Future Work

7.3.1 Arbitrary Uncertainty Quantification and Propagation

The original homogeneous polynomial chaos expansion (Ghanem & Spanos, 1991) and the modified generalized polynomial chaos expansion (Xiu D. , 2010) can result in high computational efficiency and fast convergence. Both methods are based on an appropriate selection of orthogonal polynomials. For example, the gPC uses the Wiener-Askey polynomial chaos framework based on several orthogonal polynomials including the Hermite polynomial. For uncertain input distributions outside of the Wiener-Askey scheme, the Wiener-Askey polynomial chaos does converge, but the convergence rate might be slow for high dimensional complex system.

Thus, the appropriate selection of polynomial basis function for efficient quantification of uncertainty may be one of the possible direction to improve computational efficiency. For example, a multi-element generalized polynomial chaos has been recently developed to deal with stochastic input with arbitrary probability measures (Wan & Karniadakis, 2006). Based on a decomposition of the random space, a set of optimal orthogonal

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polynomials using Stieltjes three-term recurrence procedure can be formulated. Another option is to combine Gram-Schmidt polynomial chaos with the polynomial chaos expansion (Witteveen & Bijl, 2006), in which the optimal set of orthogonal polynomials is computed for any type of input distribution. (The application of this method is illustrated as a case study in this work as can be seen in Appendix B). Moreover, uncertainty quantification of time varying uncertainty still poses a significant challenge, despite the success of the gPC methods (Gerritsma, Van der Steen, Vos, & Karniadakis, 2010). It is necessary to investigate the problem that the probability density function of uncertainty evolves as a function of time.

7.3.2 Integration of Plant Design, Control and Fault Diagnosis

The trade-off between control and fault detectability is investigated in this work to achieve a balance between these two activities, since they have competing objectives in particular in the presence of uncertainty. However, the objective is to seek the optimal controller’s parameters to improve the detectability of intermittent faults. Further work should be conducted in the area of fault tolerant control, i.e., the optimal reconfiguration of control law in the event of faults to ensure the system to continue operating at a suboptimal levels, rather than breaking down completely.

Additionally, the consideration of the dynamic and control aspects during the early state of the plant design may lead to the improved controllability and operability. For example, appropriate design of chemical plants may reduce the effort in identifying and diagnosing the possible faults. Plant design criteria can be incorporated into the optimization to evaluate the effect of sensors’ selection and distribution. This may maximize the information that can be ultimately used for the detection and control.

7.3.3 Image Segmentation and Classification

As an extension of active contour without edges method, the gPC expansion is combined with the level set functions to evolve the cells’ boundaries. Such stochastic image segmentation can propagate the information about the gray value errors and uncertainty from the input image to the final segmentation results. With this tool, it is possible to provide information about the reliability and confidence intervals of the boundary. From the mathematical point of view, there are still areas for further improvement, since the gPC method tends to be computational demanding when the number of random variables increases. Another challenge is the visualization and segmentation of high dimensional stochastic color images. For example, the application of gPC in combination with image processing method in this work is a starting point and can be further improved.

In terms of feature extraction and selection, the identification of the most important feature is critical to minimize the classification error. For example, features can be selected based on mutual information criteria of maximize dependency and relevance (Peng, Long, & Ding, 2005).

Appendix A

Comparison of Stochastic Fault Diagnosis Algorithms

(Adopted from Du et al., 2015, Chemometrics and Intelligent Laboratory Systems, ready to submit)

A.1 Overview

This appendix presents a comparison study to identify and diagnose intermittent stochastic faults occurring in a dynamic multimode nonlinear process. The main objective is to develop efficient fault diagnosis algorithms in the presence of parametric uncertainty and to show the capabilities of each method. For the first principles’ model based fault detection and diagnosis (FDD), a generalized polynomial chaos (gPC) expansion representing the stochastic input faults is employed to propagate the uncertainty onto the measured quantities. The resulting probability density functions (PDFs) of the measured variables can then be approximated and further used for fault diagnosis. For the statistical monitoring method, Gaussian process (GP) is used to map multivariate inputs into a univariate response, from which the fault can be inferred based on a minimum distance criterion. The performance of these methods is evaluated in terms of fault detection rate by applying them to a chemical plant of two continuously stirred tank reactors (CSTRs) and a flash tank separator. The proposed methods are successful in detecting and diagnosing intermittent faults in the presence of uncertainty.

A.2 Introduction

Early detection of abnormal events and malfunctions defined as faults is of great interest, since faults may affect the product quality and lead to economic losses (Gerlter, 1998). If a fault is detectable, the fault detection and diagnosis (FDD) system will provide symptomatic fingerprints, which in turn can be referred back to the FDD scheme to identify the root cause of the anomalous behaviour. Most of the available fault diagnosis algorithms can be broadly classified into three main classes (Isermann R. , 2005; Venkatasubramanian V. , Rengaswamy, Yin, & Kavuri, 2003): (i) Analytical methods that are solely based on first principles’ models of process; (ii) Empirical models that use the historical process data; and (iii) Semi-empirical algorithms that combine these aforementioned two classes. Each of these methods has its own advantages and disadvantages depending on the specific problem (Isermann R. , 2006).

In terms of applications, many industrial processes are intrinsically nonlinear systems and they are operated at different operating conditions according to economic considerations (Haghani, Jeinsch, & Ding, 2014). Due to nonlinearity, the performance of linear FDD algorithms reported in literature (Li & Yang, 2012) may be inaccurate and lead to missed detection of faults, since the process model will change from one operating conditions to another. It is critical to develop new methodologies for the detection of faults in the context of nonlinear chemical processes with multiple operating conditions (Haghani, Jeinsch, & Ding, 2014).

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Since most of the FDD schemes are invariably based on either first principle models or empirical models (Venkatasubramanian V. , Rengaswamy, Yin, & Kavuri, 2003), a main restrictive factor of an efficient FDD system is the model uncertainty. Such uncertainty may originate from either intrinsic time varying phenomena of model parameters or may result from inaccurate measurements due to noise. Models with large uncertainties make the detection and isolation of small faults very difficult. However, the step of quantifying and propagating the uncertainties onto the measured quantities that can be used for fault detection is typically omitted in reported FDD studies, leading to a loss of useful information arising from these uncertainties (Patton, Frank, & Clark, 2010). Moreover, the quantitative analysis of faults detectability in the presence of uncertainty provides more information to improve FDD algorithms. For example, engineering effort can be saved, if it is impossible to detect a fault due to uncertainties such as large measurement noise (Eriksson, Frisk, & Krysander, 2013).

To evaluate the effect of uncertainty on FDD, one possibility is to propagate stochastic variations with Monte Carlo (MC) simulations (Harrison, 2010), which involve drawing a large number of samples and running the models with each of these samples. However, approaches such as MC simulations are computationally prohibitive especially for complex processes as shown later in the manuscript. To improve the computational efficiency, this paper presents and compares two FDD algorithms in the presence of uncertainties. The uncertainty includes the parametric uncertainty of a process and measurement noise. In addition, the faults in this current work are stochastic perturbations superimposed on intermittent step changes in specific input variables for a nonlinear chemical plant. For the first FDD method, generalized polynomial chaos (gPC) (Ghanem & Spanos, 1991; Xiu D. , 2010) in combination with first principles’ process models are used to quantify and propagate the uncertainty onto the measured quantities, which can be used for the detection of faults. For the second method, a surrogate metamodel is developed with Gaussian Process (GP) (Rasmussen & Williams, 2006), which is calibrated with a minimal model adjustment algorithm and can be used estimate the value of fault.

The objective in this work is to address the capabilities of these methods and propose a possible strategy to overcome their limitation by combing their outcomes. For this purpose, the performance of each method is evaluated in terms of fault detection rate in the context of stochastic parametric input faults. These faults occur intermittently with stochastic perturbations, i.e., the mean value of faults switch between the non-faulty and faulty operating conditions in a random fashion. For simplicity, the stochastic perturbations are assumed to be time- invariant uncertainties. Thus, the key is to identify and diagnose these step changes in the presence of the random perturbations in the parametric input faults, using available measurements corrupted with measurement noise.

To summarize, the contributions in this work are: (i) the use, in the context of fault detection and diagnosis, of a gPC model and a GP model for uncertainty propagation and quantification for a complex nonlinear system; (ii) the comparison of analytical and empirical methods for the detection of faults of a stochastic nature; and (iii) an ensemble of these methods to overcome limitations instead of the standalone application of each method.

This appendix is organized as follows. In Section A.3, the formulation of a fault detection problem is presented followed by the theoretical background of the gPC and GP theories. The fault detection and diagnosis (FDD) algorithms are explained in Section A.4. A nonlinear chemical plant with two continuously stirred tank reactors and a flash tank separator is introduced as a case study in Section A.5. Analysis and discussion of the results are given in Section A.6 followed by conclusions in Section A.7.