Fault isolation in manufacturing systems based on learning algorithm and fuzzy rule selection

Abstract

This paper addresses the fault detection and isolation problem in manufacturing systems. Some of these systems can be affected by several faults, a first way of determining them is to use classification and rule-based reasoning methods. In the present work, a new hybrid algorithm, based on fuzzy Levenberg–Marquardt and genetic algorithm for both training and fault isolation in the high-dimensional setting, is developed. The genetic algorithm-based approach aims at selecting an optimal number of production rules. The developed approach consists then to minimize training time and to find accurate and interpretable fuzzy systems with an appropriate production rules subset. It is put into practice for a real manufacturing system for binary classes and also its advantage is demonstrated on multi-class system. Obtained results show that the approach can be more accurate and fast to make fault diagnosis for binary and multi-class problems compared to those reported in the literature.

Appendix: Bayes’ maximum likelihood formula

Consider a set of n patterns of known classes $\{x_{1}, x_{2},\ldots , x_{n},\}$ in N dimensional feature space, where it is assumed that each pattern corresponds to one of the possible classes.

We denote $P_i$ as the priori probability, while $p_i(x)$ is the class conditional density corresponding to the class $C_i (i = 1, 2, \ldots , M)$. If the classifier decides x to be from the class $C_i$, when it currently comes from unknown class (unknown fault) called $C_l$, it incurs a loss equal to $L_{li}$. The conditional average loss incurred in assigning a pattern x to the fault $C_i$ is presented according to the following equation:

$$\begin{aligned} r_{i}(x)= \sum _{l=1}^{m} L_{li} P(C_{l}/x) \end{aligned}$$

(19)

where $P(C_{l}/x)$ gives the probability that pattern x is from $C_l$. Based on the formula of Bayes, Eq. 19 can be expressed as:

$$\begin{aligned} r_{i}(x)= \frac{1}{P(x)}\sum _{l=1}^{m} L_{li} P_{l} (x) P_{l} \end{aligned}$$

(20)

where

$$\begin{aligned} P(x)= \sum _{l=1}^{m} P_{l}(x) P_{l} \end{aligned}$$

(21)

For each dataset, the pattern x is affected to the class with the minimum probability of misclassification.