Fuzzy Rules and Fuzzy Inference – AI with Fuzzy Logic (Senior 6 ICT) | Ganzaa

AI with Fuzzy Logic

Lessons

Senior 6 · ICT

AI with Fuzzy Logic

5 lessons

Lesson 4 of 5

Fuzzy Rules and Fuzzy Inference

25 min

Rules an expert would say aloud

Ask an experienced farmer on a school vegetable plot in Bugesera how long to water the tomatoes, and you will not get an equation. You will get rules: "If the soil is dry and the day is hot, water for a long time. If the soil is moist and the day is mild, a short watering is enough. If it is wet, hardly water at all." Each rule uses vague words, and several rules may apply at once to some degree. A fuzzy inference system takes rules exactly like these, combines them, and produces one number: how many minutes to open the valve.

Definition

Fuzzy rule: a statement of the form IF antecedent THEN consequent, where both parts use linguistic terms. Example: IF soil is dry AND temperature is hot THEN watering time is long.

Firing strength: the degree to which a rule's IF part is true for the current inputs, found with the fuzzy operators.

Rule base: the full set of rules of a system, often written as a table.

Defuzzification: turning the combined fuzzy output into one crisp number. The most common method is the centroid (centre of area):

$z^{*} = \frac{\int z\,\mu(z)\,dz}{\int \mu(z)\,dz}$

The Mamdani method

The method used here is named after Ebrahim Mamdani, whose 1975 steam-engine controller used it. It has five steps.

The irrigation controller

Two inputs and one output, each a linguistic variable:

Variable	Terms and membership functions
Soil moisture (%)	dry: trapezoid (0, 0, 20, 40); moist: triangle (20, 40, 60); wet: trapezoid (40, 60, 100, 100)
Air temperature (°C)	mild: trapezoid (0, 0, 20, 30); hot: trapezoid (20, 30, 50, 50)
Watering time (min)	short: triangle (0, 5, 10); medium: triangle (8, 15, 22); long: triangle (20, 25, 30)

The rule base, written as a table (soil down the side, temperature across the top, watering time in the cells):

Soil \ Temperature	mild	hot
dry	medium (R2)	long (R1)
moist	short (R4)	medium (R3)
wet	short (R5)	short (R6)

Worked example 1: steps 1 to 4 for 26% moisture and 27 °C

Step 1, fuzzify. Soil: dry $= \frac{40 - 26}{40 - 20} = 0.7$ ; moist $= \frac{26 - 20}{40 - 20} = 0.3$ ; wet $= 0$ . Temperature: mild $= \frac{30 - 27}{30 - 20} = 0.3$ ; hot $= \frac{27 - 20}{30 - 20} = 0.7$ .

Step 2, firing strengths (AND = min):

Rule	IF part	Strength	THEN
R1	dry AND hot	$\min(0.7, 0.7) = 0.7$	long
R2	dry AND mild	$\min(0.7, 0.3) = 0.3$	medium
R3	moist AND hot	$\min(0.3, 0.7) = 0.3$	medium
R4	moist AND mild	$\min(0.3, 0.3) = 0.3$	short
R5, R6	wet AND …	0	short

Step 3, clip. Each output set is cut off at its rule's strength. Two rules point to medium, both at 0.3; when several rules share an output, the strongest one counts (OR = max), so medium is clipped at 0.3. Short is clipped at 0.3 and long at 0.7.

Step 4, aggregate. The clipped sets are joined with max into one shape, shaded below.

Worked example 2: step 5, the centroid

Each clipped triangle is a trapezoid. A triangle of base $w$ clipped at height $h$ has a top of width $w (1 - h)$ , so its area is

$A = \frac{h\,\bigl(w + w(1 - h)\bigr)}{2}$

Short: $w = 10,\ h = 0.3$ : $A = \frac{0.3 \times (10 + 7)}{2} = 2.55$ , centred at 5.
Medium: $w = 14,\ h = 0.3$ : $A = \frac{0.3 \times (14 + 9.8)}{2} = 3.57$ , centred at 15.
Long: $w = 10,\ h = 0.7$ : $A = \frac{0.7 \times (10 + 3)}{2} = 4.55$ , centred at 25.

Ignoring the small overlaps between neighbouring sets, the centroid is the area-weighted mean of the centres:

$z^{*} \approx \frac{2.55 \times 5 + 3.57 \times 15 + 4.55 \times 25}{2.55 + 3.57 + 4.55} = \frac{180.05}{10.67} \approx 16.9\ \text{min}$

The program in the next lesson, which integrates the exact shape, also gives 16.9 minutes. The controller waters for about 17 minutes: longer than "medium", because the strongest rule (dry and hot) points to "long", but pulled down by the three weaker rules.

MATLAB: Fuzzy Inference System Walkthrough (Fuzzy Logic, Part 2). Follow how the inputs are fuzzified, the rules fire and clip their outputs, and the result is defuzzified; it is the same five-step process as the irrigation example.

Warning

Check that the rules cover every case. If some combination of inputs makes every rule's strength 0 (for example, a gap between two sets), the aggregated shape is empty and the centroid is $\frac{0}{0}$ , undefined. A complete rule table with overlapping sets avoids this, and a program should still guard against it.

Tip

Quick estimate, then exact. The weighted mean of the peaks, $\frac{0.3 \times 5 + 0.3 \times 15 + 0.7 \times 25}{0.3 + 0.3 + 0.7} \approx 18.1$ , is a fast sanity check. It differs from the centroid because the clipped areas are not proportional to the strengths, but if your centroid is far from it, look for an error.

✦

Hands-on: run the controller by hand

Copy the three membership-function graphs and the rule table onto paper.
Choose the input 34% moisture and 24 °C. Fuzzify both inputs.
Find the strength of all six rules, then the clip level of short, medium and long.
Shade the aggregated shape and estimate the centroid with the trapezoid-area method.
Compare with a classmate who used different inputs: does a drier, hotter input always give a longer time?

Check: for 34% and 24 °C your strengths should be dry 0.3, moist 0.7, mild 0.6, hot 0.4, and the strongest rule should be R4 (moist AND mild, 0.6).

Practice

Name the five steps of Mamdani inference in order.
Fuzzify 50% soil moisture and 22 °C with the sets in this lesson.
For those inputs, find the strength of each rule and the clip level of each output set.
Why is medium clipped at 0.3 in worked example 1, although two rules point to it?
A triangle of base 14 is clipped at 0.5. Find the area of the clipped shape.
Use the trapezoid-area method to estimate the centroid when short is clipped at 0, medium at 0.6 and long at 0.2.
A designer removes rule R6 (wet AND hot → short). Analyse what happens for 70% moisture on a hot day.

Show answersHide answers

Fuzzify the inputs; evaluate each rule's IF part (firing strength); clip each rule's output set; aggregate the clipped sets with max; defuzzify (centroid).
Soil: dry 0; moist $\frac{60 - 50}{20} = 0.5$ ; wet $\frac{50 - 40}{20} = 0.5$ . Temperature: mild $\frac{30 - 22}{10} = 0.8$ ; hot $\frac{22 - 20}{10} = 0.2$ .
R1, R2: 0 (dry is 0). R3 moist AND hot: 0.2. R4 moist AND mild: 0.5. R5 wet AND mild: 0.5. R6 wet AND hot: 0.2. Short: $\max(0.5, 0.5, 0.2) = 0.5$ ; medium: 0.2; long: 0.
When several rules give the same output term, their strengths are combined with OR (max): $\max(0.3, 0.3) = 0.3$ .
$\frac{0.5 \times (14 + 7)}{2} = 5.25$ .
Medium: $\frac{0.6 \times (14 + 5.6)}{2} = 5.88$ at 15. Long: $\frac{0.2 \times (10 + 8)}{2} = 1.8$ at 25. $z^{*} \approx \frac{5.88 \times 15 + 1.8 \times 25}{7.68} = \frac{133.2}{7.68} \approx 17.3$ min.
At 70% only wet is non-zero (1), and on a hot day only hot is non-zero (1, if at least 30 °C). R5 needs mild (0) and R6 is gone, so every rule has strength 0: the output is empty and the centroid undefined. The rule table must stay complete.

Sources and credits

Fuzzy control system, Wikipedia contributors, Wikipedia, licensed under CC BY-SA 4.0. Adapted by Ganzaa.
Defuzzification, Wikipedia contributors, Wikipedia, licensed under CC BY-SA 4.0. Adapted by Ganzaa.
E. H. Mamdani and S. Assilian, "An experiment in linguistic synthesis with a fuzzy logic controller", International Journal of Man-Machine Studies, 1975 (cited, not copied).
Video: Fuzzy Inference System Walkthrough | Fuzzy Logic, Part 2, MATLAB, embedded from YouTube.
The irrigation controller, its rules, graphs, worked examples, hands-on task and practice items were written by Ganzaa. The controller is a teaching design, not agronomic advice.

This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1Rules an expert would say aloud

2The Mamdani method

3The irrigation controller

4Hands-on: run the controller by hand

Rules an expert would say aloud

The Mamdani method

The irrigation controller

Hands-on: run the controller by hand