A Hybrid Model through the Fusion of Type-2 Fuzzy Logic Systems and Sensitivity-Based Linear Learning Method for Modeling PVT Properties of Crude Oil Systems

Algorithm 3

Back propagation algorithm for FLS.

(1) Initialize the parameters of all the membership functions for all the rules, 𝑚_𝐹^𝑙_𝑘(0),𝑦^𝑙(0), and 𝜎_𝐹^𝑙_𝑘(0).

(2) Set an end criterion to achieve convergence.

(3) Repeat

 (i) for all data points (𝑥^(𝑖)∶𝑦^(𝑖)) only 𝑖=1,…,𝑁

  (a) Propagate the next data point through the FLS.

  (b) Compute error.

  (c) Update the parameters of the membership functions using (10), (11), and (12).

 (ii) end for (*end for each input-output pair*)

 (iii) Compute the root mean square relative error (RMSRE) as in (13).

 (iv) Test the end criterion. If satisfied break.

Until (*end for each epoch*)