Sugeno Fuzzy Inference System
Interactive demonstration of the Sugeno fuzzy inference system. Unlike Mamdani, Sugeno uses singleton (single spike) membership functions for rule consequents and mathematical functions instead of fuzzy sets, making it computationally more efficient and eliminating the need for defuzzification.
Key Differences from Mamdani
1. Singleton Consequents
Sugeno uses singleton membership functions (single spikes) for rule consequents instead of fuzzy sets. A singleton is a membership function that has a value of 1 at a single point and 0 everywhere else.
2. Mathematical Functions
Rule consequents use mathematical functions of input variables: IF x is A AND y is B THEN z = f(x,y). The most common form is zero-order Sugeno: IF x is A AND y is B THEN z = k, where k is a constant.
3. No Defuzzification Needed
Since rule consequents are already crisp values (constants or functions), the output is calculated directly as a weighted average: Output = Σ(firing_strength × rule_output) / Σ(firing_strength). This eliminates the computationally expensive defuzzification step required in Mamdani systems.
Input: Person's Height
Adjust the slider to change the input height value and observe how it affects the Sugeno fuzzy inference process.
Step 1: Fuzzification (Input Membership Functions)
Same as Mamdani: Convert the crisp input (height) into fuzzy values using membership functions. Each height value belongs to multiple fuzzy sets (Short, Average, Tall) with different degrees of membership.
Membership Values at 172cm:
μshort(172) = 0.00
μaverage(172) = 0.80
μtall(172) = 0.20
Step 2: Rule Evaluation
Evaluate fuzzy rules to determine the firing strength of each rule. The firing strength equals the membership value of the input in the rule's antecedent. Key difference: Rule consequents are constants (zero-order Sugeno) instead of fuzzy sets.
Rule 1: IF height is Short THEN size = 36 (constant)
Firing Strength = 0.00
Weighted Output = 0.00 × 36 = 0.00
Rule 2: IF height is Average THEN size = 40 (constant)
Firing Strength = 0.80
Weighted Output = 0.80 × 40 = 32.00
Rule 3: IF height is Tall THEN size = 44 (constant)
Firing Strength = 0.20
Weighted Output = 0.20 × 44 = 8.80
Step 3: Singleton Consequents (Rule Outputs)
Unlike Mamdani which uses fuzzy sets, Sugeno uses singleton membership functions (single spikes) at constant values. Each rule produces a crisp output value that is then weighted by the firing strength.
The solid lines show singleton membership functions at constant values (36, 40, 44). The dashed lines show these values weighted by their firing strengths. The final output is calculated as a weighted average of these singleton outputs.
Step 4: Weighted Average (No Defuzzification Needed)
Since rule consequents are already crisp values (constants), we calculate the final output directly as a weighted average. No defuzzification step is required, making Sugeno computationally more efficient than Mamdani.
Final Crisp Output (Weighted Average):
Recommended Size = 40.8
Calculation Method: Weighted Average (No defuzzification needed)
Formula: Output = Σ(firing_strength × rule_output) / Σ(firing_strength)
Calculation Breakdown:
Numerator = Σ(firing_strength × rule_output)
= 0.00 × 36 + 0.80 × 40 + 0.20 × 44
= 0.00 + 32.00 + 8.80
= 40.80
Denominator = Σ(firing_strength)
= 0.00 + 0.80 + 0.20
= 1.00
Output = 40.80 / 1.00 = 40.80
The weighted average directly gives us the final crisp output without needing a defuzzification step. This makes Sugeno systems computationally more efficient, especially in real-time applications and expert systems.
Comparison: Sugeno vs Mamdani
| Sugeno Method | Mamdani Method |
|---|---|
| Computationally more efficient (no defuzzification needed) | More intuitive and interpretable (uses fuzzy sets) |
| Works well with optimization and adaptive techniques | Better for human decision-making systems |
| Ensures continuity of output surface | Rules are more natural and easier to understand |
| Well-suited for mathematical analysis and linear control systems | Better for systems where linguistic interpretation is important |
Summary
- Fuzzification: Height 172cm → μ values for Short, Average, Tall (same as Mamdani)
- Rule Evaluation: Each rule fires with strength equal to its input membership. Consequents are constants (zero-order Sugeno)
- Singleton Consequents: Rule outputs are crisp values (singletons) instead of fuzzy sets
- Weighted Average: Final output = Σ(firing_strength × rule_output) / Σ(firing_strength) → Size 40.8