Summary
We have a natural quest to know input-output relationships of a process. A fuzzy logic controller (FLC) will be an appropriate choice to tackle the said problem, if there are some imprecisions and uncertainties associated with the data. The present chapter deals with Takagi and Sugeno approach of FLC, in which a better accuracy is generally obtained compared to Mamdani approach but at the cost of interpretability. For most of the real-world processes, the input-output relationships might be nonlinear in nature. Moreover, the degree of non-linearity could not be the same over the entire range of the variables. Thus, the same set of response equations (obtained through statistical regression analysis for the entire range of the variables) might not hold equally good at different regions of the variable-space. Realizing the above fact, an attempt has been made to cluster the data based on their similarity among themselves and then cluster-wise linear regression analysis has been carried out, to determine the response equations. Moreover, attempts have been made to develop Takagi and Sugeno approach of FLC clusterwise, by utilizing the above linear response equations as the consequent part of the rules, which have been further optimized to improve the performances of the controllers. Moreover, two types of membership function distributions, namely linear (i.e., triangular) and non-linear (i.e., 3rd order polynomial) have been considered for the input variables, which have been optimized during the training. The performances of the developed three approaches (i.e., Approach 1: Cluster-wise linear regression; Approach 2: Cluster-wise Takagi and Sugeno model of FLC with linear membership function distributions for the input variables; Approach 3: Cluster-wise Takagi and Sugeno model of FLC with nonlinear (polynomial) membership function distributions for the input variables) have been tested on two physical problems. Approach 3 is found to outperform the other two approaches.
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Abbreviations
- a :
-
Abrasive mesh size (micro-m)
- a 0, a 1,…, a p :
-
Coefficients
- A 1, A 2, …, A p :
-
Membership function distributions corresponding to the linguistic terms
- c :
-
% concentration of the abrasive
- C 1 :
-
Welding speed (cm/min)
- C 2 :
-
Wire feed rate (cm/min)
- C 3 :
-
% cleaning
- C 4 :
-
Arc gap (mm)
- C 5 :
-
Welding current (Amp)
- CO ij :
-
Calculated value ofj-th output for i-th case
- D ij :
-
Euclidean distance between i and j
- ¯D :
-
Mean distance
- E i :
-
Entropy of i-th data point
- gk :
-
Half base-width of membership function distribution of x k variable
- G :
-
Generation number
- h ijk :
-
Variation of the value of a co-efficient
- n :
-
No. of clusters
- n′:
-
No. of cycles
- N :
-
No. of data points
- O 1 :
-
Front height of weld bead (mm)
- O 2 :
-
Front width of weld bead (mm)
- O 3 :
-
Back height of weld bead (mm)
- O 4 :
-
Back width of weld bead (mm)
- p :
-
No. of input variables
- p c :
-
Crossover probability
- p m :
-
Mutation probability
- P :
-
Population size
- q :
-
No. of output variables
- Q :
-
No. of fired rules
- R :
-
Dimension of data points
- R a :
-
Surface roughness (micro-m)
- S ij :
-
Similarity between the data points i and j
- T :
-
No. of training cases
- [T]:
-
Hyperspace
- TO ij :
-
Target value of j-th output for i-th case
- v :
-
Flow speed of abrasive media (cm/min)
- w i :
-
Control action of i-th rule
- x 1, x 2, …, x p :
-
Set of input dimensions of a variable
- x 1′, x 2′, …, x N ′:
-
Data points
- y :
-
Output
- α :
-
Constant
- β :
-
Threshold for similarity
- γ :
-
Threshold for determining a valid cluster
- μ :
-
Membership function value
- FLC :
-
Fuzzy logic controller
- GA :
-
Genetic Algorithm
- H :
-
High
- KB :
-
Knowledge base
- L :
-
Low
- M :
-
Medium
- MRR :
-
Material removal rate (mg/min)
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Kumar Pratihar, T., Kumar Pratihar, D. (2008). Cluster-wise Design of Takagi and Sugeno Approach of Fuzzy Logic Controller. In: Abraham, A., Grosan, C., Pedrycz, W. (eds) Engineering Evolutionary Intelligent Systems. Studies in Computational Intelligence, vol 82. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75396-4_8
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