A FUZZY INFERENCE SYSTEM (FIS) maps crisp inputs to crisp outputs through fuzzy rules. FOUR COMPONENTS: crisp inputs │ ▼ ┌─────────────────┐ │ FUZZIFIER │ crisp values → membership degrees └────────┬────────┘ ▼ ┌─────────────────┐ ┌──────────────────┐ │ INFERENCE │◀─────│ RULE BASE │ │ ENGINE │ │ IF … THEN … │ │ (evaluate and │ └──────────────────┘ │ aggregate) │ └────────┬────────┘ ▼ ┌─────────────────┐ │ DEFUZZIFIER │ a fuzzy output set → one number └────────┬────────┘ ▼ crisp output 1. FUZZIFIER — applies the membership functions of the previous topic to each crisp input, producing a degree for each linguistic term. 2. RULE BASE (the knowledge base) — the fuzzy IF-THEN rules, supplied by an expert or learned. The IF part is the ANTECEDENT, the THEN part the CONSEQUENT. 3. INFERENCE ENGINE — computes each rule's firing strength, applies it to the rule's output set (IMPLICATION), and combines all the rules (AGGREGATION). 4. DEFUZZIFIER — reduces the aggregated fuzzy output to a single crisp value. THE FOUR STEPS OF INFERENCE, named as the syllabus expects: FUZZIFICATION → RULE EVALUATION → AGGREGATION → DEFUZZIFICATION A RULE with several antecedents combines them with the fuzzy operators: IF temperature IS hot AND humidity IS high THEN fan IS fast firing strength = min(μ_hot, μ_high) IF temperature IS hot OR humidity IS high THEN fan IS fast firing strength = max(μ_hot, μ_high) MOST RULES USE AND, so min is the usual case. A KEY PROPERTY: SEVERAL RULES FIRE AT ONCE, each to a different degree. That is what produces smooth output — as an input drifts, one rule's strength falls while another's rises, and the defuzzified result moves continuously rather than jumping. THE OVERLAP IN THE MEMBERSHIP FUNCTIONS IS WHAT MAKES THIS WORK, as the previous topic noted.

A FAN CONTROLLER. Inputs: temperature and humidity. Output: fan speed, 0–100%. THE MEMBERSHIP FUNCTIONS (triangular): TEMPERATURE cold(0,10,20) warm(15,25,35) hot(25,35,45) HUMIDITY low(0,25,50) medium(25,50,75) high(50,75,100) FAN SPEED low(0,0,50) medium(25,50,75) high(50,100,100) THE RULE BASE: R1: IF temp = warm AND humidity = medium THEN fan = medium R2: IF temp = hot AND humidity = medium THEN fan = high R3: IF temp = hot AND humidity = high THEN fan = high R4: IF temp = warm AND humidity = high THEN fan = high THE INPUT: temperature = 28 °C, humidity = 60%. ── STEP 1: FUZZIFICATION ──────────────────────────────────── TEMPERATURE = 28: cold: 28 ≥ 20, so 0.00 warm: (35 − 28)/(35 − 25) = 7/10 = 0.70 hot: (28 − 25)/(35 − 25) = 3/10 = 0.30 HUMIDITY = 60: low: 60 ≥ 50, so 0.00 medium: (75 − 60)/(75 − 50) = 15/25 = 0.60 high: (60 − 50)/(75 − 50) = 10/25 = 0.40 ── STEP 2: RULE EVALUATION (firing strengths, using min) ───── R1: min(warm 0.70, medium 0.60) = 0.60 → fan = medium R2: min(hot 0.30, medium 0.60) = 0.30 → fan = high R3: min(hot 0.30, high 0.40) = 0.30 → fan = high R4: min(warm 0.70, high 0.40) = 0.40 → fan = high ALL FOUR RULES FIRE, to different degrees. Note that R2, R3 and R4 all conclude "high" with strengths 0.30, 0.30 and 0.40. ── STEP 3: AGGREGATION (combine with max, per output set) ──── fan = medium : max(0.60) = 0.60 fan = high : max(0.30, 0.30, 0.40) = 0.40 fan = low : nothing fired = 0.00 THE MAX IS THE OR: several rules supporting the same conclusion do not add up — the strongest evidence stands. This is the fuzzy union of the clipped output sets. ── STEP 4: DEFUZZIFICATION ────────────────────────────────── The aggregated output is a fuzzy set: the "medium" triangle clipped at 0.60 unioned with the "high" triangle clipped at 0.40. It must become one number. CENTROID METHOD (centre of gravity) — the most common: crisp output = Σ x · μ(x) / Σ μ(x) over the output range. Computing this over fan speed 0–100 gives FAN SPEED = 62.5% WEIGHTED AVERAGE METHOD — simpler, using each output set's peak: peaks: medium = 50, high = 75 (0.60 × 50 + 0.40 × 75) / (0.60 + 0.40) = (30 + 30) / 1.00 = 60.0% THE TWO ANSWERS DIFFER SLIGHTLY (62.5 against 60.0) because the centroid accounts for the actual SHAPE of the clipped sets while the weighted average uses only their peaks. Both are defensible; the centroid is standard, and the weighted average is what you compute by hand in an exam. READ WHAT THE SYSTEM DID: from "28 degrees and 60% humidity" it produced "run the fan at about 60%", by way of four rules stated in ordinary words. AND IF THE TEMPERATURE RISES TO 29, every membership shifts slightly, every firing strength shifts, and the output moves smoothly upward — NO THRESHOLD IS CROSSED AND NOTHING JUMPS. That is the property a crisp rule base cannot provide.

TWO STANDARD FIS TYPES, and the distinction is examined directly. MAMDANI (Mamdani–Assilian, 1975) — what the example above used THE CONSEQUENT IS A FUZZY SET. IF temp IS hot THEN fan IS high · output is a fuzzy set, so DEFUZZIFICATION IS REQUIRED · INTUITIVE and readable — the rules look like human statements throughout · widely accepted for capturing EXPERT KNOWLEDGE · computationally heavier, because aggregation and defuzzification operate over the output range → USE IT when the rules come from a human and interpretability matters. SUGENO (Takagi–Sugeno–Kang, 1985) THE CONSEQUENT IS A FUNCTION OF THE INPUTS. IF temp IS hot THEN fan = 2·temp + 0.5·humidity − 20 · a zero-order Sugeno system uses a constant: IF temp IS hot THEN fan = 80 · output is computed as the WEIGHTED AVERAGE of the rule outputs, so NO DEFUZZIFICATION STEP IS NEEDED: output = Σ wᵢ · fᵢ / Σ wᵢ where wᵢ is rule i's firing strength · COMPUTATIONALLY EFFICIENT, and well suited to OPTIMISATION and ADAPTIVE techniques — which is why ANFIS from the previous topic is built on Sugeno systems, since the consequents' parameters are differentiable and can be tuned by gradient descent · works well with LINEAR control theory · LESS INTUITIVE: the consequent is an equation, not a phrase → USE IT when the system will be TUNED FROM DATA or must run fast. THE COMPARISON: MAMDANI SUGENO ───────────────────────────────────────────────────────────── consequent a fuzzy set a function of the inputs defuzzification REQUIRED not needed interpretability high lower computation heavier lighter knowledge source human expert data / optimisation best for expert systems, adaptive control, readable control neuro-fuzzy ───────────────────────────────────────────────────────────── (A third, TSUKAMOTO, uses monotonic membership functions in the consequent so each rule's output is a single value; it is rarely used and worth knowing only by name.) DEFUZZIFICATION METHODS — the list, with what each does: CENTROID (centre of gravity) the balance point of the aggregated area. Standard, smooth, most computation. BISECTOR the line dividing the area into two equal halves. MEAN OF MAXIMUM (MOM) the average of the points at which membership is maximal. SMALLEST / LARGEST OF the extremes of the maximal MAXIMUM region — used where a conservative or aggressive response is wanted. WEIGHTED AVERAGE each output set's peak, weighted by its strength. Cheap; the hand-calculation method. THE CHOICE AFFECTS THE OUTPUT, as the 62.5 versus 60.0 above showed, and it is a design decision rather than a detail — two engineers using different defuzzifiers on the same rule base get different controllers. THE ASSESSMENT — advantages: · captures expert knowledge in the expert's own vague terms · SMOOTH, gradual control action with no threshold jumps · robust to imprecise inputs and sensor noise · rules remain READABLE and individually modifiable · no mathematical model of the plant is required, which is the decisive advantage where the plant is too complex to model LIMITATIONS: · MEMBERSHIP FUNCTIONS AND RULES MUST BE CHOSEN, and there is no principled derivation — the design is expert judgement · the RULE COUNT EXPLODES with inputs: with n inputs each having m terms, a complete rule base has m^n rules 2 inputs × 3 terms → 9 rules 3 inputs × 5 terms → 125 rules 4 inputs × 5 terms → 625 rules THIS IS THE CURSE OF DIMENSIONALITY, appearing in fuzzy form, and it is why fuzzy control is used on problems with few inputs. · NO LEARNING in the basic form — hence neuro-fuzzy · no formal stability guarantees comparable to classical control theory, which is a genuine objection in safety-critical use · foundations weaker than probability's, as the previous topic conceded THE HONEST VERDICT: A FUZZY INFERENCE SYSTEM IS A WAY OF INTERPOLATING BETWEEN EXPERT-STATED OPERATING POINTS. That is less grand than "fuzzy reasoning" suggests and more useful than it sounds — interpolating smoothly between a handful of rules an operator can state is exactly what a great many control problems need, and it requires no model of the system being controlled.