Fuzzy Logic Systems: Continuous Membership, Fuzzy Inference & Defuzzification
Pioneered by Lotfi A. Zadeh in 1965, Fuzzy Logic generalizes classical Aristotelian binary logic{0, 1} into a continuous spectrum of truth values [0, 1], allowing algorithms to reason mathematically under epistemic vagueness, linguistic ambiguity, and incomplete sensor data.
1. Crisp Sets vs. Fuzzy Sets: The Membership Function
In classical set theory, an element either belongs to a set or does not: χ_A(x) ∈ {0, 1}. In a fuzzy set A over a universe of discourse X, membership is defined by a continuous characteristic membership function:
Standard Zadeh Operations
- Intersection (AND):
μ_(A ∩ B)(x) = min(μ_A(x), μ_B(x)) - Union (OR):
μ_(A ∪ B)(x) = max(μ_A(x), μ_B(x)) - Complement (NOT):
μ_(¬A)(x) = 1 - μ_A(x)
Common Geometries
- Triangular: Defined by 3 parameters (a, m, b); computationally minimal.
- Trapezoidal: Defined by 4 parameters (a, b, c, d); models plateau regions.
- Gaussian: Smooth, infinitely differentiable:
e^(-(x - c)^2 / 2σ^2).
Linguistic Variables
Variables whose values are words rather than numbers. E.g., "Temperature" decomposes into fuzzy linguistic terms: Freezing, Cold, Warm, Boiling.
2. Fuzzy Inference Systems (FIS): Mamdani vs. Takagi-Sugeno-Kang (TSK)
Intuitive Linguistic Consequents
Both antecedents and consequents are defined by fuzzy linguistic sets:
Output: A continuous aggregated fuzzy set surface requiring geometric defuzzification (e.g. Centroid method). Highly interpretable for expert human control rules.
Mathematical Functional Consequents
Antecedents are fuzzy sets, but consequents are mathematical linear polynomial functions:
Output: Computed directly as a weighted average. Computationally ultra-fast, perfectly suited for adaptive neuro-fuzzy inference systems (ANFIS) and linear control theory.
3. Defuzzification: Converting Fuzzy Sets to Crisp Control Actions
Physical actuators (motors, valves, brake pedals) cannot execute a fuzzy set; they demand an exact crisp scalar value. The Center of Gravity (Centroid / COG) method evaluates the center of area under the aggregated membership curve:
