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Tham khảo tài liệu 'frontiers in robotics, automation and control part 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Fault Detection with Bayesian Network 353 Or equivalently pln c _ CL 1-1 c 1 c CPL ln c 0 30 31 Equation 31 admits two solutions c 1 not acceptable and a second solution numerically computable which depends of p and a . With the coefficient c correctly computed we obtain the equivalence between the bayesian network and the multivariate control charts. We precise that as univariate charts are simply a particular case of multivariate control chart the proof given is also available for univariate control charts. In order to demonstrate the proposed approach we illustrate it on a simple system with two variables. 4.2 Detection with bayesian network We will study a T2 control chart and a MEWMA control chart with Ấ 0.1 modelized by bayesian networks. We choose a false alarm rate a 1 . When the system is in-control it follows a multivariate Gaussian distribution with parameters n and E such as M 5 10 1 1 2i E l I Ự.2 2 J 32 33 In order to monitor this process we apply the proposed method of detection with bayesian network. So for a T control chart we obtain the bayesian network of the figure 6. We have also given the conditional probability table of each node and where c is equal to 95.28 solution of equation 31 for a 1 and p 2 . C Class C IC OC 1-a a X C X IC X N g E OC X N g cxE Fig. 6. Bayesian Network similar to T2 control chart 354 Frontiers in Robotics Automation and Control In the same way we can also monitor the process with a MEWMA control chart modelized by the bayesian network of the figure 7 where c is equal to 90.29 solution of equation 31 for a 1 and p 2 in the MEWMA case . C Clas s C IC OC 1-a a Y C Y IC Y N yjt OC Y N n 2 Fig. 7. Bayesian Network similar to MEWMA control chart We have simulated this system on 30 observations. But a fault has been introduced from observation 6 to 30. This fault is a mean step of magnitude 0.5 on the first variable. The figure 8 represents the decision taken at each instant respectively for the T2 chart left graphs and for .