变分贝叶斯鲁棒线性动态系统：动态建模与故障检测的新方法
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本文主要探讨了动态过程建模与故障检测领域的创新方法，即变分贝叶斯鲁棒线性动态系统（Variational Bayesian Robust Linear Dynamic System, VBRLDS）。这项研究发表在IFAC-PapersOnLine期刊的第48卷第21期（2015年），535-540页，可在ScienceDirect网站上获取，DOI为10.1016/j.ifacol.2015.09.581，由Elsevier Ltd.托管，受国际自动控制联合会（International Federation of Automatic Control, IFAC）的同行评审。
传统的动态过程建模通常采用卡尔曼滤波（Kalman Filter）构建线性动态系统（LDS），这种模型假设噪声是高斯分布的，这在某些情况下可能不完全准确，特别是面对复杂工业环境中的非线性和不确定性时。变分贝叶斯方法引入了一种更为灵活且鲁棒的处理方式，它利用贝叶斯统计理论来处理不确定性，并通过优化技术（如变分推断）来简化复杂的后验分布计算。
VBRLDS方法的优势在于它能够更好地适应现实世界中的复杂动态过程，即使在存在噪声和模型不精确的情况下，也能提供更准确的预测和故障检测性能。该方法通过非参数化的方法，允许数据驱动的模型学习，减少了对先验知识的依赖，并且能自适应地调整模型参数以应对未知的系统行为。
研究者们，Jinlin Zhu、Zhiqiang Ge 和 Zhihuan Song，来自浙江大学工业控制技术国家重点实验室和工业过程控制研究所，他们通过控制科学与工程系，提出了这一新颖的建模框架。他们的工作地址位于中国杭州310027，可以通过电话+86-87951442或电子邮件gezhiqiang@zju.edu.cn联系。
这篇论文的核心贡献在于提供了一种实用的工具，使得动态过程的建模和故障检测在面对不确定性和复杂性时更加稳健和有效。通过将变分贝叶斯技术和鲁棒线性动态系统相结合，研究人员不仅改进了模型的适应性，还提高了故障检测的准确性，这对于工业自动化和控制系统领域具有重要的实际应用价值。
ScienceDirect
IFAC-PapersOnLine 48-21 (2015) 535–540
ScienceDirect
Available online at www.sciencedirect.com
2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Peer review under responsibility of International Federation of Automatic Control.
10.1016/j.ifacol.2015.09.581
Jinlin Zhu et al. / IFAC-PapersOnLine 48-21 (2015) 535–540
©
2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
A variational Bayesian robust linear dynamic system approach for dynamic
process modelling and fault detection
Jinlin Zhu, Zhiqiang Ge*, Zhihuan Song
* State Key Laboratory of Industrial Control Technology, Institute of Industrial Process Control, Department of Control
Science and Engineering, Zhejiang University, Hangzhou 310027, Zhejiang, P. R. China
(Tel: +86-87951442; e-mail: gezhiqiang@zju.edu.cn).
Abstract: In this work, a variational Bayesian robust linear dynamic system (VBRLDS) approach is
proposed for dynamic process modeling and monitoring. Traditional linear dynamic system (LDS)
constructed with Kalman filter is designed by Gaussian assumption which can be easily violated in
outlier contaminated modeling situations. To deal with this issue, the conventional Gaussian-based
Kalman filter is modified with heavy tailed Student’s t-distribution so as to deal with modeling outliers.
Then, a variational Bayesian expectation maximization (VBEM) algorithm is developed for learning
parameters of the robust linear dynamic system. For process monitoring, traditional residual space
modeling method of SPE statistics are modified. To explore the feasibility and effectiveness, our
proposed method is applied into fault detection and is comparatively studied with several other methods
on the Tennessee Eastman benchmark.
Keywords: Robust LDS; Variational Bayesian; Kalman filter; Student’s t-distribution; Fault detection.
1. INTRODUCTION
With the development of modern process control technology,
industrial processes have become more and more
complex(Chiang et al., 2001). In order to maintain the safe
operation of manufacturing system, process monitoring is
particularly essential. Among the various monitoring
schemes, data driven multivariate statistical process
monitoring (MSPM) models assume little requirement for
accurate kinematic equations and are easy to be
established(Yin et al., 2012, Ge et al., 2013). MSPM models
such as principal component analysis (PCA) are widely
studied and applied over the past few decades. PCA can
detect faults effectively by constructing T
2
statistic using
latent space information; meanwhile, SPE statistic is also
constructed by the residual space so as to monitor the change
of measurement noise. Although traditional PCA has been
popular and effective for many industrial applications, a main
drawback is that the original model is lack of natural
representation of the process uncertainties. For this reason,
probabilistic PCA (PPCA) has been proposed(Kim and Lee,
2003).
As static modelling methods, both PCA and PPCA take data
samples as independently collected from sensors with no time
serial correlations. However, it is well known that most
industrial processes evolve from past operation situations to
potential future events. Therefore, dynamic behaviour should
be one of the most important characteristics for industrial
process data (Russell et al., 2000). To consider the time serial
related property, dynamic PCA (DPCA) has been developed
by augmenting each measurement with a fixed length of
several previous measurements and aligned to a stacking
matrix (Luo et al., 1999). After that, similar PCA projections
and statistics are then constructed. Another commonly used
dimensionality reduction based method is canonical variate
analysis (CVA). CVA considers correlations by maximizing
the related correlation index among variables, some studies
show that compared with PCA and DPCA, CVA provides
more desirable monitoring performance (Russell et al., 2000).
It should be noted that both DPCA and CVA are built with
deterministic manner and no probabilistic interpretations for
noise uncertainties have been considered. As a probabilistic
alternative, recently, a data-based linear Gaussian state space
model (LGSSM) has been developed (Wen et al., 2012).
LGSSM constructs a linear Gaussian state space model and
then uses Kalman filter and common EM for estimating
model parameters. Results show that compared with
traditional PCA and PPCA, LDS is more suitable for
dynamic process modeling and monitoring. However, one
common issue for such Gaussian based dynamic method is
that the assumed Gaussian noise assumption can be violated
by potentially sampling outliers (Archambeau et al., 2008).
Outliers can be hardly avoided and may cause model
misspecification for Gaussian distributions (Zhu et al., 2014).
The reason is that Gaussian distributions assume the tail
section drop exponentially which is known as the three sigma
principle. However, outliers usually occur in or beyond the
three sigma region. In this condition, estimated significant
parameters like mean and covariance can be skewed.
In this article, a Student’s t-based noise assumption has been
employed for LDS observation space so as to tolerate
sampling outliers. Besides mean and covariance, the
Student’s t-distribution is defined with a tail adjust parameter
called degree of freedom. Thus, the Student’s t-distribution
provides an elegant interpretation for outliers without
distorting the entire distribution. In this sense, the derived
9th IFAC Symposium on Fault Detection, Supervision and
Safety of Technical Processes
September 2-4, 2015. Arts et Métiers ParisTech, Paris, France
Copyright © 2015 IFAC 535
A variational Bayesian robust linear dynamic system approach for dynamic
process modelling and fault detection
Jinlin Zhu, Zhiqiang Ge*, Zhihuan Song
* State Key Laboratory of Industrial Control Technology, Institute of Industrial Process Control, Department of Control
Science and Engineering, Zhejiang University, Hangzhou 310027, Zhejiang, P. R. China
(Tel: +86-87951442; e-mail: gezhiqiang@zju.edu.cn).
Abstract: In this work, a variational Bayesian robust linear dynamic system (VBRLDS) approach is
proposed for dynamic process modeling and monitoring. Traditional linear dynamic system (LDS)
constructed with Kalman filter is designed by Gaussian assumption which can be easily violated in
outlier contaminated modeling situations. To deal with this issue, the conventional Gaussian-based
Kalman filter is modified with heavy tailed Student’s t-distribution so as to deal with modeling outliers.
Then, a variational Bayesian expectation maximization (VBEM) algorithm is developed for learning
parameters of the robust linear dynamic system. For process monitoring, traditional residual space
modeling method of SPE statistics are modified. To explore the feasibility and effectiveness, our
proposed method is applied into fault detection and is comparatively studied with several other methods
on the Tennessee Eastman benchmark.
Keywords: Robust LDS; Variational Bayesian; Kalman filter; Student’s t-distribution; Fault detection.
1. INTRODUCTION
With the development of modern process control technology,
industrial processes have become more and more
complex(Chiang et al., 2001). In order to maintain the safe
operation of manufacturing system, process monitoring is
particularly essential. Among the various monitoring
schemes, data driven multivariate statistical process
monitoring (MSPM) models assume little requirement for
accurate kinematic equations and are easy to be
established(Yin et al., 2012, Ge et al., 2013). MSPM models
such as principal component analysis (PCA) are widely
studied and applied over the past few decades. PCA can
detect faults effectively by constructing T
2
statistic using
latent space information; meanwhile, SPE statistic is also
constructed by the residual space so as to monitor the change
of measurement noise. Although traditional PCA has been
popular and effective for many industrial applications, a main
drawback is that the original model is lack of natural
representation of the process uncertainties. For this reason,
probabilistic PCA (PPCA) has been proposed(Kim and Lee,
2003).
As static modelling methods, both PCA and PPCA take data
samples as independently collected from sensors with no time
serial correlations. However, it is well known that most
industrial processes evolve from past operation situations to
potential future events. Therefore, dynamic behaviour should
be one of the most important characteristics for industrial
process data (Russell et al., 2000). To consider the time serial
related property, dynamic PCA (DPCA) has been developed
by augmenting each measurement with a fixed length of
several previous measurements and aligned to a stacking
matrix (Luo et al., 1999). After that, similar PCA projections
and statistics are then constructed. Another commonly used
dimensionality reduction based method is canonical variate
analysis (CVA). CVA considers correlations by maximizing
the related correlation index among variables, some studies
show that compared with PCA and DPCA, CVA provides
more desirable monitoring performance (Russell et al., 2000).
It should be noted that both DPCA and CVA are built with
deterministic manner and no probabilistic interpretations for
noise uncertainties have been considered. As a probabilistic
alternative, recently, a data-based linear Gaussian state space
model (LGSSM) has been developed (Wen et al., 2012).
LGSSM constructs a linear Gaussian state space model and
then uses Kalman filter and common EM for estimating
model parameters. Results show that compared with
traditional PCA and PPCA, LDS is more suitable for
dynamic process modeling and monitoring. However, one
common issue for such Gaussian based dynamic method is
that the assumed Gaussian noise assumption can be violated
by potentially sampling outliers (Archambeau et al., 2008).
Outliers can be hardly avoided and may cause model
misspecification for Gaussian distributions (Zhu et al., 2014).
The reason is that Gaussian distributions assume the tail
section drop exponentially which is known as the three sigma
principle. However, outliers usually occur in or beyond the
three sigma region. In this condition, estimated significant
parameters like mean and covariance can be skewed.
In this article, a Student’s t-based noise assumption has been
employed for LDS observation space so as to tolerate
sampling outliers. Besides mean and covariance, the
Student’s t-distribution is defined with a tail adjust parameter
called degree of freedom. Thus, the Student’s t-distribution
provides an elegant interpretation for outliers without
distorting the entire distribution. In this sense, the derived
9th IFAC Symposium on Fault Detection, Supervision and
Safety of Technical Processes
September 2-4, 2015. Arts et Métiers ParisTech, Paris, France
Copyright © 2015 IFAC 535
A variational Bayesian robust linear dynamic system approach for dynamic
process modelling and fault detection
Jinlin Zhu, Zhiqiang Ge*, Zhihuan Song
* State Key Laboratory of Industrial Control Technology, Institute of Industrial Process Control, Department of Control
Science and Engineering, Zhejiang University, Hangzhou 310027, Zhejiang, P. R. China
(Tel: +86-87951442; e-mail: gezhiqiang@zju.edu.cn).
Abstract: In this work, a variational Bayesian robust linear dynamic system (VBRLDS) approach is
proposed for dynamic process modeling and monitoring. Traditional linear dynamic system (LDS)
constructed with Kalman filter is designed by Gaussian assumption which can be easily violated in
outlier contaminated modeling situations. To deal with this issue, the conventional Gaussian-based
Kalman filter is modified with heavy tailed Student’s t-distribution so as to deal with modeling outliers.
Then, a variational Bayesian expectation maximization (VBEM) algorithm is developed for learning
parameters of the robust linear dynamic system. For process monitoring, traditional residual space
modeling method of SPE statistics are modified. To explore the feasibility and effectiveness, our
proposed method is applied into fault detection and is comparatively studied with several other methods
on the Tennessee Eastman benchmark.
Keywords: Robust LDS; Variational Bayesian; Kalman filter; Student’s t-distribution; Fault detection.
1. INTRODUCTION
With the development of modern process control technology,
industrial processes have become more and more
complex(Chiang et al., 2001). In order to maintain the safe
operation of manufacturing system, process monitoring is
particularly essential. Among the various monitoring
schemes, data driven multivariate statistical process
monitoring (MSPM) models assume little requirement for
accurate kinematic equations and are easy to be
established(Yin et al., 2012, Ge et al., 2013). MSPM models
such as principal component analysis (PCA) are widely
studied and applied over the past few decades. PCA can
detect faults effectively by constructing T
2
statistic using
latent space information; meanwhile, SPE statistic is also
constructed by the residual space so as to monitor the change
of measurement noise. Although traditional PCA has been
popular and effective for many industrial applications, a main
drawback is that the original model is lack of natural
representation of the process uncertainties. For this reason,
probabilistic PCA (PPCA) has been proposed(Kim and Lee,
2003).
As static modelling methods, both PCA and PPCA take data
samples as independently collected from sensors with no time
serial correlations. However, it is well known that most
industrial processes evolve from past operation situations to
potential future events. Therefore, dynamic behaviour should
be one of the most important characteristics for industrial
process data (Russell et al., 2000). To consider the time serial
related property, dynamic PCA (DPCA) has been developed
by augmenting each measurement with a fixed length of
several previous measurements and aligned to a stacking
matrix (Luo et al., 1999). After that, similar PCA projections
and statistics are then constructed. Another commonly used
dimensionality reduction based method is canonical variate
analysis (CVA). CVA considers correlations by maximizing
the related correlation index among variables, some studies
show that compared with PCA and DPCA, CVA provides
more desirable monitoring performance (Russell et al., 2000).
It should be noted that both DPCA and CVA are built with
deterministic manner and no probabilistic interpretations for
noise uncertainties have been considered. As a probabilistic
alternative, recently, a data-based linear Gaussian state space
model (LGSSM) has been developed (Wen et al., 2012).
LGSSM constructs a linear Gaussian state space model and
then uses Kalman filter and common EM for estimating
model parameters. Results show that compared with
traditional PCA and PPCA, LDS is more suitable for
dynamic process modeling and monitoring. However, one
common issue for such Gaussian based dynamic method is
that the assumed Gaussian noise assumption can be violated
by potentially sampling outliers (Archambeau et al., 2008).
Outliers can be hardly avoided and may cause model
misspecification for Gaussian distributions (Zhu et al., 2014).
The reason is that Gaussian distributions assume the tail
section drop exponentially which is known as the three sigma
principle. However, outliers usually occur in or beyond the
three sigma region. In this condition, estimated significant
parameters like mean and covariance can be skewed.
In this article, a Student’s t-based noise assumption has been
employed for LDS observation space so as to tolerate
sampling outliers. Besides mean and covariance, the
Student’s t-distribution is defined with a tail adjust parameter
called degree of freedom. Thus, the Student’s t-distribution
provides an elegant interpretation for outliers without
distorting the entire distribution. In this sense, the derived
9th IFAC Symposium on Fault Detection, Supervision and
Safety of Technical Processes
September 2-4, 2015. Arts et Métiers ParisTech, Paris, France
Copyright © 2015 IFAC 535
A variational Bayesian robust linear dynamic system approach for dynamic
process modelling and fault detection
Jinlin Zhu, Zhiqiang Ge*, Zhihuan Song
* State Key Laboratory of Industrial Control Technology, Institute of Industrial Process Control, Department of Control
Science and Engineering, Zhejiang University, Hangzhou 310027, Zhejiang, P. R. China
(Tel: +86-87951442; e-mail: gezhiqiang@zju.edu.cn).
Abstract: In this work, a variational Bayesian robust linear dynamic system (VBRLDS) approach is
proposed for dynamic process modeling and monitoring. Traditional linear dynamic system (LDS)
constructed with Kalman filter is designed by Gaussian assumption which can be easily violated in
outlier contaminated modeling situations. To deal with this issue, the conventional Gaussian-based
Kalman filter is modified with heavy tailed Student’s t-distribution so as to deal with modeling outliers.
Then, a variational Bayesian expectation maximization (VBEM) algorithm is developed for learning
parameters of the robust linear dynamic system. For process monitoring, traditional residual space
modeling method of SPE statistics are modified. To explore the feasibility and effectiveness, our
proposed method is applied into fault detection and is comparatively studied with several other methods
on the Tennessee Eastman benchmark.
Keywords: Robust LDS; Variational Bayesian; Kalman filter; Student’s t-distribution; Fault detection.
1. INTRODUCTION
With the development of modern process control technology,
industrial processes have become more and more
complex(Chiang et al., 2001). In order to maintain the safe
operation of manufacturing system, process monitoring is
particularly essential. Among the various monitoring
schemes, data driven multivariate statistical process
monitoring (MSPM) models assume little requirement for
accurate kinematic equations and are easy to be
established(Yin et al., 2012, Ge et al., 2013). MSPM models
such as principal component analysis (PCA) are widely
studied and applied over the past few decades. PCA can
detect faults effectively by constructing T
2
statistic using
latent space information; meanwhile, SPE statistic is also
constructed by the residual space so as to monitor the change
of measurement noise. Although traditional PCA has been
popular and effective for many industrial applications, a main
drawback is that the original model is lack of natural
representation of the process uncertainties. For this reason,
probabilistic PCA (PPCA) has been proposed(Kim and Lee,
2003).
As static modelling methods, both PCA and PPCA take data
samples as independently collected from sensors with no time
serial correlations. However, it is well known that most
industrial processes evolve from past operation situations to
potential future events. Therefore, dynamic behaviour should
be one of the most important characteristics for industrial
process data (Russell et al., 2000). To consider the time serial
related property, dynamic PCA (DPCA) has been developed
by augmenting each measurement with a fixed length of
several previous measurements and aligned to a stacking
matrix (Luo et al., 1999). After that, similar PCA projections
and statistics are then constructed. Another commonly used
dimensionality reduction based method is canonical variate
analysis (CVA). CVA considers correlations by maximizing
the related correlation index among variables, some studies
show that compared with PCA and DPCA, CVA provides
more desirable monitoring performance (Russell et al., 2000).
It should be noted that both DPCA and CVA are built with
deterministic manner and no probabilistic interpretations for
noise uncertainties have been considered. As a probabilistic
alternative, recently, a data-based linear Gaussian state space
model (LGSSM) has been developed (Wen et al., 2012).
LGSSM constructs a linear Gaussian state space model and
then uses Kalman filter and common EM for estimating
model parameters. Results show that compared with
traditional PCA and PPCA, LDS is more suitable for
dynamic process modeling and monitoring. However, one
common issue for such Gaussian based dynamic method is
that the assumed Gaussian noise assumption can be violated
by potentially sampling outliers (Archambeau et al., 2008).
Outliers can be hardly avoided and may cause model
misspecification for Gaussian distributions (Zhu et al., 2014).
The reason is that Gaussian distributions assume the tail
section drop exponentially which is known as the three sigma
principle. However, outliers usually occur in or beyond the
three sigma region. In this condition, estimated significant
parameters like mean and covariance can be skewed.
In this article, a Student’s t-based noise assumption has been
employed for LDS observation space so as to tolerate
sampling outliers. Besides mean and covariance, the
Student’s t-distribution is defined with a tail adjust parameter
called degree of freedom. Thus, the Student’s t-distribution
provides an elegant interpretation for outliers without
distorting the entire distribution. In this sense, the derived
9th IFAC Symposium on Fault Detection, Supervision and
Safety of Technical Processes
September 2-4, 2015. Arts et Métiers ParisTech, Paris, France
Copyright © 2015 IFAC 535
A variational Bayesian robust linear dynamic system approach for dynamic
process modelling and fault detection
Jinlin Zhu, Zhiqiang Ge*, Zhihuan Song
* State Key Laboratory of Industrial Control Technology, Institute of Industrial Process Control, Department of Control
Science and Engineering, Zhejiang University, Hangzhou 310027, Zhejiang, P. R. China
(Tel: +86-87951442; e-mail: gezhiqiang@zju.edu.cn).
Abstract: In this work, a variational Bayesian robust linear dynamic system (VBRLDS) approach is
proposed for dynamic process modeling and monitoring. Traditional linear dynamic system (LDS)
constructed with Kalman filter is designed by Gaussian assumption which can be easily violated in
outlier contaminated modeling situations. To deal with this issue, the conventional Gaussian-based
Kalman filter is modified with heavy tailed Student’s t-distribution so as to deal with modeling outliers.
Then, a variational Bayesian expectation maximization (VBEM) algorithm is developed for learning
parameters of the robust linear dynamic system. For process monitoring, traditional residual space
modeling method of SPE statistics are modified. To explore the feasibility and effectiveness, our
proposed method is applied into fault detection and is comparatively studied with several other methods
on the Tennessee Eastman benchmark.
Keywords: Robust LDS; Variational Bayesian; Kalman filter; Student’s t-distribution; Fault detection.
1. INTRODUCTION
With the development of modern process control technology,
industrial processes have become more and more
complex(Chiang et al., 2001). In order to maintain the safe
operation of manufacturing system, process monitoring is
particularly essential. Among the various monitoring
schemes, data driven multivariate statistical process
monitoring (MSPM) models assume little requirement for
accurate kinematic equations and are easy to be
established(Yin et al., 2012, Ge et al., 2013). MSPM models
such as principal component analysis (PCA) are widely
studied and applied over the past few decades. PCA can
detect faults effectively by constructing T
2
statistic using
latent space information; meanwhile, SPE statistic is also
constructed by the residual space so as to monitor the change
of measurement noise. Although traditional PCA has been
popular and effective for many industrial applications, a main
drawback is that the original model is lack of natural
representation of the process uncertainties. For this reason,
probabilistic PCA (PPCA) has been proposed(Kim and Lee,
2003).
As static modelling methods, both PCA and PPCA take data
samples as independently collected from sensors with no time
serial correlations. However, it is well known that most
industrial processes evolve from past operation situations to
potential future events. Therefore, dynamic behaviour should
be one of the most important characteristics for industrial
process data (Russell et al., 2000). To consider the time serial
related property, dynamic PCA (DPCA) has been developed
by augmenting each measurement with a fixed length of
several previous measurements and aligned to a stacking
matrix (Luo et al., 1999). After that, similar PCA projections
and statistics are then constructed. Another commonly used
dimensionality reduction based method is canonical variate
analysis (CVA). CVA considers correlations by maximizing
the related correlation index among variables, some studies
show that compared with PCA and DPCA, CVA provides
more desirable monitoring performance (Russell et al., 2000).
It should be noted that both DPCA and CVA are built with
deterministic manner and no probabilistic interpretations for
noise uncertainties have been considered. As a probabilistic
alternative, recently, a data-based linear Gaussian state space
model (LGSSM) has been developed (Wen et al., 2012).
LGSSM constructs a linear Gaussian state space model and
then uses Kalman filter and common EM for estimating
model parameters. Results show that compared with
traditional PCA and PPCA, LDS is more suitable for
dynamic process modeling and monitoring. However, one
common issue for such Gaussian based dynamic method is
that the assumed Gaussian noise assumption can be violated
by potentially sampling outliers (Archambeau et al., 2008).
Outliers can be hardly avoided and may cause model
misspecification for Gaussian distributions (Zhu et al., 2014).
The reason is that Gaussian distributions assume the tail
section drop exponentially which is known as the three sigma
principle. However, outliers usually occur in or beyond the
three sigma region. In this condition, estimated significant
parameters like mean and covariance can be skewed.
In this article, a Student’s t-based noise assumption has been
employed for LDS observation space so as to tolerate
sampling outliers. Besides mean and covariance, the
Student’s t-distribution is defined with a tail adjust parameter
called degree of freedom. Thus, the Student’s t-distribution
provides an elegant interpretation for outliers without
distorting the entire distribution. In this sense, the derived
9th IFAC Symposium on Fault Detection, Supervision and
Safety of Technical Processes
September 2-4, 2015. Arts et Métiers ParisTech, Paris, France
Copyright © 2015 IFAC 535
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