Monitoring and diagnosis of complex production process based on free energy of Gaussian–Bernoulli restricted Boltzmann machine

Abstract

Online monitoring and diagnosis of production processes face great challenges due to the nonlinearity and multivariate of complex industrial processes. Traditional process monitoring methods employ kernel function or multilayer neural networks to solve the nonlinear mapping problem of data. However, the above methods increase the model complexity and are not interpretable, leading to difficulties in subsequent fault recognition/diagnosis/location. A process monitoring and diagnosis method based on the free energy of Gaussian–Bernoulli restricted Boltzmann machine (GBRBM-FE) was proposed. Firstly, a GBRBM network was established to make the probability distribution of the reconstructed data as close as possible to the probability distribution of the raw data. On this basis, the weights and biases in GBRBM network were used to construct F statistics, which represents the free energy of the sample. The smaller the energy of the sample is, the more normal the sample is. Therefore, F statistics can be used to monitor the production process. To diagnose fault variables, the F statistic for each sample was decomposed to obtain the Fv statistic for each variable. By analyzing the deviation degree between the corresponding variables of abnormal samples and normal samples, the cause of process abnormalities can be accurately located. The application of converter steelmaking process demonstrates that the proposed method outperforms the traditional methods, in terms of fault monitoring and diagnosis performance.

Appendix

1.1 A. Deduction of relationship between F and Fv statistic

There is a linear correlation between the sample free energy F and the variable free energy Fv. The proof is as follows.

The first term of Eq. (5) can be written as Eq. (14), that is, the first term of Eq. (5) is the linear addition of the first term of p Eq. (9).

$$\sum\limits_{i = 1}^{p} {\frac{{\left( {v_{i} - a_{i} } \right)^{2} }}{{2\sigma_{i}^{2} }}} = \frac{{\left( {v_{1} - a_{1} } \right)^{2} }}{{2\sigma_{1}^{2} }} + \frac{{\left( {v_{2} - a_{2} } \right)^{2} }}{{2\sigma_{2}^{2} }} + \cdots \frac{{\left( {v_{p} - a_{p} } \right)^{2} }}{{2\sigma_{p}^{2} }}$$

(14)

The exponential term in the second term of Eq. (5) can be expanded as:

$$\begin{gathered} \sum\limits_{j = 1}^{q} {\left( {b_{j} + \sum\limits_{i = 1}^{p} {\frac{{w_{ji} v_{i} }}{{\sigma_{i} }}} } \right)} = b_{1} + w_{11} v_{1} /\sigma_{1} + w_{12} v_{2} /\sigma_{2} + \cdots + w_{1p} v_{{p}} /\sigma_{{p}} + b_{2} + w_{21} v_{1} /\sigma_{1} + w_{22} v_{2} /\sigma_{2} + \cdots + w_{2p} v_{{p}} /\sigma_{{p}} \\ + \cdots + b_{q} + w_{q1} v_{1} /\sigma_{1} + w_{q2} v_{2} /\sigma_{2} + \cdots + w_{qp} v_{{p}} /\sigma_{{p}} \\ \end{gathered}$$

(15)

The exponential term in the second term of Eq. (9) can be unfolded as:

$$\sum\limits_{j = 1}^{q} {\left( {b_{j} + \frac{{w_{ji} v_{i} }}{{\sigma_{i} }}} \right)} = b_{1} + w_{1i} v_{i} /\sigma_{i} + b_{2} + w_{2i} v_{i} /\sigma_{i} + \cdots + b_{q} + w_{qi} v_{i} /\sigma_{i}$$

(16)

Therefore, the relationship between Eqs. (15) and (16) is shown in Eq. (17):

$$\sum\limits_{j = 1}^{q} {\left( {b_{j} + \sum\limits_{i = 1}^{p} {\frac{{w_{ji} v_{i} }}{{\sigma_{i} }}} } \right)} { = }\sum\limits_{i = 1}^{p} {\sum\limits_{{{j} = 1}}^{{q}} {\left( {{b}_{{j}} + \frac{{{w}_{{{ji}}} {v}_{{i}} }}{{\sigma_{i} }}} \right)} - \left( {p - 1} \right)} \sum\limits_{{{\it{j}} = 1}}^{{\it{q}}} {{b}_{j}}$$

(17)

Let $A = - \left( {p - 1} \right)\sum\limits_{{{j}} = 1}^{{q}}b_{j}$, and then the exponential term of Eq. (5) is the linear addition of p exponent terms of Eq. (9) plus a constant A.

In summary, the F statistic is the linear addition of the Fv statistics of all variables plus a constant C.

$$F{ = }\sum\limits_{i = 1}^{p} {Fv\left( {v_{i} } \right)} + C$$

(18)

where constant $C \approx - \left( {p - 1} \right)\sum\limits_{{{j}} = 1}^{{q}}b_{j}$.

1.2 B. Monitoring results of traditional methods

See Fig. 7.