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A variable selection aided residual generator design approach for process control and monitoring
Neurocomputing 171 (2016) 1013–1020
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Neurocomputing journal homepage: www.elsevier.com/locate/neucom
A variable selection aided residual generator design approach for process control and monitoring$ Chaojing Duan a, Zhongyang Fei b,n, Jiachen Li a a b
Honors School, Harbin Institute of Technology, Harbin 150001, PR China Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 April 2015 Received in revised form 3 June 2015 Accepted 15 July 2015 Communicated by Xudong Zhao Available online 29 July 2015
This paper investigates the design of residual generator based on data-driven techniques, which is optimized by variable selection and can be applied to control and monitoring in industrial process to satisfy the high performance requirements of systems. The basic idea is designing the residual generator based on the foundation of the Luenberger equations as well as the relationship between diagnosis observer (DO) and the parity vector. The crucial and innovative part of the scheme is to acquire the parity space by the Subspace Identification Method (SIM) and obtain the simplified and optimized data from a new variable selection system according to partial least squares (PLS) method during the procedure of SIM. The design and optimization of the residual generator make it independent of model and effective to deal with big data, and avoid the challenges of handling bulk data. After the realization of the Youla parameterization, the proposed approach of residual generator can not only be applied to control purposes, but also monitoring objective of practical industrial systems. A numerical example is used to demonstrate the performance and effectiveness of the proposed scheme. & 2015 Elsevier B.V. All rights reserved.
Keywords: Variable selection Patrial least square Residual generator Control Monitoring
1. Introduction During the past two decades, the techniques of model-based controller design have been established and greatly developed in different systems, such as switched system, Markovian system, singular system, etc [1–4]. Moreover, the field of model based fault detection and isolation techniques have been widely investigated during this period, especially for linear time invariant (LTI) systems [12,13]. However, with the development of industrial techniques, the requirements on product quality and system performance increase significantly. Since the industry process is becoming more and more complicated, to establish a model for the whole system is arduous in most applications and to employ the high-dimensional model is even impractical in pragmatic systems. In another hand, a lot of data has been recorded and tremendous information of the system could be extracted with the advanced technology of real data, which will be definitely
☆ This research is partially supported by the Fundamental Research Funds for the Central Universities (AUGA5710056615) and by China Postdoctoral Science Foundation (AUGA4110033015). n Corresponding author. E-mail addresses: [email protected] (C. Duan), [email protected] (Z. Fei), [email protected] (J. Li).
http://dx.doi.org/10.1016/j.neucom.2015.07.042 0925-2312/& 2015 Elsevier B.V. All rights reserved.
beneficial to improve the performance reliability and system safety. The study of designing and updating the controllers by the input as well as output of the system has attracted widespread attention from industry and academia. The main approaches of data-driven system can be divided into 3 categories according to the utilization of data: on-line data, off-line data and the combination of both [5]. Therefore, some data-driven approaches have nowadays gained more attention and improvement. For instance, proportionalintegral-derivative (PID) method may be the earliest and the most widely used technique in industrial processes [6,7,20]. Meanwhile, approaches in multivariate statistical analysis, which can deal with abundant highly correlated measured data, have been modified since it was applied into actual industry. For example, principal component analysis (PCA) method is designed to decrease dimensional principal components [8,9], and partial least squares (PLS) method can predict key indicator from processes measurements directly [11,10]. It is reasonable that the more intricate and multifunctional the industrial process is, the more easily and frequently the fault will occur. Since the fault not only impacts the efficiency and quality of production, but also threatens the security of industrial manufacture, detecting and isolating the fault while it exists in the system are important for modern industry. However, it is very difficult to establish quantitative model for large-scale industrial process with
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model based FDI method. Furthermore, the challenges of big data are unable to be solved effectively by traditional mathematic and statistical approaches. This dilemma motivates the investigation of new techniques that can be more independent of model and more effective to utilize and deal with big data. In addition, the noteworthy development of the PLS method and subspace identification methods (SIM) which can generate parity vector directly from process measurements inspire the study in this paper. An approach of data-driven control and monitoring for LTI system and the technique to optimize it will be proposed in this paper. The foundation of this achievement is embedding a residual generator with the purpose of FDI in the feedback control loop. The main contribution of this paper will be divided into two parts. One is a variable selection procedure according to PLS method, which can simplify the SIM process by analyzing the connections between the inputs and reducing the quantity among them. The other is constructing a residual generator based on the relationship between diagnosis observer (DO) and the parity vector [14]. The rest of this paper is organized as follows. In Section 2.1, the basic description of system and preliminary factorization are presented. A variable selection approach based on PLS method is introduced in Section 3. In Section 4, the approaches of DO and parity space are explained respectively firstly, then the method of residual generator design is analyzed and provided. The method of the residual generator application in both control and monitoring is discussed in Section 5. To illustrate the performance of the design, a numerical example on the model is presented in Section 6. Finally, some conclusions of this paper are made in Section 7.
2. Problem formulation
The linear time invariant (LTI) system in discrete state space can be described by
ð2Þ
where GðzÞ and G1 ðzÞ can be described from the relationship between the transfer function matrix and the state space equation.
ð4Þ
^ ^ with MðzÞ and NðzÞ defined as follows: ^ MðzÞ ¼ I CðzI AF Þ 1 F; ^ NðzÞ ¼ M þ CðzI AF Þ 1 BF ;
ð5Þ
in which AF ¼ A FC and BF ¼ B FM. 2.3. Youla parameterizations According to the Youla Parameterization [29,30], all the stabilizing controllers V(z) can be expressed in a unified form with a classical output feedback control loop. Suppose an appropriate parameter matrix V(z) that makes ðA þ BV Þ stable, which satisfies the following equation: 1 ^ ^ ^ VðzÞ ¼ ðX^ ðzÞ Q ðzÞNðzÞÞ ðY ðzÞ Q ðzÞMðzÞÞ;
ð6Þ
in which X^ ðzÞ ¼ I V ðzI AF Þ 1 BF ; Y^ ðzÞ ¼ VðzI AF Þ 1 F:
ð7Þ
It is reasonable to assume that the matrix V, which is stable to simplify the controller design, is equal to zero, so a simplified controller can be obtained from (6): ð8Þ
The equation about the residual vectors and the inputs of the process can be governed by ^ rðzÞ ¼ MðzÞyðzÞ N^ ðzÞiðzÞ; iðzÞ ¼ VðzÞðλðzÞ yðzÞÞ:
ð1Þ
where the plant inputs defined as iðkÞ A Rl and outputs defined as yðkÞ A Rm at every discrete time point. xðkÞ A Rn stands for the status variables, mðkÞ A Rn and nðkÞ A Rm denote white noise with uncertainty. However, the system matrices and orders are all unknown parameters. The charter for the classical output feedback control method is shown in Fig. 1 [28]. According to Fig. 1, the transfer function can be written into yðzÞ ¼ GðzÞiðzÞ þ G1 ðzÞmðzÞ þ H n nðzÞ;
^ 1 ðzÞN^ 1 ðzÞ; GðzÞ ¼ M
^ VðzÞ ¼ ðI Q ðzÞN^ ðzÞÞ 1 Q ðzÞMðzÞ:
2.1. System descriptions
xðk þ 1Þ ¼ AxðkÞ þ BiðkÞ þH m mðkÞ; yðkÞ ¼ CðxÞ þDiðkÞ þ H n nðkÞ;
^ N^ are left coprimes to each other. Assume then the matrixes M; that there is a proper matrix of parameters F and all the eigenvalues of the matrix ðA FCÞ are less than 1. Then, a left coprime factorization of the system transfer function matrix G(z) can be given by
ð9Þ
Combining Eqs. (8) and (9), we have ^ ^ λðzÞ ðMðzÞyðzÞ N^ ðzÞiðzÞÞÞ: iðzÞ ¼ Q ðzÞðMðzÞ
ð10Þ
Then the feedback control loop is restructured and arranged as sketched in Fig. 2. By rearranging the blocks, the feedback control system can be arranged as shown in Fig. 3.
3. Variable selection based on PLS
^ N^ ; X^ and Y^ satisfy the following relationship: If matrixes M; " # h i ^ ^ N^ X ¼ M ^ X^ þ N^ Y^ ¼ I; M ð3Þ Y^
As discussed previously, the number of data which is significant for controlling and monitoring system performance will be quite large during the present practical industrial process. Too much data hampers the residual generator design inevitably or even make it impossible. On the other hand, some errors or nonsignificant data which will impact the effectiveness of the residual generator. As a result, a variable selection method is necessary to reduce redundant information and useless variables, not only for decreasing the number of inputs simplifying the computation during the construction of residual generator, but also for improving the stability and performance of it [25–27].
Fig. 1. The classical system.
Fig. 2. Youla parameterization.
2.2. Left coprime factorization
C. Duan et al. / Neurocomputing 171 (2016) 1013–1020
Fig. 3. Arrangement of control system.
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3.1.2. Root mean square error The root mean square error (RMSE) is generally used to calculate the differences between the real values and their predicted ones. Fundamentally, the RMSE index indicates the sample standard deviation of the differences, which are also called residuals on the condition that the measurements are performed over the data sample. During this work, the RMSE index can be computed as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN ^ 2 i ¼ 1 ðyi y i Þ RMSE ¼ ; ð11Þ N in which yi is the i-th element of the real output variables and y^ i denotes the corresponding predicted value. 3.2. Input variables classification The input variables need to be classified according to their weighting in the regression vector obtained. The weight stands for the significance of the corresponding input variable, and each input variable corresponds to one weighting in the regression vector. As a result, the absolute values of related elements in regression vector will be the basis to decide how to realign the original input variables and the construction of new input variable matrix will be realized at the same time. For instance, if the regression vector is a row vector, then the elements should be realigned in descending order from left to right. Otherwise, the order will be from top to bottom. 3.3. Test procedure correlation
Fig. 4. Variable selection procedure.
The further step, which is aimed at examining the correlations between variables, should be taken after the realignment of the input variable matrix. The examination should begin from the least crucial variable to the most significant one, which means it is from the last column of realigned input variable matrix to the first column. During this process, if a certain input variable is highly correlated with others, then it will be cancelled. This work can be described as follows:
Step 1. Obtain the correlation coefficient matrix of realigned input variable matrix from the following equations: The data pre-processing with PLS-based variable selection will find out the characteristics of the input data matrix and PLS regression, which provides an effective method for multiple linear regression modeling. After that, the variable selection method proposed will be divided into four parts, which are displayed in Fig. 4.
Cði; jÞ ¼ E½X i EðX i Þ½X j EðX j Þ; Cði; jÞ Rði; jÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: Cði; iÞCðj; jÞ
Step 2. Compare the absolute values of elements in the
3.1. The regression vector generation The regression vector for sorting variables, which is crucial for the success of variable selection, can be derived from PLS regression. Meanwhile, the number of latent variables property that will influences the regression vector should be decided to find out the correct complexity of model. The method of cross-validation (CV) and root mean square error (RMSE) are applied to select proper latent variables so that the risks of “over-fitting” and “underfitting” can be avoided effectively. 3.1.1. Cross validation In general, not all the existing m components t 1 ; t 2 ; …; t m but just the first l components are used to build the regression model. The function of CV method is to select the latent variables. The i-th experimental data will be abandoned and the rest data will be used to make a PLS regression model each time.
ð12Þ
correlation coefficient matrix with each other. In the situation that the differences between two elements are less than 0.1, one of the corresponding input variables with less weighting in regression vector would be cancelled. Step 3. Repeat Step 2 until all of the elements have been considered.
3.4. Subset selection with the best estimation performance In above steps, the uncorrelated input variable matrix has been obtained. The next step is to calculate the estimation performance of each subset and complete the subset selection with the best estimation performance [15]. The uncorrelated input variables are aligned and several variable subsets are selected according to their weightings in the regression vector and priorities, respectively. Based on the results that have been achieved, the subset selection can be implemented with the following steps.
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Step 1. Define the initial subset which has ki columns in X or . For
instance, if ki is given as 4, the initial subset X 1 consists of the first four columns of X or . Step 2. Expand the second subset X 2 by adding a fixed number of variables kj . For example, if kj is 3, then the second subset X 2 will include the most seven significant input variables in X or . Step 3. Iterate the steps above till all the columns in X or are taken into consideration. At the same time, the PLS regression model is constructed and the corresponding RMSE index is calculated, which is combined with CV method for each selected subset.
So far, all of the variable subsets have been evaluated by the RMSE indices. Besides, the subset with the smallest RMSE value represents the best prediction performance. Meanwhile, the variables in subset should be the selected ones. What has been discussed is the core of the variable selection method based on PLS method. It can characterize the appropriate complexity of fitting and prediction model in residual generator design. Furthermore, the variable selection method will be applied to simplify the construction of residual generator in the following sections.
4. Residual generator design Residual is supposed to be the error between the measured and the predicted values, which is significant for the system process because of the important and fundamental information of the system model it contains. The residual generator can be designed directly by the input and output data without identification of system parameters and the generator can also be inserted to the feedback control system to be applied in both control and monitoring in practical industry [13,23,24]. In this section, the method of residual generator design will be given. 4.1. Residual signals generation
^ þ BiðkÞ þ LðyðkÞ yðkÞÞ; ^ ^ þ 1Þ ¼ AxðkÞ xðk ^ þ DiðkÞ: yðkÞ ¼ C xðkÞ
ð13Þ
The state space equations of the method based on DO can be expressed by zðk þ 1Þ ¼ Az zðkÞ þ Bz iðkÞ þ Lz yðkÞ; ^ yðkÞ ¼ C z zðkÞ þ Dz iðkÞ þGz yðkÞ; ss
sm
ð14Þ sm
ls
1l
; Lz A R ; C z A R ; Dz A R and in which z A R ; Az A R ; Bz A R Gz A R1m . T and s represent the transformation matrix and the order of DO with s Z n respectively. Moreover, the matrixes Az ; Bz ; Lz ; C z ; Dz ; Gz and T satisfy the Luenberger equations: TA Az T ¼ Lz C; TB Lz D ¼ Bz ; C z T þ Gz C ¼ C; Gz D þ D ¼ Dz :
ð15Þ
To simplify the design process, the residual vector can be defined in the following form: ^ rðkÞ ¼ VðyðkÞ yðkÞÞ;
ð16Þ
then the form of DO can be transferred as follows: zðk þ 1Þ ¼ Az zðkÞ þ Bz iðkÞ þ Lz yðkÞ; ^ z iðkÞ; rðkÞ ¼ G^ z yðkÞ C^ z zðkÞ D
TA Az T ¼ Lz C; TB Lz D ¼ Bz ; ^ z ¼ G^ z D; D C^ z T ¼ G^ z C:
ð17Þ
ð18Þ
Residual generator based on parity space approach: Parity space was proposed and modified by Chow and Willsky and has been widely used in FDI during these years [17]. In general, the system can be described by recursive method in [18]: yðkÞ ¼ CAs 1 xðk s þ 1Þ þ CAs 2 Biðk s þ 1Þ; þ ⋯ þ CBiðk 1Þ þ DiðkÞ:
ð19Þ
The inputs and outputs can be built in the following data structure: 2 3 yðk sÞ 6 7 6 yðk s þ 1Þ 7 7 A Rsm ; ys ðkÞ ¼ 6 6 7 ⋮ 4 5 yðkÞ 2 3 iðk sÞ 6 7 6 iðk s þ 1Þ 7 7 A Rsl : ð20Þ is ðkÞ ¼ 6 6 7 ⋮ 4 5 iðkÞ Besides, if we define Λ; Φ as 2 2 3 C D 0 6 CB 6 CA 7 D 6 7 Λ¼6 6 7; Φ ¼ 6 4 ⋮ 5 s;i 4 ⋮ ⋱ CA
A large number of proper methods are available to generate the residual signals in industries for their crucial function. Two respective approaches of the methods will be discussed based on diagnostic observer (DO) and parity space [16]. Residual generator based on DO approach: The full state observer equations can be described as follows:
s
^ z ¼ VDz ; all the matrices satisfy the where G^ z ¼ VðI Gz Þ; C^ z ¼ VC z ; D Luenberger equations:
s
CA
s1
B
⋯
follows: 3 ⋯ 0 ⋯ ⋮7 7 7: ⋱ ⋮5 CB
ð21Þ
C
The relationship between system inputs and outputs can be expressed with the past plant inputs iðkÞ and past state vectors xðk s þ 1Þ by the following form: ys ðkÞ ¼ Λs xðk s þ 1ÞΦs;i is ðkÞ:
ð22Þ
Since we assume the system is observable and s 4 n, we can obtain a parity vector αs ð a 0Þ A R1ðs þ 1Þm , satisfying that
αs Λs ¼ 0:
ð23Þ
All the vectors that meet the condition above are defined as parity vectors. The set of the vectors that can be described by Ψ ¼ α s j j α s Λs ¼ 0 ; ð24Þ is named as parity space. If αs times both sides of (22) simultaneously, the residual signal sequence can be expressed in the following form: rðkÞ ¼ αs ðys ðkÞ Φs;i iðkÞÞ:
ð25Þ
The relationship between DO and parity space: Even the methods of diagnostic observer (DO) and the parity space are expressed in different forms, the relationship between them has been established quiet well [16,19]. For a known parity vector αs , which can be described as αs ¼ ½αs;0 αs;1 ⋯αs;s ; αs;i A Rm ; i ¼ 0; 1; ⋯; s, the parameter matrixes of the DO can be decided as follows [12]: 2 3 0 0 ⋯ 0 61 0 ⋯ 07 6 7 Az ¼ 6 7; 4⋮ ⋱ ⋱ ⋮5 0
⋯
1
0
C. Duan et al. / Neurocomputing 171 (2016) 1013–1020
2
3
αs;0 6 7 6 αs;1 7
L ¼ 6 6 4 2
⋮
αs;s 1
[21]. Furthermore, the variable selection method based on PLS should be applied to select the practical inputs during this process. Algorithm 1 Subspace identification method
7; 7 5
Step 1. Construct Γ f ; Γ p :
3
αs;1 αs;2 ⋯ αs;s 1 αs;s 6 7 0 6 αs;2 ⋯ ⋯ 7
T ¼6 6 ⋮ 4
αs;s
2
⋯ 0
αs Φs;0
6 6 αs Φs;1 Bz ¼ 6 6 ⋮ 4
αs Φs;s 1
3
⋯
⋮
⋯
⋯
"
Γf ¼
7; ⋮ 7 5 0
7 7 7 Λs : 7 5
C^ z ¼ ½0⋯ 01 A Rss ; G^ z ¼ αs;s A Rm ; ^ z ¼ αs Φs;s ; Φs;i ¼ ½Φs;0 ⋯Φs;s : D
ð26Þ
⋮
7 5;
þ
If
Φs;w wf þ vf 0
:
ð31Þ
Λs Φs;i 0
I
;
Xi If
ð32Þ
#
Γ Tp :
ð33Þ
ð34Þ
in which I z and V z A Rsl ðs þ mÞsl ðl þ mÞ . In addition, Δz;1 is diagonal matrix with sf l þ n nonzero eigenvalues. In other words " # rankðΔz;1 Þ ¼ rankð
Λs Φs;i 0
I
Þ ¼ sf l þ n;
ð35Þ
Δz;1 is also a diagonal matrix with ðsf m nÞ zero eigenvalues. Hence, Iz can be described as follows: " # I z;11 I z z; 12 Iz ¼ ; ð36Þ I z;22 I z;21 in which I z;11 A Rsl mðsl ml þ nÞ , I z;12 A Rsl mμ , I z;22 A Rsl μ , sl 1 ¼ s, μ ¼ sl m n. Therefore, the value of n can be calculated. ? ? Step 4. Define Λs and Λs Φs;u as the following forms:
Λs? ¼ I Tz;12 ; Λs? Φs;i ¼ ITz;22 :
ð37Þ
Step 5. Do SVD on Λs? , we can get the following equation: h
i
Λs? ¼ I Λs? ΔΛs? 0 V TΛs? ; ð28Þ
I p ¼ ½ip ðkÞið k þ1Þ⋯ip ðk þ N 1Þ; I f ¼ ½if ðkÞif ðk þ 1Þ⋯if ðk þ N 1Þ;
I
#
Step 3. Do SVD on ð1=NÞΓ f Γ Tp : " # Δz;1 0 1 T Γ Γ ¼ Iz VT ; Δz;2 z 0 N f p
3
yðk þ sf Þ 2 3 wðkÞ 6 7 ⋮ wf ðkÞ ¼ 4 5; wðk þ sf Þ 2 3 vðkÞ 6 7 ⋮ vf ðkÞ ¼ 4 5; vðk þ sf Þ
;
Xi
We can choose the past data of Γ p to reduce the impact of noise here. T Step 2. Calculate Γ f Γ p : if the length of data N is large enough, it is reasonable to make the conclusion that
Γ f Γ Tp
ð27Þ
yðkÞ yðkÞ
0
# "
so it is obtained that " # "
According to the state space of the process listed in problem formulation and the relationship between DO and parity space mentioned above, it is reasonable and convincing to construct data structures as follows: 2 3 iðk sp Þ 6 7 ⋮ ip ðkÞ ¼ 4 5; iðkÞ 2 3 iðkÞ 6 7 ⋮ if ðkÞ ¼ 4 5; iðk þ sf Þ 2 3 yðk sp Þ 6 7 ⋮ yp ðkÞ ¼ 4 5;
6 yf ðkÞ ¼ 4
Λs Φs;i
1 T lim ðΦs;w wf þvf ÞΓ p ¼ 0; N
4.2. Realization of the parity space approach
2
1017
ð29Þ
where N is parameter defined by users and always much larger than s, which is also user-defined parameters. Besides, both sp and sf are larger than n. To simplify the problem, we suppose sp ¼ sf ¼ s. Therefore, combining the equations in Eqs. (28) and (29) with Hankel structure, we can complete the construction of matrixes Γ p and Γ f for the process inputs and outputs: " # Yp Γ p ¼ I A Rðsp l þ sp m þ l þ mÞN ; p " # Yf Γ f ¼ I A Rðsp l þ sp m þ l þ mÞN : ð30Þ f The subspace identification method (SIM) provided by Wang and Qin will be introduced in algorithm form to identify the parity space
ð38Þ
in which V Λ ? ¼ ½V Λ ? ;1 V Λ ? ;2 , V Λ ? ;2 A Rðsl þ 1Þmn ; so the following s s s s equation about Λs can be obtained :
Λs ¼ V Λs? ;2 :
ð39Þ
From the necessary conditions shown in [21,22], we can use SIM method to identify the parity space. On the condition that d and n are absolute white noise, Algorithm 1 can be employed to ? ? identify Λs ; Λs Φs;i , and Λs . During the identification process, PLS method discussed above is applied to analyze the relationships between the input data and reduce its quantity. As a result, it is convincible that the design of residual generator design can be simplified and optimized for dealing with the inputs and erroneous data. However, when the condition does not meet the requirements discussed above, the following method should be applied to identify n, which is from [31]. Suppose that the order of a system n A ½0⋯20, then the order of the system is the one that makes the AIC index minimum: AIC N ðnÞ ¼ Nðmð1 þ ln 2π Þ þ lnjΩjÞ þ2δΥ ;
ð40Þ
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^ design the state equation of the filter DðzÞ as follows:
in which
Ω¼
xðk þ 1Þ ¼ Ax xðkÞ þ Lx iðkÞ;
N 1X eðkÞeðkÞT ; Ni¼1
yðkÞ ¼ G 1 C x xðkÞ þ iðkÞ:
^ eðkÞ ¼ yðkÞ yðkÞ ? T ¼ ðΛs Þ I Φs;i zf ðk nÞ;
(2) Parameter matrix design: Most work has been finished to design the controller except the parameter matrix Q ðzÞ, which is based on the system performance. It is apparent that the system error can be defined as follows:
mðm þ 1Þ þ nl þ ml; 2 N : δn ¼ N ððΥ =mÞ þ ððm þ1Þ=2ÞÞ
Υ ¼ 2 nm þ
ð41Þ
eðzÞ ¼ λðzÞ yðzÞ:
ð48Þ
The dynamics of the system can be described by the following equation: ^ MðzÞyðzÞ ¼ VðzÞwðzÞ þ WðzÞSðzÞ;
4.3. Basic residual generator design ? s ;
? s
According to the result in the subsection above, the Λ Λ Φs;i can be identified and residual generator can be constructed with the parity space method. ? Select a parity vector αs A Rmðs þ 1Þ from Λs which satisfies that αs ¼ αs;0 αs;1 ⋯αs;s ; αs;i A Rm ; i ¼ 0; 1; …; s: ð42Þ Then, we can calculate the parameter matrixes of DO ^ z according to Eqs. (26) and (27) and Az ; Bz ; L; G^ z ; C^ z , and D construct a DO by solving (13). The residual signal as follows can be obtained by the residual generator [16]: ^ rðkÞ ¼ Q ðyðkÞ yðkÞÞ A Rx ;
ð47Þ
ð43Þ
^ is an in which Q A Rxm denotes a parameter matrix and yðkÞ estimated value of the process outputs. To enhance the performance of the system in order to use it in more widespread industrial area, it is necessary to construct a residual generator that can transmit an m-dimensional residual vector to meet the higher requirement of the system. As a result, extending the order of residual generator based on data-driven method from single one to multiple one is also applied in the practical industry [18].
ð49Þ
where ^ VðzÞ ¼ N^ ðzÞQ ðzÞMðzÞ; ^ WðzÞ ¼ I N ðzÞQ ðzÞ; SðzÞ ¼ ðICðzI AÞ 1 ÞmðzÞ:
ð50Þ
Therefore, the problem of designing the parameter matrix has become a norm optimization one that is relatively easy to solve. In general, the parameter matrix Q(z) should be not only simple enough to make itself practical, but also suitable for both the performance and the stability of the whole system. Generally speaking, Q ðzÞ can be chosen as constant matrixes to meet with the requirements above. We can assume that U ¼ Dx þ C x ðI Ax Þ 1 Bx :
ð51Þ
Since U is an invertible one, the parameter matrix Q(z) is Q ¼ U 1 G;
ð52Þ
otherwise, Q ¼ U G; with ðYÞ
ð53Þ
denoting the pseudo-inverse of the matrix Y.
6. An academic example 5. Residual generator application As discussed in the sections above, the variable selection aided residual generator design is completed. The following work will be dedicated to its application in control and monitoring. 5.1. Residual generator embedding into control loop (1) Filter design: According to the classical and modern control theory and their relationship, the residual can be described in following form: ^ rðzÞ ¼ MðzÞyðzÞ N^ ðzÞiðzÞ;
ð44Þ
In this section, a numerical example will be given to demonstrate the practicability and effectiveness of the controller based on residual generator method developed in this paper. Since we only need the system model to generate the inputs and outputs which is indispensable as the test data, the system matrixes A,B,C,D and the noises will not influence the result of controller design. Assume the system matrixes as follows: mðkÞ Nð0; 0:0012 Þ; nðkÞ Nð0; 0:012 Þ; 2 3 2 0 0 0:0058 0:97 6 7 6 0:12 5; B ¼ 4 0:19 A ¼ 4 0:95 0:01 0:01
0:97
0:58
0:51
3
7 0:31 5; 0:79
where
C ¼ ½1:1
^ MðzÞ ¼ I CðzI AL Þ 1 L; AL ¼ A LC; ^ NðzÞ ¼ D þ CðzI AL Þ 1 BL ; BL ¼ B LD:
where Nðμ; σ 2 Þ denotes the normal distribution with the mean μ and the variance σ 2 . The simulation process of the controller based on data-driven will be described as follows:
ð45Þ
Combining the state equations of multidimensional residual generator, the equations above can be expressed in another form: ^ MðzÞ ¼ I G 1 C x ðzI Ax Þ 1 Lx ; ¼ I CðzI AF Þ 1 L; ¼ D þ CðzI AF Þ
1
BL :
ð46Þ
Therefore, the matrix that satisfies the equations above can be applied as the filter in the system described in Fig. 4. We can also
ð54Þ
Step 1. PLS method Application: Apply the PLS method to analyze
^ NðzÞ ¼ G 1 ðDx þ C x ðzI Ax Þ 1 Bx Þ;
1:65
0:55; D ¼ zerosð1; 2Þ;
0:01
the inputs during the identification process to reduce the number of them. Step 2. Model Pretreatment: Simulate the academic example with the program and collect 1000 samples as test data under the reference stimulation. Step 3. Identify the Parity Space: The method of identifying the parity space has been shown in Algorithm 1.
C. Duan et al. / Neurocomputing 171 (2016) 1013–1020
1.2
1019
0.35 0.3
1
0.25
0.8 1.1
1.1
1.05
1.05
1
1
0.95 450
0.95 550
0.2
0.6
0.15 0.1
0.4 0.2
0.05 0
0 −0.2
500
550
600
650
−0.05 −0.1
0
200
400
600
800
1000
0
400
600
800
1000
Fig. 8. The residual generator s ¼9.
Fig. 5. The output of system s ¼ 3.
Step 5. Time Setting: Considering the quality and reliability of
0.4 0.35
0.3 0.25 0.2 0.15 0.1 0.05 0 −0.05
200
0
200
400
600
800
1000
Fig. 6. The residual generator s ¼3.
1.2 1 0.8 1.1
1.1
1.05
1.05
1
1
0.6
the simulation, we set the simulation time as T ¼ 1000 s and sample time as t ¼ 1 s to demonstrate the performance and application of controller. Step 6. Interference Introduce: After finishing the work above, the interferences mðkÞ ¼ nðkÞ ¼ 0:1 should be introduced at the time t ¼ 500 s and t ¼ 600 s respectively to verify the controller that we have designed.
Fig. 5 shows the output signal obtained from experiment with the condition that s¼n. Fig. 6 displays the data of residual generator on the same condition. It is obvious that the performance of the controller is good enough from the aspect of the response speed and disturbances attenuation. Without loss of generality, we also simulate a system in which s ¼ 9 4 n as another example. The output signal is shown in Fig. 7 and the data of residual generator is displayed in Fig. 8 From the figures above, it is easy to learn that the disturbances can be resistant effectively. However, comparing Figs. 5 and 7, we realize that the overshoot of the latter one will be a bit larger when the reference stimulation is a unit step signal. Therefore, the relationship of size between s and n is significant for the performance of controller. With the method of optimizing the residual generator based on PLS given in this paper, it becomes possible to determine the parameter s in the SIM method more precisely. As a result, ensuring s¼n leads to a better performance and practical design of controller.
0.4 0.2
0.95 450
0 −0.2
0
200
500
550
400
0.95 550
7. Conclusion
600
600
650
800
1000
Fig. 7. The output of system s ¼ 9.
Step 4. Residual Generator Design: According to the relationship between the parity space and DO, the residual generator based on observer can be constructed without any problems, which will be the advanced work of designing controller.
In this paper, we have studied the problem of residual generation based on data-driven approach and its optimization by applying the PLS method to the SIM segment. Furthermore, the residual generator can be employed in the process of control and monitoring after the realization of the Youla parameterization for the feedback control loop and other relevant work. The design of residual generator is based on the relationship between DO as well as the parity vector, and PLS method is utilized to analyze the relationships among the input variables and decrease the number of them at the same time. The applications of data-driven approach and PLS method make the residual generator proposed avoid the difficulties of building models and more effective to deal with big data in modern industry. Moreover, a numerical example
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demonstrates the performance and effectiveness of the residual generator design approach for the process control and monitoring. References [1] X. Zhao, L. Zhang, P. Shi, M. Liu, Stablilization of switched linear systems with mode-dependent average dwell time, IEEE Trans. Autom. Control 57 (7) (2012) 1809–1815. [2] A. Lopez, D. Patino, R. Diez, G. Perilla, An equivalent continuous model for switched systems, Syst. Control Lett. 66 (2) (2013) 124–131. [3] Z. Feng, L. James, Reliable dissipative control for singular Markovian systems, Asian J. Control 15 (3) (2013) 901–910. [4] L. Boccardo, L. Orsina, A variational semilinear singular system, Nonlinear Anal. 74 (12) (2011) 3849–3860. [5] J. Xu, Z. Hou, Notes on data-driven system approaches, Acta Autom. Sin. 35 (6) (2009) 668–675. [6] T. Yamamoto, K. Takao, T. Yamada, Design of a data-driven PID controller, IEEE Trans. Control Syst. Technol. 17 (1) (2009) 29–39. [7] S. Wakitani, K. Nishida, M. Nakamoto, T. Yamamoto, Design of a data-driven PID controller using operating data, IFAC Proc. 11 (2013) 587–592 . [8] P. Brito, C. Vieira, G. Sousa, R. Neves-Silva, Process control based on PCA models, in: 2010 IEEE 15th Conference on Emerging Technologies & Factory Automation, Bilbao, 2010. [9] V. Nguyen, J. Golinval, Fault detection based on Kernel principal component cnalysis, Eng. Struct. 32 (11) (2010) 3683–3691. [10] G. Lee, C. Han, E. Yoon, Multiple-fault diagnosis of the Tennessee Eastman process based on system decomposition and dynamic PLS, Ind. Eng. Chem. Res. 43 (25) (2004) 8037–8048. [11] R. Muradore, P. Fiorini, A PLS-based statistical approach for fault detection and isolation of robotic manipulators, IEEE Trans. Ind. Electron. 59 (8) (2012) 3167–3175. [12] S. Ding, P. Zhang, A. Naik, E. Ding, B. Huang, Subspace method aided datadriven design of fault detection and isolation systems, J. Process Control 19 (9) (2009) 1496–1510. [13] S. Ding, N. Weinhold, P. Zhang, E. Ding, T. Jeinsch, M. Schultalbers, Integration of FDI functional units into embedded tracking control loops and its application to FDI in engine control systems, in: Proceedings of the IEEE International Conference on Control Applications, Toronto, Canada, August 2005, pp. 1299–1304 . [14] S. Ding, G. Yang, P. Zhang, E. Ding, T. Jeinsch, N. Weinhold, M. Schultalbers, Feedback control structures, embedded residual signals, and feedback control schemes with an integrated residual access, IEEE Trans. Control Syst. Technol. 18 (2) (2010) 352–366. [15] S. Ding, P. Zhang, E. Ding, S. Yin, A. Naik, P. Deng, W. Gui, On the application of PCA technique to fault diagnosis, Tsinghua Sci. Technol. 15 (2) (2010) 138–144. [16] S. Ding, Mode-Based Fault Diagnosis Techniques, Springer, Berlin, Germany, 2008. [17] E. Chow, A. Willsky, Analytical redundancy and the design of robust failure detection systems, IEEE Trans. Autom. Control 29 (7) (1984) 603–614. [18] S. Yin, Data-driven design of fault diagnosis systems, VDIVerlag, Düsseldorf, Germany, 2012. [19] P. Zhang, S. Ding, Disturbance decoupling in fault detection of linear periodic systems, Automatica 43 (8) (2007) 1410–1417. [20] S. Yin, G. Wang, H. Karimi, Data-driven design of robust fault detection system for wind turbines, Mechatronics 24 (4) (2014) 298–306. [21] J. Wang, S. Qin, A new subspace identification approach based on principal component analysis, J. Process Control 12 (8) (2002) 841–855. [22] J. Wang, S. Qin, Closed-loop subspace identification using the parity space, Automatica 42 (2) (2006) 315–320. [23] G. Ma, Z. Yang, Q. Hu, Multiobjective optimum control based on finite dimensional Youla parameterization, Electr. Mach. Control 8 (4) (2004) 345–388. [24] N. Weinhold, S. Ding, T. Jeinsch, M. Schulalbers, Embedded model-based fault diagnosis for on-board diagnosis of engine management systems, in: Proceedings of 2005 IEEE Conference on Control Applications, Toronto, Canada, 2005, pp. 1206–1211.
[25] R. Gerlach, B. Kowalski, H. Wold, Partial least-squares path modeling with latent variables, Anal. Acta 112 (1979) 411–427. [26] A. Höskuldsson, Variable and subset selection in PLS regression, Chemometr. Intell. Lab. Syst. 55 (1) (2001) 23–38. [27] D. Haaland, E. Thomas, Partial least-squares methods for spectral analyses. 2. Application to simulated and glass spectral data, Anal. Chem. 60 (11) (1988) 1202–1208. [28] H. Li, X. Jing, H. Karimi, Output-feedback-based H-infinity control for vehicle suspension systems with control delay, IEEE Trans. Ind. Electron. 61 (1) (2014) 436–446. [29] G. Conte, A course in H1 Control Theory, IEEE Trans. Autom. Control 32 (12) (1987) 1144–1145. [30] P. Thomas, Fault detection and diagnosis in engineering systems, Control Eng. Pract. 10 (9) (2002) 1037–1038. [31] J. Wang, S. Qin, A new subspace identification approach based on principal component analysis, J. Process Control 12 (8) (2002) 841–855.
Chaojing Duan was enrolled in Harbin Institute of Technology (HIT), Harbin, China, in 2012, and is studying toward B.E. degree in Control science and Engineering in Honors School of HIT. Her research interests involve data-driven fault-diagnosis, robust control and their applications.
Zhongyang Fei received the B.E. degree and the M.S. degree in Control Science and Engineering from Harbin Institute of Technology, PR China in 2007 and 2009, respectively; and the Ph.D. degree in Mechanical Engineering & Material Science Department, Washington University in St. Louis, USA in 2013. He joint Harbin Institute of Technology in 2014. His research interests involve complex networks, robust control, time-delay systems, fault detection and diagnosis.
Jiachen Li was enrolled in Harbin Institute of Technology, Harbin, China, in 2012, and is currently studying toward the B.E. degree in Control theory and Engineering in Honors School of HIT. His research interests include data-driven fault diagnosis and prognosis in process control and their applications.
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
A variable selection aided residual generator design approach for process control and monitoring$ Chaojing Duan a, Zhongyang Fei b,n, Jiachen Li a a b
Honors School, Harbin Institute of Technology, Harbin 150001, PR China Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 April 2015 Received in revised form 3 June 2015 Accepted 15 July 2015 Communicated by Xudong Zhao Available online 29 July 2015
This paper investigates the design of residual generator based on data-driven techniques, which is optimized by variable selection and can be applied to control and monitoring in industrial process to satisfy the high performance requirements of systems. The basic idea is designing the residual generator based on the foundation of the Luenberger equations as well as the relationship between diagnosis observer (DO) and the parity vector. The crucial and innovative part of the scheme is to acquire the parity space by the Subspace Identification Method (SIM) and obtain the simplified and optimized data from a new variable selection system according to partial least squares (PLS) method during the procedure of SIM. The design and optimization of the residual generator make it independent of model and effective to deal with big data, and avoid the challenges of handling bulk data. After the realization of the Youla parameterization, the proposed approach of residual generator can not only be applied to control purposes, but also monitoring objective of practical industrial systems. A numerical example is used to demonstrate the performance and effectiveness of the proposed scheme. & 2015 Elsevier B.V. All rights reserved.
Keywords: Variable selection Patrial least square Residual generator Control Monitoring
1. Introduction During the past two decades, the techniques of model-based controller design have been established and greatly developed in different systems, such as switched system, Markovian system, singular system, etc [1–4]. Moreover, the field of model based fault detection and isolation techniques have been widely investigated during this period, especially for linear time invariant (LTI) systems [12,13]. However, with the development of industrial techniques, the requirements on product quality and system performance increase significantly. Since the industry process is becoming more and more complicated, to establish a model for the whole system is arduous in most applications and to employ the high-dimensional model is even impractical in pragmatic systems. In another hand, a lot of data has been recorded and tremendous information of the system could be extracted with the advanced technology of real data, which will be definitely
☆ This research is partially supported by the Fundamental Research Funds for the Central Universities (AUGA5710056615) and by China Postdoctoral Science Foundation (AUGA4110033015). n Corresponding author. E-mail addresses: [email protected] (C. Duan), [email protected] (Z. Fei), [email protected] (J. Li).
http://dx.doi.org/10.1016/j.neucom.2015.07.042 0925-2312/& 2015 Elsevier B.V. All rights reserved.
beneficial to improve the performance reliability and system safety. The study of designing and updating the controllers by the input as well as output of the system has attracted widespread attention from industry and academia. The main approaches of data-driven system can be divided into 3 categories according to the utilization of data: on-line data, off-line data and the combination of both [5]. Therefore, some data-driven approaches have nowadays gained more attention and improvement. For instance, proportionalintegral-derivative (PID) method may be the earliest and the most widely used technique in industrial processes [6,7,20]. Meanwhile, approaches in multivariate statistical analysis, which can deal with abundant highly correlated measured data, have been modified since it was applied into actual industry. For example, principal component analysis (PCA) method is designed to decrease dimensional principal components [8,9], and partial least squares (PLS) method can predict key indicator from processes measurements directly [11,10]. It is reasonable that the more intricate and multifunctional the industrial process is, the more easily and frequently the fault will occur. Since the fault not only impacts the efficiency and quality of production, but also threatens the security of industrial manufacture, detecting and isolating the fault while it exists in the system are important for modern industry. However, it is very difficult to establish quantitative model for large-scale industrial process with
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model based FDI method. Furthermore, the challenges of big data are unable to be solved effectively by traditional mathematic and statistical approaches. This dilemma motivates the investigation of new techniques that can be more independent of model and more effective to utilize and deal with big data. In addition, the noteworthy development of the PLS method and subspace identification methods (SIM) which can generate parity vector directly from process measurements inspire the study in this paper. An approach of data-driven control and monitoring for LTI system and the technique to optimize it will be proposed in this paper. The foundation of this achievement is embedding a residual generator with the purpose of FDI in the feedback control loop. The main contribution of this paper will be divided into two parts. One is a variable selection procedure according to PLS method, which can simplify the SIM process by analyzing the connections between the inputs and reducing the quantity among them. The other is constructing a residual generator based on the relationship between diagnosis observer (DO) and the parity vector [14]. The rest of this paper is organized as follows. In Section 2.1, the basic description of system and preliminary factorization are presented. A variable selection approach based on PLS method is introduced in Section 3. In Section 4, the approaches of DO and parity space are explained respectively firstly, then the method of residual generator design is analyzed and provided. The method of the residual generator application in both control and monitoring is discussed in Section 5. To illustrate the performance of the design, a numerical example on the model is presented in Section 6. Finally, some conclusions of this paper are made in Section 7.
2. Problem formulation
The linear time invariant (LTI) system in discrete state space can be described by
ð2Þ
where GðzÞ and G1 ðzÞ can be described from the relationship between the transfer function matrix and the state space equation.
ð4Þ
^ ^ with MðzÞ and NðzÞ defined as follows: ^ MðzÞ ¼ I CðzI AF Þ 1 F; ^ NðzÞ ¼ M þ CðzI AF Þ 1 BF ;
ð5Þ
in which AF ¼ A FC and BF ¼ B FM. 2.3. Youla parameterizations According to the Youla Parameterization [29,30], all the stabilizing controllers V(z) can be expressed in a unified form with a classical output feedback control loop. Suppose an appropriate parameter matrix V(z) that makes ðA þ BV Þ stable, which satisfies the following equation: 1 ^ ^ ^ VðzÞ ¼ ðX^ ðzÞ Q ðzÞNðzÞÞ ðY ðzÞ Q ðzÞMðzÞÞ;
ð6Þ
in which X^ ðzÞ ¼ I V ðzI AF Þ 1 BF ; Y^ ðzÞ ¼ VðzI AF Þ 1 F:
ð7Þ
It is reasonable to assume that the matrix V, which is stable to simplify the controller design, is equal to zero, so a simplified controller can be obtained from (6): ð8Þ
The equation about the residual vectors and the inputs of the process can be governed by ^ rðzÞ ¼ MðzÞyðzÞ N^ ðzÞiðzÞ; iðzÞ ¼ VðzÞðλðzÞ yðzÞÞ:
ð1Þ
where the plant inputs defined as iðkÞ A Rl and outputs defined as yðkÞ A Rm at every discrete time point. xðkÞ A Rn stands for the status variables, mðkÞ A Rn and nðkÞ A Rm denote white noise with uncertainty. However, the system matrices and orders are all unknown parameters. The charter for the classical output feedback control method is shown in Fig. 1 [28]. According to Fig. 1, the transfer function can be written into yðzÞ ¼ GðzÞiðzÞ þ G1 ðzÞmðzÞ þ H n nðzÞ;
^ 1 ðzÞN^ 1 ðzÞ; GðzÞ ¼ M
^ VðzÞ ¼ ðI Q ðzÞN^ ðzÞÞ 1 Q ðzÞMðzÞ:
2.1. System descriptions
xðk þ 1Þ ¼ AxðkÞ þ BiðkÞ þH m mðkÞ; yðkÞ ¼ CðxÞ þDiðkÞ þ H n nðkÞ;
^ N^ are left coprimes to each other. Assume then the matrixes M; that there is a proper matrix of parameters F and all the eigenvalues of the matrix ðA FCÞ are less than 1. Then, a left coprime factorization of the system transfer function matrix G(z) can be given by
ð9Þ
Combining Eqs. (8) and (9), we have ^ ^ λðzÞ ðMðzÞyðzÞ N^ ðzÞiðzÞÞÞ: iðzÞ ¼ Q ðzÞðMðzÞ
ð10Þ
Then the feedback control loop is restructured and arranged as sketched in Fig. 2. By rearranging the blocks, the feedback control system can be arranged as shown in Fig. 3.
3. Variable selection based on PLS
^ N^ ; X^ and Y^ satisfy the following relationship: If matrixes M; " # h i ^ ^ N^ X ¼ M ^ X^ þ N^ Y^ ¼ I; M ð3Þ Y^
As discussed previously, the number of data which is significant for controlling and monitoring system performance will be quite large during the present practical industrial process. Too much data hampers the residual generator design inevitably or even make it impossible. On the other hand, some errors or nonsignificant data which will impact the effectiveness of the residual generator. As a result, a variable selection method is necessary to reduce redundant information and useless variables, not only for decreasing the number of inputs simplifying the computation during the construction of residual generator, but also for improving the stability and performance of it [25–27].
Fig. 1. The classical system.
Fig. 2. Youla parameterization.
2.2. Left coprime factorization
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Fig. 3. Arrangement of control system.
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3.1.2. Root mean square error The root mean square error (RMSE) is generally used to calculate the differences between the real values and their predicted ones. Fundamentally, the RMSE index indicates the sample standard deviation of the differences, which are also called residuals on the condition that the measurements are performed over the data sample. During this work, the RMSE index can be computed as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN ^ 2 i ¼ 1 ðyi y i Þ RMSE ¼ ; ð11Þ N in which yi is the i-th element of the real output variables and y^ i denotes the corresponding predicted value. 3.2. Input variables classification The input variables need to be classified according to their weighting in the regression vector obtained. The weight stands for the significance of the corresponding input variable, and each input variable corresponds to one weighting in the regression vector. As a result, the absolute values of related elements in regression vector will be the basis to decide how to realign the original input variables and the construction of new input variable matrix will be realized at the same time. For instance, if the regression vector is a row vector, then the elements should be realigned in descending order from left to right. Otherwise, the order will be from top to bottom. 3.3. Test procedure correlation
Fig. 4. Variable selection procedure.
The further step, which is aimed at examining the correlations between variables, should be taken after the realignment of the input variable matrix. The examination should begin from the least crucial variable to the most significant one, which means it is from the last column of realigned input variable matrix to the first column. During this process, if a certain input variable is highly correlated with others, then it will be cancelled. This work can be described as follows:
Step 1. Obtain the correlation coefficient matrix of realigned input variable matrix from the following equations: The data pre-processing with PLS-based variable selection will find out the characteristics of the input data matrix and PLS regression, which provides an effective method for multiple linear regression modeling. After that, the variable selection method proposed will be divided into four parts, which are displayed in Fig. 4.
Cði; jÞ ¼ E½X i EðX i Þ½X j EðX j Þ; Cði; jÞ Rði; jÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: Cði; iÞCðj; jÞ
Step 2. Compare the absolute values of elements in the
3.1. The regression vector generation The regression vector for sorting variables, which is crucial for the success of variable selection, can be derived from PLS regression. Meanwhile, the number of latent variables property that will influences the regression vector should be decided to find out the correct complexity of model. The method of cross-validation (CV) and root mean square error (RMSE) are applied to select proper latent variables so that the risks of “over-fitting” and “underfitting” can be avoided effectively. 3.1.1. Cross validation In general, not all the existing m components t 1 ; t 2 ; …; t m but just the first l components are used to build the regression model. The function of CV method is to select the latent variables. The i-th experimental data will be abandoned and the rest data will be used to make a PLS regression model each time.
ð12Þ
correlation coefficient matrix with each other. In the situation that the differences between two elements are less than 0.1, one of the corresponding input variables with less weighting in regression vector would be cancelled. Step 3. Repeat Step 2 until all of the elements have been considered.
3.4. Subset selection with the best estimation performance In above steps, the uncorrelated input variable matrix has been obtained. The next step is to calculate the estimation performance of each subset and complete the subset selection with the best estimation performance [15]. The uncorrelated input variables are aligned and several variable subsets are selected according to their weightings in the regression vector and priorities, respectively. Based on the results that have been achieved, the subset selection can be implemented with the following steps.
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Step 1. Define the initial subset which has ki columns in X or . For
instance, if ki is given as 4, the initial subset X 1 consists of the first four columns of X or . Step 2. Expand the second subset X 2 by adding a fixed number of variables kj . For example, if kj is 3, then the second subset X 2 will include the most seven significant input variables in X or . Step 3. Iterate the steps above till all the columns in X or are taken into consideration. At the same time, the PLS regression model is constructed and the corresponding RMSE index is calculated, which is combined with CV method for each selected subset.
So far, all of the variable subsets have been evaluated by the RMSE indices. Besides, the subset with the smallest RMSE value represents the best prediction performance. Meanwhile, the variables in subset should be the selected ones. What has been discussed is the core of the variable selection method based on PLS method. It can characterize the appropriate complexity of fitting and prediction model in residual generator design. Furthermore, the variable selection method will be applied to simplify the construction of residual generator in the following sections.
4. Residual generator design Residual is supposed to be the error between the measured and the predicted values, which is significant for the system process because of the important and fundamental information of the system model it contains. The residual generator can be designed directly by the input and output data without identification of system parameters and the generator can also be inserted to the feedback control system to be applied in both control and monitoring in practical industry [13,23,24]. In this section, the method of residual generator design will be given. 4.1. Residual signals generation
^ þ BiðkÞ þ LðyðkÞ yðkÞÞ; ^ ^ þ 1Þ ¼ AxðkÞ xðk ^ þ DiðkÞ: yðkÞ ¼ C xðkÞ
ð13Þ
The state space equations of the method based on DO can be expressed by zðk þ 1Þ ¼ Az zðkÞ þ Bz iðkÞ þ Lz yðkÞ; ^ yðkÞ ¼ C z zðkÞ þ Dz iðkÞ þGz yðkÞ; ss
sm
ð14Þ sm
ls
1l
; Lz A R ; C z A R ; Dz A R and in which z A R ; Az A R ; Bz A R Gz A R1m . T and s represent the transformation matrix and the order of DO with s Z n respectively. Moreover, the matrixes Az ; Bz ; Lz ; C z ; Dz ; Gz and T satisfy the Luenberger equations: TA Az T ¼ Lz C; TB Lz D ¼ Bz ; C z T þ Gz C ¼ C; Gz D þ D ¼ Dz :
ð15Þ
To simplify the design process, the residual vector can be defined in the following form: ^ rðkÞ ¼ VðyðkÞ yðkÞÞ;
ð16Þ
then the form of DO can be transferred as follows: zðk þ 1Þ ¼ Az zðkÞ þ Bz iðkÞ þ Lz yðkÞ; ^ z iðkÞ; rðkÞ ¼ G^ z yðkÞ C^ z zðkÞ D
TA Az T ¼ Lz C; TB Lz D ¼ Bz ; ^ z ¼ G^ z D; D C^ z T ¼ G^ z C:
ð17Þ
ð18Þ
Residual generator based on parity space approach: Parity space was proposed and modified by Chow and Willsky and has been widely used in FDI during these years [17]. In general, the system can be described by recursive method in [18]: yðkÞ ¼ CAs 1 xðk s þ 1Þ þ CAs 2 Biðk s þ 1Þ; þ ⋯ þ CBiðk 1Þ þ DiðkÞ:
ð19Þ
The inputs and outputs can be built in the following data structure: 2 3 yðk sÞ 6 7 6 yðk s þ 1Þ 7 7 A Rsm ; ys ðkÞ ¼ 6 6 7 ⋮ 4 5 yðkÞ 2 3 iðk sÞ 6 7 6 iðk s þ 1Þ 7 7 A Rsl : ð20Þ is ðkÞ ¼ 6 6 7 ⋮ 4 5 iðkÞ Besides, if we define Λ; Φ as 2 2 3 C D 0 6 CB 6 CA 7 D 6 7 Λ¼6 6 7; Φ ¼ 6 4 ⋮ 5 s;i 4 ⋮ ⋱ CA
A large number of proper methods are available to generate the residual signals in industries for their crucial function. Two respective approaches of the methods will be discussed based on diagnostic observer (DO) and parity space [16]. Residual generator based on DO approach: The full state observer equations can be described as follows:
s
^ z ¼ VDz ; all the matrices satisfy the where G^ z ¼ VðI Gz Þ; C^ z ¼ VC z ; D Luenberger equations:
s
CA
s1
B
⋯
follows: 3 ⋯ 0 ⋯ ⋮7 7 7: ⋱ ⋮5 CB
ð21Þ
C
The relationship between system inputs and outputs can be expressed with the past plant inputs iðkÞ and past state vectors xðk s þ 1Þ by the following form: ys ðkÞ ¼ Λs xðk s þ 1ÞΦs;i is ðkÞ:
ð22Þ
Since we assume the system is observable and s 4 n, we can obtain a parity vector αs ð a 0Þ A R1ðs þ 1Þm , satisfying that
αs Λs ¼ 0:
ð23Þ
All the vectors that meet the condition above are defined as parity vectors. The set of the vectors that can be described by Ψ ¼ α s j j α s Λs ¼ 0 ; ð24Þ is named as parity space. If αs times both sides of (22) simultaneously, the residual signal sequence can be expressed in the following form: rðkÞ ¼ αs ðys ðkÞ Φs;i iðkÞÞ:
ð25Þ
The relationship between DO and parity space: Even the methods of diagnostic observer (DO) and the parity space are expressed in different forms, the relationship between them has been established quiet well [16,19]. For a known parity vector αs , which can be described as αs ¼ ½αs;0 αs;1 ⋯αs;s ; αs;i A Rm ; i ¼ 0; 1; ⋯; s, the parameter matrixes of the DO can be decided as follows [12]: 2 3 0 0 ⋯ 0 61 0 ⋯ 07 6 7 Az ¼ 6 7; 4⋮ ⋱ ⋱ ⋮5 0
⋯
1
0
C. Duan et al. / Neurocomputing 171 (2016) 1013–1020
2
3
αs;0 6 7 6 αs;1 7
L ¼ 6 6 4 2
⋮
αs;s 1
[21]. Furthermore, the variable selection method based on PLS should be applied to select the practical inputs during this process. Algorithm 1 Subspace identification method
7; 7 5
Step 1. Construct Γ f ; Γ p :
3
αs;1 αs;2 ⋯ αs;s 1 αs;s 6 7 0 6 αs;2 ⋯ ⋯ 7
T ¼6 6 ⋮ 4
αs;s
2
⋯ 0
αs Φs;0
6 6 αs Φs;1 Bz ¼ 6 6 ⋮ 4
αs Φs;s 1
3
⋯
⋮
⋯
⋯
"
Γf ¼
7; ⋮ 7 5 0
7 7 7 Λs : 7 5
C^ z ¼ ½0⋯ 01 A Rss ; G^ z ¼ αs;s A Rm ; ^ z ¼ αs Φs;s ; Φs;i ¼ ½Φs;0 ⋯Φs;s : D
ð26Þ
⋮
7 5;
þ
If
Φs;w wf þ vf 0
:
ð31Þ
Λs Φs;i 0
I
;
Xi If
ð32Þ
#
Γ Tp :
ð33Þ
ð34Þ
in which I z and V z A Rsl ðs þ mÞsl ðl þ mÞ . In addition, Δz;1 is diagonal matrix with sf l þ n nonzero eigenvalues. In other words " # rankðΔz;1 Þ ¼ rankð
Λs Φs;i 0
I
Þ ¼ sf l þ n;
ð35Þ
Δz;1 is also a diagonal matrix with ðsf m nÞ zero eigenvalues. Hence, Iz can be described as follows: " # I z;11 I z z; 12 Iz ¼ ; ð36Þ I z;22 I z;21 in which I z;11 A Rsl mðsl ml þ nÞ , I z;12 A Rsl mμ , I z;22 A Rsl μ , sl 1 ¼ s, μ ¼ sl m n. Therefore, the value of n can be calculated. ? ? Step 4. Define Λs and Λs Φs;u as the following forms:
Λs? ¼ I Tz;12 ; Λs? Φs;i ¼ ITz;22 :
ð37Þ
Step 5. Do SVD on Λs? , we can get the following equation: h
i
Λs? ¼ I Λs? ΔΛs? 0 V TΛs? ; ð28Þ
I p ¼ ½ip ðkÞið k þ1Þ⋯ip ðk þ N 1Þ; I f ¼ ½if ðkÞif ðk þ 1Þ⋯if ðk þ N 1Þ;
I
#
Step 3. Do SVD on ð1=NÞΓ f Γ Tp : " # Δz;1 0 1 T Γ Γ ¼ Iz VT ; Δz;2 z 0 N f p
3
yðk þ sf Þ 2 3 wðkÞ 6 7 ⋮ wf ðkÞ ¼ 4 5; wðk þ sf Þ 2 3 vðkÞ 6 7 ⋮ vf ðkÞ ¼ 4 5; vðk þ sf Þ
;
Xi
We can choose the past data of Γ p to reduce the impact of noise here. T Step 2. Calculate Γ f Γ p : if the length of data N is large enough, it is reasonable to make the conclusion that
Γ f Γ Tp
ð27Þ
yðkÞ yðkÞ
0
# "
so it is obtained that " # "
According to the state space of the process listed in problem formulation and the relationship between DO and parity space mentioned above, it is reasonable and convincing to construct data structures as follows: 2 3 iðk sp Þ 6 7 ⋮ ip ðkÞ ¼ 4 5; iðkÞ 2 3 iðkÞ 6 7 ⋮ if ðkÞ ¼ 4 5; iðk þ sf Þ 2 3 yðk sp Þ 6 7 ⋮ yp ðkÞ ¼ 4 5;
6 yf ðkÞ ¼ 4
Λs Φs;i
1 T lim ðΦs;w wf þvf ÞΓ p ¼ 0; N
4.2. Realization of the parity space approach
2
1017
ð29Þ
where N is parameter defined by users and always much larger than s, which is also user-defined parameters. Besides, both sp and sf are larger than n. To simplify the problem, we suppose sp ¼ sf ¼ s. Therefore, combining the equations in Eqs. (28) and (29) with Hankel structure, we can complete the construction of matrixes Γ p and Γ f for the process inputs and outputs: " # Yp Γ p ¼ I A Rðsp l þ sp m þ l þ mÞN ; p " # Yf Γ f ¼ I A Rðsp l þ sp m þ l þ mÞN : ð30Þ f The subspace identification method (SIM) provided by Wang and Qin will be introduced in algorithm form to identify the parity space
ð38Þ
in which V Λ ? ¼ ½V Λ ? ;1 V Λ ? ;2 , V Λ ? ;2 A Rðsl þ 1Þmn ; so the following s s s s equation about Λs can be obtained :
Λs ¼ V Λs? ;2 :
ð39Þ
From the necessary conditions shown in [21,22], we can use SIM method to identify the parity space. On the condition that d and n are absolute white noise, Algorithm 1 can be employed to ? ? identify Λs ; Λs Φs;i , and Λs . During the identification process, PLS method discussed above is applied to analyze the relationships between the input data and reduce its quantity. As a result, it is convincible that the design of residual generator design can be simplified and optimized for dealing with the inputs and erroneous data. However, when the condition does not meet the requirements discussed above, the following method should be applied to identify n, which is from [31]. Suppose that the order of a system n A ½0⋯20, then the order of the system is the one that makes the AIC index minimum: AIC N ðnÞ ¼ Nðmð1 þ ln 2π Þ þ lnjΩjÞ þ2δΥ ;
ð40Þ
1018
C. Duan et al. / Neurocomputing 171 (2016) 1013–1020
^ design the state equation of the filter DðzÞ as follows:
in which
Ω¼
xðk þ 1Þ ¼ Ax xðkÞ þ Lx iðkÞ;
N 1X eðkÞeðkÞT ; Ni¼1
yðkÞ ¼ G 1 C x xðkÞ þ iðkÞ:
^ eðkÞ ¼ yðkÞ yðkÞ ? T ¼ ðΛs Þ I Φs;i zf ðk nÞ;
(2) Parameter matrix design: Most work has been finished to design the controller except the parameter matrix Q ðzÞ, which is based on the system performance. It is apparent that the system error can be defined as follows:
mðm þ 1Þ þ nl þ ml; 2 N : δn ¼ N ððΥ =mÞ þ ððm þ1Þ=2ÞÞ
Υ ¼ 2 nm þ
ð41Þ
eðzÞ ¼ λðzÞ yðzÞ:
ð48Þ
The dynamics of the system can be described by the following equation: ^ MðzÞyðzÞ ¼ VðzÞwðzÞ þ WðzÞSðzÞ;
4.3. Basic residual generator design ? s ;
? s
According to the result in the subsection above, the Λ Λ Φs;i can be identified and residual generator can be constructed with the parity space method. ? Select a parity vector αs A Rmðs þ 1Þ from Λs which satisfies that αs ¼ αs;0 αs;1 ⋯αs;s ; αs;i A Rm ; i ¼ 0; 1; …; s: ð42Þ Then, we can calculate the parameter matrixes of DO ^ z according to Eqs. (26) and (27) and Az ; Bz ; L; G^ z ; C^ z , and D construct a DO by solving (13). The residual signal as follows can be obtained by the residual generator [16]: ^ rðkÞ ¼ Q ðyðkÞ yðkÞÞ A Rx ;
ð47Þ
ð43Þ
^ is an in which Q A Rxm denotes a parameter matrix and yðkÞ estimated value of the process outputs. To enhance the performance of the system in order to use it in more widespread industrial area, it is necessary to construct a residual generator that can transmit an m-dimensional residual vector to meet the higher requirement of the system. As a result, extending the order of residual generator based on data-driven method from single one to multiple one is also applied in the practical industry [18].
ð49Þ
where ^ VðzÞ ¼ N^ ðzÞQ ðzÞMðzÞ; ^ WðzÞ ¼ I N ðzÞQ ðzÞ; SðzÞ ¼ ðICðzI AÞ 1 ÞmðzÞ:
ð50Þ
Therefore, the problem of designing the parameter matrix has become a norm optimization one that is relatively easy to solve. In general, the parameter matrix Q(z) should be not only simple enough to make itself practical, but also suitable for both the performance and the stability of the whole system. Generally speaking, Q ðzÞ can be chosen as constant matrixes to meet with the requirements above. We can assume that U ¼ Dx þ C x ðI Ax Þ 1 Bx :
ð51Þ
Since U is an invertible one, the parameter matrix Q(z) is Q ¼ U 1 G;
ð52Þ
otherwise, Q ¼ U G; with ðYÞ
ð53Þ
denoting the pseudo-inverse of the matrix Y.
6. An academic example 5. Residual generator application As discussed in the sections above, the variable selection aided residual generator design is completed. The following work will be dedicated to its application in control and monitoring. 5.1. Residual generator embedding into control loop (1) Filter design: According to the classical and modern control theory and their relationship, the residual can be described in following form: ^ rðzÞ ¼ MðzÞyðzÞ N^ ðzÞiðzÞ;
ð44Þ
In this section, a numerical example will be given to demonstrate the practicability and effectiveness of the controller based on residual generator method developed in this paper. Since we only need the system model to generate the inputs and outputs which is indispensable as the test data, the system matrixes A,B,C,D and the noises will not influence the result of controller design. Assume the system matrixes as follows: mðkÞ Nð0; 0:0012 Þ; nðkÞ Nð0; 0:012 Þ; 2 3 2 0 0 0:0058 0:97 6 7 6 0:12 5; B ¼ 4 0:19 A ¼ 4 0:95 0:01 0:01
0:97
0:58
0:51
3
7 0:31 5; 0:79
where
C ¼ ½1:1
^ MðzÞ ¼ I CðzI AL Þ 1 L; AL ¼ A LC; ^ NðzÞ ¼ D þ CðzI AL Þ 1 BL ; BL ¼ B LD:
where Nðμ; σ 2 Þ denotes the normal distribution with the mean μ and the variance σ 2 . The simulation process of the controller based on data-driven will be described as follows:
ð45Þ
Combining the state equations of multidimensional residual generator, the equations above can be expressed in another form: ^ MðzÞ ¼ I G 1 C x ðzI Ax Þ 1 Lx ; ¼ I CðzI AF Þ 1 L; ¼ D þ CðzI AF Þ
1
BL :
ð46Þ
Therefore, the matrix that satisfies the equations above can be applied as the filter in the system described in Fig. 4. We can also
ð54Þ
Step 1. PLS method Application: Apply the PLS method to analyze
^ NðzÞ ¼ G 1 ðDx þ C x ðzI Ax Þ 1 Bx Þ;
1:65
0:55; D ¼ zerosð1; 2Þ;
0:01
the inputs during the identification process to reduce the number of them. Step 2. Model Pretreatment: Simulate the academic example with the program and collect 1000 samples as test data under the reference stimulation. Step 3. Identify the Parity Space: The method of identifying the parity space has been shown in Algorithm 1.
C. Duan et al. / Neurocomputing 171 (2016) 1013–1020
1.2
1019
0.35 0.3
1
0.25
0.8 1.1
1.1
1.05
1.05
1
1
0.95 450
0.95 550
0.2
0.6
0.15 0.1
0.4 0.2
0.05 0
0 −0.2
500
550
600
650
−0.05 −0.1
0
200
400
600
800
1000
0
400
600
800
1000
Fig. 8. The residual generator s ¼9.
Fig. 5. The output of system s ¼ 3.
Step 5. Time Setting: Considering the quality and reliability of
0.4 0.35
0.3 0.25 0.2 0.15 0.1 0.05 0 −0.05
200
0
200
400
600
800
1000
Fig. 6. The residual generator s ¼3.
1.2 1 0.8 1.1
1.1
1.05
1.05
1
1
0.6
the simulation, we set the simulation time as T ¼ 1000 s and sample time as t ¼ 1 s to demonstrate the performance and application of controller. Step 6. Interference Introduce: After finishing the work above, the interferences mðkÞ ¼ nðkÞ ¼ 0:1 should be introduced at the time t ¼ 500 s and t ¼ 600 s respectively to verify the controller that we have designed.
Fig. 5 shows the output signal obtained from experiment with the condition that s¼n. Fig. 6 displays the data of residual generator on the same condition. It is obvious that the performance of the controller is good enough from the aspect of the response speed and disturbances attenuation. Without loss of generality, we also simulate a system in which s ¼ 9 4 n as another example. The output signal is shown in Fig. 7 and the data of residual generator is displayed in Fig. 8 From the figures above, it is easy to learn that the disturbances can be resistant effectively. However, comparing Figs. 5 and 7, we realize that the overshoot of the latter one will be a bit larger when the reference stimulation is a unit step signal. Therefore, the relationship of size between s and n is significant for the performance of controller. With the method of optimizing the residual generator based on PLS given in this paper, it becomes possible to determine the parameter s in the SIM method more precisely. As a result, ensuring s¼n leads to a better performance and practical design of controller.
0.4 0.2
0.95 450
0 −0.2
0
200
500
550
400
0.95 550
7. Conclusion
600
600
650
800
1000
Fig. 7. The output of system s ¼ 9.
Step 4. Residual Generator Design: According to the relationship between the parity space and DO, the residual generator based on observer can be constructed without any problems, which will be the advanced work of designing controller.
In this paper, we have studied the problem of residual generation based on data-driven approach and its optimization by applying the PLS method to the SIM segment. Furthermore, the residual generator can be employed in the process of control and monitoring after the realization of the Youla parameterization for the feedback control loop and other relevant work. The design of residual generator is based on the relationship between DO as well as the parity vector, and PLS method is utilized to analyze the relationships among the input variables and decrease the number of them at the same time. The applications of data-driven approach and PLS method make the residual generator proposed avoid the difficulties of building models and more effective to deal with big data in modern industry. Moreover, a numerical example
1020
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Chaojing Duan was enrolled in Harbin Institute of Technology (HIT), Harbin, China, in 2012, and is studying toward B.E. degree in Control science and Engineering in Honors School of HIT. Her research interests involve data-driven fault-diagnosis, robust control and their applications.
Zhongyang Fei received the B.E. degree and the M.S. degree in Control Science and Engineering from Harbin Institute of Technology, PR China in 2007 and 2009, respectively; and the Ph.D. degree in Mechanical Engineering & Material Science Department, Washington University in St. Louis, USA in 2013. He joint Harbin Institute of Technology in 2014. His research interests involve complex networks, robust control, time-delay systems, fault detection and diagnosis.
Jiachen Li was enrolled in Harbin Institute of Technology, Harbin, China, in 2012, and is currently studying toward the B.E. degree in Control theory and Engineering in Honors School of HIT. His research interests include data-driven fault diagnosis and prognosis in process control and their applications.