Prof. Pushpak Bhattacharyya, IIT Bombay1 CS 621 Artificial Intelligence Lecture /08/05 Prof. Pushpak Bhattacharyya Fuzzy Inferencing

Prof. Pushpak Bhattacharyya, IIT Bombay IIT Bombay2 Example: INVERTED PENDULUM MOTOR θ θ = angular displacement θ` = dθ/dt = angular velocity i = current

Prof. Pushpak Bhattacharyya, IIT Bombay IIT Bombay3 Rules are of the form If displacement is large, then current is large in the opposite direction. If velocity is small, then current needed is small. Any numerically expressed rule base will be of infinite size. For every θ and θ ` we specify a rule.

Prof. Pushpak Bhattacharyya, IIT Bombay IIT Bombay4 Fuzzy Rule Base Expressed as a table 0 – region Large Med Small Positive Negative θ θ`

Prof. Pushpak Bhattacharyya, IIT Bombay IIT Bombay5 The Cell – contents i - value Z-S-M +SZ-S +M+ SZ Z – Zero S – Small M - Med L - Large θ θ` If θ is Z and θ` is +S then i is -S

Prof. Pushpak Bhattacharyya, IIT Bombay IIT Bombay6 Rules Centre: If θ is z and θ ` is z then i is z. If θ is z and θ ` is +s then i is –s. If θ is z and θ ` is –s then i is +s.

Prof. Pushpak Bhattacharyya, IIT Bombay IIT Bombay7 Rules (Contd 1) 4) If θ is +s and θ ` is z then i is –s. 5) If θ is –s and θ ` is z then i is +s. 6) If θ is +s and θ ` is –s then i is z.

Prof. Pushpak Bhattacharyya, IIT Bombay IIT Bombay8 Rules (Contd 2) 7) If θ is –s and θ ` is +s then i is z. 8) If θ is +s and θ ` is +s then i is –M. 9) If θ is –s and θ ` is –s then i is +M.

Prof. Pushpak Bhattacharyya, IIT Bombay IIT Bombay9 Scenario At every moment 1)θ and θ ` are read by sensors. 2)These numerical values are passed through profiles to get μ – Fuzzification. 3)Rules are computed on the LHS using primitive First order logic operations. 4)Truth values transferred to the RHS (Luk. System). 5)From the μ values in RHS compute the i-value. 5 th step - Computing the target variable - numeric value DEFUZZIFICATION - Centroid Method

Prof. Pushpak Bhattacharyya, IIT Bombay IIT Bombay10 Fundamentally the operation is FORWARD CHAINING. Unlike precise logic (crisp logic). Many rules apply, potentially all. Corresponds to VERDICT BY MULTIPLE JURY, where each rule is a judge. Forward Chaining

Prof. Pushpak Bhattacharyya, IIT Bombay IIT Bombay11 Obtain Profiles 1)Profiles of small, medium and large quantities for θ, θ ` and i. 2)Even Zero (z) needs a profile. DOMAIN EXPERTS, EMPIRICAL.

Prof. Pushpak Bhattacharyya, IIT Bombay IIT Bombay12 Zero - Medium + Medium Alternate Form Of Zero

Prof. Pushpak Bhattacharyya, IIT Bombay IIT Bombay13 Consider θ = 0.2 degree θ ` = 0.8 degree/sec Work in terms of Units θ = 0.2 unit, θ ` = 0.8 unit

Prof. Pushpak Bhattacharyya, IIT Bombay14 Each of θ and θ ` values can be used to read μ s from all profiles θ