Projection to latent structures (PLS) model has been widely used in qualityrelated process monitoring, as it can establish a mapping relationship between process variables and quality index variables. To enhance the adaptivity of PLS, kernel PLS (KPLS) as an advanced version has been proposed for nonlinear processes. In this paper, we discuss a new total kernel PLS (TKPLS) for nonlinear qualityrelated process monitoring. The new model divides the input spaces into four parts instead of two parts in KPLS, where an individual subspace is responsible in predicting quality output, and two parts are utilized for monitoring the qualityrelated variations. In addition, fault detection policy is developed based on the TKPLS model, which is more well suited for nonlinear qualityrelated process monitoring. In the case study, a nonlinear numerical case, the typical Tennessee Eastman Process (TEP) and a real industrial hot strip mill process (HSMP) are employed to access the utility of the present scheme.
Multivariate statistic process monitoring (MSPM) is effective for detecting and diagnosing abnormal operating situations in many industrial processes, which helps by improve products' quality a lot. In MSPM, projection to latent structures (PLS) model pays more attention to qualityrelated faults while principal component analysis (PCA) considers all faults in a process [
For many physical and chemical processes, the nonlinearity lying in the process data and quality data is too obvious to be neglected. To deal with this problem, many nonlinear PLS methods have been proposed [
In the aforementioned literature [
In order to improve the KPLS model, a new total kernel PLS (TKPLS) is proposed for nonlinear qualityrelated process monitoring in this paper. First of all, we reveled and summarized the existing KPLS model and corresponding process monitoring techniques. Then TKPLS is developed. The properties of the new model and the process monitoring strategies are discussed then. TKPLS model can describe the nonlinear process according to quality data effectively and also give a further decomposition on the feature spaces in KPLS. Actually, besides nonlinearity, traditional MSPM approaches also possess the assumption that the processes operate under a Gaussian distribution and in a single mode. Also, increasing number of studies can be found in this area. However, due to the scope in this paper, these issues will be considered in the subsequent researches [
This paper is organized as follows. KPLSrelated algorithm and process monitoring methods are introduced in Section
For a nonlinear process, the input matrix can be defined as
For a test sample
The algorithm of KPLS modeling has been illustrated in Appendix
Let
The derivation of (
The determination of kernel function
Usually
The residuals of
KPLS divides the feature space
The TKPLS model is a further decomposition on the KPLS model. It can be thought as a postprocessing method to decompose the
(1) Perform KPLS algorithm on
(2) Run PCA on
(3) Define
(4) Run PCA on
(5) Perform PCA on
(6)
In step
Obtain
(1) After KPLS model:
(2) Run eigenvector decomposition on
(3) Perform eigenvector decomposition on
(4) Perform eigenvector decomposition on
In TKPLS model, we can model
The meanings of different sections of
Meaning of different sections of
Section  Description 


The 

The part of 

The principal part of 

The residual part which is not excited in 
In TKPLS, the orthogonality among all score vectors holds. Meanwhile,
In multivariate statistical process monitoring, two types of statistics are widely used for fault detection. One is the
According to TKPLS model, three score vectors can be calculated as follows:
Motivated by total PLS (TPLS) based methods [
Monitoring statistics and control limits.
Statistic  Calculation  Control limit 













Implementation of the TKPLSbased qualityrelated detection scheme involves offline training model and online testing model. As sketched in Figure
Training model TKPLSbased monitoring.
Flowchart of testing model for TKPLSbased monitoring.
In this section, two detailed simulation examples are carried out to demonstrate the advantage of TKPLS.
Firstly, a synthetic nonlinear numerical process without feedback is presented as follows:
We used 200 samples generated from the above process as a training dataset. The faulty dataset with 400 samples was also generated according to the following faults:
Fault 1: a step bias in
Fault 2: a ramp change in
Fault 3: a step bias in
Fault 4: a ramp change in
Training samples are applied to perform a KPLS model on
According to the descriptions of Faults 1 and 2, they are qualityunrelated faults. Let
False alarm rates of faults unrelated to
Fault value ( 
KPLS ( 
TKPLS ( 
TKPLS ( 
TKPLS ( 


Fault 1  0.2  26.8  0  4.7  4.7 
0.4  31.7  0  11.6  11.6  
0.6  53.3  0  26.6  26.6  
0.8  77.2  0  43.3  43.3  
 
Fault 2  0.002  24.5  0  8.3  8.3 
0.003  37.4  0  18.6  18.6  
0.004  44.5  0  27.8  27.8  
0.005  56.3  0  36.8  36.8 
KPLSbased monitoring with 99% control limit when qualityunrelated Fault 1 occurs.
TKPLSbased monitoring when qualityunrelated Fault 1 occurs.
KPLSbased monitoring with 99% control limit when qualityunrelated Fault 2 occurs.
TKPLSbased monitoring when qualityunrelated Fault 2 occurs.
The predefined Faults 3 and 4 are qualityrelated. For Fault 3 with
False detection rates of faults related to
Fault value ( 
KPLS ( 
TKPLS ( 
TKPLS ( 
TKPLS ( 


Fault 3  0.2  63.6  80.5  57.3  83.6 
0.4  79.8  88.5  74.4  88.7  
0.6  90.3  99.2  86.3  99.2  
0.8  99.4  100  99.5  100  
 
Fault 4  0.002  4.1  0  29.5  29.5 
0.003  4.3  0  43.1  43.1  
0.004  3.5  0  54.2  54.2  
0.005  4.6  0  62.1  62.1 
KPLSbased monitoring with 99% control limit when qualityrelated Fault 3 occurs.
TKPLSbased monitoring when qualityrelated Fault 3 occurs.
KPLSbased monitoring with 99% control limit when qualityrelated Fault 4 occurs.
TKPLSbased monitoring when qualityrelated Fault 4 occurs.
The Tennessee Eastman (TE) Process was provided by Eastman Chemical Company which is a realistic industrial process for evaluating different process control and monitoring technologies [
The TEP contains two blocks of variables: 12 manipulated variables and 41 measured variables. Process measurements are sampled with interval of 3 min, while nineteen composition measurements are sampled with time delays which vary from 6 min to 15 min. The time delay has a potentially critical impact on product quality control in this process, because the closedloop control works when the next sample of quality variable is available [
PLS and KPLSbased monitoring methods can detect the fault correlated to
In this case study, the component G in steam 9, that is, the 35th measured variable, is chosen as the output quality variable
Fault detection rate of TEP using TPLS, KPLS, and TKPLS (%).
Faults ID  Type  TPLS  KPLS  TKPLS 

IDV(1)  Step  99.3  88.6 

IDV(2)  Step  97.6  98.6 

IDV(5)  Step 

48.2  97.4 
IDV(6)  Step 

99.5 

IDV(8)  Random variation  93.4  95.6 

IDV(12)  Random variation  95.6 

98.3 
IDV(13)  Slow drift  95.3  96.4 

False alarm rates of TEP using TPLS, KPLS, and TKPLS (%).
Faults ID  Type  TPLS  KPLS  TKPLS 

IDV(0)  — 

8.6  5.9 
IDV(3)  Step 

9.8 

IDV(4)  Step  33.5  25.3 

IDV(9)  Step  5.3  8.2 

IDV(11)  Random variation  32.3  32.7 

IDV(14)  Random variation  12.4  22.7 

IDV(15)  Slow drift 

28.0  10.0 
Detection of IDV
Detection of IDV
From the detection results, it is observed that TKPLSbased method gives a higher detection rate and lower false alarm rate than KPLSbased method. Compared with linear TPLS, TKPLS performs better in most cases. In Table
Hot strip mill process (HSMP) is an extremely complex process in iron and steel industry. A schematic layout of the hot strip mill is illustrated in Figure
Schematic layout of the hot strip mill.
The demand of dimensional precision, especially thickness precision of hot strip mill, has become stricter in recent years, which makes the improvement of thickness precision be a hot topic. In general, the thickness in exit of finishing mill is closely related to gap and rolling force and has little connection with bending force. In this paper, two classes of strips' manufacturing process are taken for this test with thicknesses, where their thickness targets are 3.95 mm and 2.70 mm, respectively. Based on historical dataset, the new proposed framework can be constructed with the measured process variables and quality variable which are listed in Table
Process and quality variables in finishing mill.
Variable  Type  Description  Unit 

1~7  Measured 

mm 
8~14  Measured 

MN 
15~20  Measured 

MN 

Quality  Finishing mill exit strip thickness  mm 
Typical faults in finishing mill.
No.  Description  Fault type  Quality related 

1  Sensor fault of bending force measurement in 
Sensor fault  No 
2  Malfunction of hydraulic gap control loop in 
Process fault  Yes 
3  Actuator fault of cooling valve between 
Actuator fault  Yes 
The results of thickness qualityrelated process monitoring are given by Table
Detection rate or false alarm rate for hot strip mill (%).
Fault No.  Type of detection  PLS ( 
KPLS ( 
TPLS 
TKPLS 

1  False alarm rate  0.104  0.117  0.366 

2  Detection rate  0.998 



3  Detection rate  0.656  0.870  0.900 

Regarding HSMP, the following should be noted.
We clarify that the data considered about finishing mill process are acquired from real steel industrial field, namely, Ansteel Corporation, China. The faults occur occasionally and were eliminated manually.
In this implementation, only thickness has been concerned as the quality variable, whereas TKPLS model can handle multioutput cases.
In this paper, the TKPLS algorithm is proposed by further decomposing KPLS. The purpose of TKPLS is to perform a further decomposition on the high dimension space induced by KPLS, which is more suitable for qualityrelated process monitoring. The process monitoring methods based on TKPLS are developed to monitor the operating performance. Both theoretical analysis and simulation results show better performance of TKPLS than KPLS. TKPLSbased methods can give lower false alarm rates and missing alarm rates than KPLSbased methods in most simulated cases. However, there are still some problems needed to be considered in the modeling with TKPLS, such as how to select an appropriate kernel function for a given process data and establish a framework for precisely choosing the kernel parameters. Due to the scope of this paper, further studies for these issues will be concerned in the future.
The nonlinear iterative KPLS algorithm is shown in Algorithm
(1) Set
(2)
(3) Scale
(4)
(5) Scale
Repeat (2)–(5) until
(6) Deflate matrices
(7) Set
Based on Algorithm
First of all, setting
Motivated by the calculation in TPLS model,
This work was supported by national 973 projects under Grants 2010CB731800 and 2009CB32602 and by NSFC under Grants (61074084 and 61074085), China, and Beijing Key Discipline Development Program (no. XK100080537). We also appreciate the data support from Ansteel Corporation in Liaoning Province, China.