Monitoring of Operating Condition and Process Dynamics with Slow Feature Analysis

Abstract

Latent variable (LV) models have been widely used in multivariate statistical process monitoring. However, whatever deviation from nominal operating condition is detected, an alarm is triggered based on classical monitoring methods. Consequently, they cannot distinguish real faults incurring dynamics anomalies from normal deviations in operating conditions. In this chapter, a new process monitoring strategy based on slow feature analysis (SFA) is proposed for the concurrent monitoring of operating point deviations and process dynamics anomalies. Slow features as LVs are developed to describe slowly varying dynamics, yielding improved physical interpretation. In addition to classical statistics for monitoring deviation from design conditions, two novel indices are proposed to detect anomalies in process dynamics through the slowness of LVs. The proposed approach can distinguish whether normal changes in operating conditions or real faults occur. Two case studies show the validity of the SFA-based process monitoring approach.

Appendices

Appendix A: Proof of Theorem 2.1

In order to calculate the value of $\varDelta (\cdot )$, we need to first normalize $x_{ {j}}$ to unit variance:

$$\begin{aligned} x_{ {j}}^\mathrm{norm} \doteq \frac{x_{ {j}}}{\sqrt{\langle x_{ {j}}^2 \rangle _t}} = \frac{\mathbf {r}^\mathrm{T}_{ {j}}\mathbf {s}}{||\mathbf {r}_{ {j}}||_2}. \end{aligned}$$

(2.47)

Then the its value of $\varDelta$ can be calculated as:

$$\begin{aligned} \varDelta (x_{ {j}}) \doteq \langle (\dot{x}_{ {j}}^\mathrm{norm})^2 \rangle _t = \frac{\mathbf {r}_{ {j}}^\mathrm{T}\langle \dot{\mathbf {s}}\dot{\mathbf {s}}^\mathrm{T} \rangle _t\mathbf {r}_{ {j}}}{\mathbf {r}_{ {j}}^\mathrm{T}\mathbf {r}_{ {j}}} = \frac{\mathbf {r}_{ {j}}^\mathrm{T}\varvec{\varOmega }\mathbf {r}_{ {j}}}{\mathbf {r}_{ {j}}^\mathrm{T}\mathbf {r}_{ {j}}}. \end{aligned}$$

(2.48)

Notice that $\varvec{\varOmega }=\text {diag}\left\{ \omega _1, \ldots , \omega _{m(d+1)} \right\} = \text {diag}\left\{ \varDelta (s_1), \ldots , \varDelta (s_{m(d+1)}) \right\}$. Therefore (2.48) can be further decomposed as:

$$\begin{aligned} \varDelta (x_{ {j}}) = \frac{\sum _i r_{ji}^2\varDelta (s_i)}{\sum _i r_{ji}^2} = \sum _i \frac{r_{ji}^2}{\sum _i r_{ji}^2}\varDelta (s_i) \doteq \sum _i \alpha _i \varDelta (s_i). \end{aligned}$$

(2.49)

where $\alpha _i=r_{ji}^2 / \sum _i r_{ji}^2$ and $\sum _i \alpha _i=1$. This completes the proof.

Appendix B: Proof of Theorem 2.2

If k satisfy $\varDelta (s_k) > \varDelta (x_{ {j}})$, then we have:

$$\begin{aligned}&\qquad \qquad \qquad \qquad \qquad \varDelta (s_k)> \frac{\sum _i r_{ji}^2\varDelta (s_i)}{\sum _i r_{ji}^2} \\&\qquad \qquad \qquad \Rightarrow \sum _i r_{ji}^2\varDelta (s_k)> \sum _i r_{ji}^2\varDelta (s_i) \\&\Rightarrow \sum _{i \ne k} r_{ji}^2\varDelta (s_k) + r_{jk}^2\varDelta (s_k)> \sum _{i \ne k} r_{ji}^2\varDelta (s_i) + r_{jk}^2\varDelta (s_k) \\&\qquad \qquad \qquad \Rightarrow \varDelta (s_k)\sum _{i \ne k} r_{ji}^2> \sum _{i \ne k} r_{ji}^2\varDelta (s_i) \\&\qquad \qquad \quad \Rightarrow r_{jk}^2\varDelta (s_k)\sum _{i \ne k} r_{ji}^2> r_{jk}^2 \sum _{i \ne k} r_{ji}^2\varDelta (s_i) \\&\qquad \qquad \qquad \Rightarrow \frac{r_{jk}^2\varDelta (s_k)}{\sum _{i \ne k} r_{ji}^2\varDelta (s_i)}> \frac{r_{jk}^2}{\sum _{i \ne k} r_{ji}^2} \\&\qquad \qquad \Rightarrow 1+\frac{r_{jk}^2\varDelta (s_k)}{\sum _{i \ne k} r_{ji}^2\varDelta (s_i)}> 1+\frac{r_{jk}^2}{\sum _{i \ne k} r_{ji}^2} \\&\qquad \qquad \Rightarrow \frac{\sum _i r_{jk}^2\varDelta (s_k)}{\sum _{i \ne k} r_{ji}^2\varDelta (s_i)}> \frac{\sum _i r_{jk}^2}{\sum _{i \ne k} r_{ji}^2} \\&\qquad \qquad \Rightarrow \frac{\sum _i r_{jk}^2\varDelta (s_k)}{\sum _i r_{jk}^2}> \frac{\sum _{i \ne k} r_{ji}^2\varDelta (s_i)}{\sum _{i \ne k} r_{ji}^2} \\&\qquad \qquad \Rightarrow \varDelta (x_{ {j}}) > \varDelta (x_{ {j}}^\mathrm{rec}). \end{aligned}$$

This completes the proof.