Two-dimensional Bayesian monitoring method for nonlinear multimode processes Zhiqiang Ge a,b , Furong Gao b,n , Zhihuan Song a a State Key Laboratory of Industrial Control Technology, Institute of Industrial Process Control, Zhejiang University, Hangzhou 310027, Zhejiang, PR China b Department of Chemical and Biomolecular engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR article info Article history: Received 19 July 2010 Received in revised form 20 June 2011 Accepted 4 July 2011 Available online 18 July 2011 Keywords: Process monitoring Nonlinear Multimode Two-dimensional Bayesian inference Linear subspace Two-step variable selection abstract Nonlinear and multimode are two common behaviors in modern industrial processes, monitoring research studies have been carried out separately for these two natures in recent years. This paper proposes a two-dimensional Bayesian method for monitoring processes with both nonlinear and multimode characteristics. In this method, the concept of linear subspace is introduced, which can efficiently decompose the nonlinear process into several different linear subspaces. For construction of the linear subspace, a two-step variable selection strategy is proposed. A Bayesian inference and combination strategy is then introduced for result combination of different linear subspaces. Besides, through the direction of the operation mode, an additional Bayesian combination step is performed. As a result, a two-dimensional Bayesian monitoring approach is formulated. Feasibility and efficiency of the method are evaluated by the Tennessee Eastman (TE) process case study. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction With growing requirements of safety and high product quality in modern industrial processes, monitoring and fault diagnosis have become increasingly important. Data-based process mon- itoring methods such as multivariate statistical process control (MSPC) have gained much attentions (Chiang et al., 2001; Qin, 2003; Singhai and Seborg, 2006). Traditional MSPC methods are under the assumption that the process variables are linear correlated, Gaussian distributed, and operated under a single mode and stationary process condition. In practice, however, the variable correlations in most industrial processes are non- linear, partly non-Gaussian, and the process may be operated under multiple conditions. In the past years, many improvements have been made for the traditional MSPC based monitoring methods (Chen and Chen, 2006; Lee et al., 2004; Thissen et al., 2005; AlGhazzawi and Lennox, 2008; Hu and Yuan, 2008; Nomikos and MacGregor, 1995; Yao and Gao, 2008). This paper is focused on nonlinear and multimode character- istics of the process, both of which are very common in practice. Compared to the linear case, faults may be more difficult to detect in nonlinear processes, this is because the correlations between process variables are much more complex. So far, several nonlinear process monitoring methods have been developed, such as principal curve, neural network, kernel PCA, etc (Dong and McAvoy, 1996; Hiden et al., 1999; Choi et al., 2005; Maulud et al., 2006; Wang et al., 2007; Zhang and Qin, 2008; Ge et al., 2009). However, due to different nonlinear transformations, most of those methods are computationally inefficient and thus difficult for online implementation, e.g. modeling, monitoring, and fault diagnosis. An alternative way for nonlinear modeling is to use several linear models to approximate the nonlinearity of the process data (Ge et al., 2010). While local linear models have been used for nonlinear approximation in the continuous process, several similar approaches have also been developed in batch processes. Instead of constructing a nonlinear model for the whole batch process, different linear models can be built in each phase of the batch process, such as Camacho and Pico (2006) and Sun et al. (2011). On the other hand, some multimode process monitoring methods have also been developed in the last few years, including adaptive methods, multiple model methods, Gaussian mixture model based approaches, external analysis, etc (Qin, 1998; Hwang and Han, 1999; Wang et al., 2005; Choi et al., 2004; Zhao et al., 2004; Yoo et al., 2007; Yu and Qin, 2008; Chen and Sun, 2009; Ge and Song, 2008; Ge et al., 2008; Ng and Srinivasan, 2009; Natarajan and Srinivasan, 2010). However, the implementation of the adaptive method can cause false alarms, especially during the transition of two operation modes. Similar problem may arise in the multiple model based method, since they always assign the Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ces Chemical Engineering Science 0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.07.001 n Corresponding author: Tel.: þ852 23587136; fax: þ852 23580054. E-mail address: kefgao@ust.hk (F. Gao). Chemical Engineering Science 66 (2011) 5173–5183