value−0.70.42.0−3.0RLSRAFF−0.70110.40162.0014−2.9984RLSM [31]−0.71570.38971.9152−2.9785LSM [30]−0.67050.36681.8951−3.0987LSRAFF [30]−0.73810.38992.1158−2.9857Truth value−0.6−0.2−4.0−3.0RLSRAFF−0.6060−0.1988−4.0408−3.0135RLSM [31]−0.6247−0.1962−3.9870−2.9425LSM [30]-0.6309-0.1755-4.0537−2.8846LSRAFF [30]−0.5882−0.2156−3.9819−2.9502Truth value0.30.48.05.5RLSRAFF0.30040.40067.99875.4976RLSM [31]0.29210.39827.97625.5475LSM [30]0.28340.38777.89145.3927LSRAFF [30]0.28950.39898.02395.4837Truth value0.3−0.5−3.01.0RLSRAFF0.2935−0.4956−3.01480.9929RLSM [31]0.3283−0.4890−2.97720.9802LSM [30]0.2553−0.5161−2.89331.0424LSRAFF [30]0.2621−0.5128−3.01511.0201By analyzing the RLSRAFF algorithm, we can see that forgetting factors and regularization parameters are the key elements for this method. When forgetting factors are the same, the identification results of the two algorithms (RLSRAFF and LSRAFF [30]) with regularization parameters are better than those without regularization parameters. Furthermore, the forgetting factor of the proposed RLSRAFF method can eliminate the data saturated phenomenon.Remark 3.
According to equation (19), we can see that the proposed RLSRAFF method is equal to LSRAFF [30] when λ=0. By analyzing the results of statistical results, it can be seen that the regularization parameters not only can solve ill-posed problems but also improve the identification results by adjusting the appropriate regularization parameters λ.


## 4.3. Application of Parameter Identification in the Grinding Process

In this section, the data in the grinding process are given to identify the parameters in the model by the proposed RLSRAFF. We set the sampling time is 10 seconds and collect 70 groups of data. The sampling data are shown in Table5 [27]. In this table, the time is from 11 : 45 : 14 to 11 : 56 : 54; the process data of the ore feeding quantity is set from 25 t/h to 233 t/h; the process data of the classifier adding water quantity is set from 5.5 m3/h to 35 m3/h; the process data of the sand return quantity is set from 22 m3/h to 105 m3/h; the process data of the classifier overflow concentration is set from 42% to 92%, the forgetting factor is 0.3, and 0.5∗randn is added in the sampling data [27].Table 5
Sampling data [27].
SequenceTimeOre feeding quantityt/hClassifier adding water quantitym3/hSand return quantitym3/hClassifier overflow concentrationm3/h111 : 45 : 1424.8845.45722.11442.05211 : 45 : 2427.1625.49630.50658.306311 : 45 : 3434.7825.76231.11862.482………………7011 : 56 : 54233.09135.059105.13192.101The observation data of input variables and output variables are selected for identifying the parameters in the grinding process model and RLSRAFF is used. The average relative error between the calculated value and the actual value with different regularization parametersλ are used to test the performance of RLSRAFF, and the results are given in Table 6. The best result is given when λ=8.Table 6
The average relative error between the calculated value and the actual value.
Regularization parametersAverage relative error (%)Regularization parametersAverage relative error (%)λ=13.87λ=73.85λ=23.94λ=83.83λ=34.02λ=93.88λ=44.12λ=103.89λ=54.00λ=113.94λ=63.93λ=124.03The experiment compares RLSRAFF and LSRAFF [27] (We select the obtained parameters θT=−0.303−0.4090.2360.136 as the parameter identification results of LSRAFF in [27].), and the output results are given in Figure 5, which compares the output contrasting curves obtained by RLSRAFF and LSRAFF. It is observed from this figure that RLSRAFF can achieve a better performance than LSRAFF.Figure 5
Output of the system.Further, the relative errors of the RLSRAFF method and the LSRAFF method are given in Figure6, and the relative error of RLSRAFF is smaller than LSRAFF. Meanwhile, the average relative error of RLSRAFF is 3.83%, and the maximum relative error is 8.02%, but for LSRAFF is 8.79% and 18.32%, respectively. Therefore, RLSRAFF gives a better performance.Figure 6
The relative error of the RLSRAFF method and the LSRAFF method [27].


## 5. Conclusion

In the actual industrial production process, a large amount of data is often obtained by online real-time measurement, and the measured data often contains some error. Thus, the data saturated phenomenon and the ill-posed problem often occur simultaneously. The original LSM seldom considers the above effects to the identification of parameters. Therefore, this paper investigates the identification of parameters in the grinding process and considers the data saturated phenomenon and the ill-posed problem.In order to solve the above problems, this paper presents a RLSRAFF algorithm, which combines the forgetting factor with regularization parameters. Furthermore, to analyze the performance of the RLSRAFF algorithm, this paper verifies the recursive calculation of criterion function and describes the effect of calculation error from the gain matrix and proves the convergence of the proposed algorithm. Finally, effectiveness of RLSRAFF is verified by the simulation experiments and grinding data. Comparing with other algorithms, the results show that RLSRAFF gives a better result and eliminates the ill-posed problem by the simulation and statistical results.RLSRAFF is a universal method, the aim of which is to solve the data saturated phenomenon and the ill-posed problem. Both problems often occurs in the parameter identification process, such as the lithium ion batteries [32] and the wastewater treatment process [5]. Therefore, the RLSRAFF method and its extension method also can be applied in other fields. Furthermore, for LSM-based identification methods, noises can cause biased identification results. Some noise-compensated methods [33–35] can be used to reduce these biases. In the future research work, the noise-compensated methods can be considered to improve the RLSRAFF method and be applied in the grinding process (see Table 7).Table 7
A List of symbols.
SymbolsDefinitionα1,α2,α3,α4,α5;β1,β2,β3,β4Coefficient of inertia linkτ1,τ2,τ3,τ4Coefficient of delay linkG′11S,G′12S,G′21S,G′22STransfer function with delay link and inertia linkG11S,G12S,G21S,G22STransfer function with inertia linkU1The actual value of ore feeding quantityt/hU3The actual value of classifier adding water quantitym3/hQDThe actual value of the return sand quantitym3/hCcThe actual value of classifier overflow concentration (%)kObserved timesk=2,3,…,1+Nii=1,2,…,NU3k, U1kU1KThe actual input signal of recorded dataCckThe actual output signal of recorded dataa1,b1,c1,d1;a2,a3,b2,b3,c2,d2aParameters to be identifiedεkUncorrelated random variable that subjects toN0,1 distributionN0,1Normal distributionθNθN=0.5a1,0.5d1,0.5b1,0.5c1TCcNCcN=Cc1+1,Cc1+2,…,Cc1+NTΦNΦN=ΦT1,ΦT2,…,ΦTNTϕTiϕTi=−Cci,−Cci,U3i,U1iTFThe sum of residuals squaresNNumber of data groupμForgetting factor0<μ≤1)ρA factorρ2=μλRegularization parameterKNGain matrixPNPN=ϕTϕ+λI−1JNCriterion function of parameter estimation atNθ0True value of model parametersηEigenvalue of matrixAN+1)xNonzero eigenvectorukSystem input of equation (43)zkSystem output of equation (43)a1′,a2′,b1′,b2′Parameters to be identified of equation (43)LLength of observation data



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*Source: 5188389-2022-09-27.xml*