Consider a Takagi-Sugeno – Kang (TSK) Model consisting of rules of the form:…
2017
Consider a Takagi-Sugeno – Kang (TSK) Model consisting of rules of the form:
If \(𝑥_1\) is \(A_{i1}\) and … and \(𝑥_𝑟\) is \(A_{ir}\)
THEN \(y=f_i(x_1, x_2, \dots , x_t)=b_{i0} + b_{i1} x_1 + \dots + b_{ir}x_r\)
assume, \(\alpha _i\) is the matching degree of rule 𝑖, then the total output of the model is given by:
- A.
\(y=\sum \limits_{i=1}^L \alpha_i f_i (x_1, x_2, \dots , x_r)\) - B.
\(y=\dfrac{\sum \limits_{i=1}^L \alpha_i f_i (x_1, x_2, \dots , x_r)}{\sum \limits_{i=1}^L \alpha _i}\) - C.
\(y=\dfrac{\sum \limits_{i=1}^L f_i (x_1, x_2, \dots , x_r)}{\sum \limits_{i=1}^L \alpha _i}\) - D.
\(y=\underset{i}{\text{max}} [\alpha _i f_i (x_1, x_2, \dots , x_r)]\)
Attempted by 41 students.
Show answer & explanation
Correct answer: B
Answer: y = (sum_{i=1}^L alpha_i f_i(x_1, ..., x_r)) / (sum_{i=1}^L alpha_i)
Reasoning:
Each rule returns a local output f_i(x_1,...,x_r) (a linear function in the Takagi–Sugeno model).
alpha_i is the matching degree (firing strength) of rule i and acts as an unnormalized weight for that rule.
Normalize the firing strengths: define w_i = alpha_i / (sum_{j=1}^L alpha_j). These normalized weights sum to 1.
Combine consequents using the normalized weights: y = sum_{i=1}^L w_i f_i = (sum_{i=1}^L alpha_i f_i) / (sum_{i=1}^L alpha_i).
Note: If the sum of alpha_i is zero (all rules have zero matching), a fallback or default output must be specified by the system to avoid division by zero.