Consider the following models: M 1: Mamdani model M 2: Takagi – Sugeno-Kang model M 3: Kosko’s additive model (SAM)
Only M 2and M 3
Understanding Additive Rule Models in Fuzzy Systems
Fuzzy logic systems use different types of models to process information and make decisions based on rules. The question asks about which of the given models are examples of additive rule models. Let's look at the characteristics of each model mentioned:
- Mamdani Model (M1): This is one of the most common types of fuzzy inference systems. In the Mamdani model, both the antecedent (IF part) and the consequent (THEN part) of the fuzzy rules are fuzzy sets. The outputs of individual rules are aggregated (often using maximum or sum, forming a combined fuzzy set) and then defuzzified to produce a crisp output. While outputs are combined, the structure isn't typically referred to as an "additive rule model" in the same sense as TSK or SAM, where rule outputs themselves are directly summed or weighted averaged before final output calculation.
- Takagi-Sugeno-Kang Model (TSK Model) (M2): In the TSK model, the antecedent is a fuzzy set, but the consequent is a crisp function of the input variables, often a linear equation. The overall output of the system is a weighted average or summation of the crisp outputs of each rule, where the weights are the firing strengths of the rules. Because the final output is obtained by summing/averaging the rule outputs, this model is considered an additive rule model.
- Kosko’s Additive Model (SAM) (M3): Kosko's Additive Model (SAM) is explicitly designed as an additive fuzzy system. It uses fuzzy rules where the consequents are typically fuzzy sets, but the overall system output is determined by summing the contributions of each rule's consequent, scaled by the rule's firing strength. This direct summation makes it an additive rule model.
Based on the nature of how the rule outputs contribute to the final system output, the TSK model (M2) and Kosko's Additive Model (M3) are examples of additive rule models because their overall output is based on a summation or weighted average of the individual rule outputs or their contributions.
The Mamdani model (M1), while aggregating fuzzy sets, does not combine rule outputs in the same direct additive or weighted additive manner as M2 and M3 to produce the final crisp output.
Comparing Fuzzy Models
| Model | Antecedent | Consequent | How rule outputs combine for final output | Considered Additive Rule Model? |
|---|---|---|---|---|
| Mamdani (M1) | Fuzzy Set | Fuzzy Set | Aggregate fuzzy sets, then defuzzify | No |
| Takagi-Sugeno-Kang (TSK) (M2) | Fuzzy Set | Crisp Function (e.g., linear) | Weighted average/summation of rule outputs | Yes |
| Kosko’s Additive Model (SAM) (M3) | Fuzzy Set | Fuzzy Set (often) | Summation of scaled rule consequent contributions | Yes |
Therefore, models M2 and M3 are examples of additive rule models.
Revision Table: Fuzzy Logic Models
Let's quickly recap the key types of fuzzy inference systems and their output mechanisms.
| Model Type | Key Characteristic | Output Combination Method |
|---|---|---|
| Mamdani | Fuzzy output per rule | Aggregation of fuzzy sets followed by defuzzification |
| Sugeno (TSK) | Crisp function output per rule | Weighted average or sum of crisp rule outputs |
| Additive (SAM) | Summation of rule contributions | Direct summation of scaled rule consequents |
Additional Information on Additive Fuzzy Systems
Additive fuzzy systems, like the TSK and SAM models, are particularly useful in applications where the relationship between inputs and outputs can be effectively represented as a sum of simpler relationships captured by the rules. The additive nature simplifies the calculation of the overall output compared to aggregating and then defuzzifying complex fuzzy sets, as done in the Mamdani model. This can lead to more computationally efficient systems, especially the TSK model which is widely used in control systems and function approximation.
Important Questions from Fuzzy Sets - Teaching
Let A α0 denotes the α-cut of a fuzzy set A at α 0. If α 1 < α 2, then
Consider the following methods:
M 1: mean of maximum
M 2: Centre of area
M 3: Height method
Which of the following is/are defuzzification method(s)?A fuzzy conjunction operator denoted as t(x,y) and fuzzy disjunction operator denoted as s(x,Y) form dual pair if they satisfy the condition:
A fuzzy conjunction operators, t(x, y), and a fuzzy disjunction operator, s(x, y), form a pair if they satisfy:
t(x, y) = 1 – s(1 – x, 1 - y).
If \(t\left( {x,\;y} \right) = \frac{{xy}}{{\left( {x + y - xy} \right)}}\) then s(x, y) is given byConsider a Takagi - Sugeno - Kanga (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, ...., x r) = b i0 + b i1 x1 + b ir xr
assume, α i is the matching degree of rule i, then the total output of the model is given by :
