EE368 Soft Computing Dr. Unnikrishnan P.C. Professor, EEE
Module III • Hybrid Intelligent Systems • Fuzzy Expert Systems
Hybrid IntelligentSystems: • Neural expert systems and Neuro-fuzzysystems • Introduction • Neural expertsystems • Fuzzy expert Systems • Neuro-Fuzzysystems • ANFIS: Adaptive Neuro-FuzzyInference System
Fuzzy Rules • In 1973, Lotfi Zadeh published his second most influential paper. He suggested capturing human knowledge in fuzzy rules. • A fuzzy rule can be defined as a conditional statement in the form: IFx is A, THEN y is B where • xand y are linguistic variables; • Aand B are linguistic values determined by fuzzy sets on the universe of discourses X and Y, respectively. • Antecedent (or condition): x is A • Consequent (or conclusion): y is B
Classical vs. Fuzzy Rules • Classical rule: Rule 1:Rule 2: IF speed is > 100 (km/h) IF speed is < 40 (km/h) THEN stopping_distanceis > 100m THEN stopping_distance is < 40m • Fuzzy rule: Rule 1:Rule 2: IF speed is fast IF speed is slow THEN stopping_distance is long THEN stopping_distance is short • Fuzzy rules relate fuzzy sets. • In a fuzzy system, all rules fire partially.
Firing Fuzzy Rules IF height is tall THEN weight is heavy
Firing Fuzzy Rules • If the antecedent is true to some degree of membership, then the consequent is also true to that same degree. • This form of fuzzy inference is called monotonic selection.
Firing Fuzzy Rules • A fuzzy rule can have multiple antecedents, for example: IF project_duration is long AND project_staffing is large AND project_funding is inadequate THEN risk is high IF service is excellent OR food is delicious THEN tip is generous
Firing Fuzzy Rules • The consequent of a fuzzy rule can also include multiple parts, for instance: IF temperature is hot THEN hot_water is reduced; cold_water is increased • Solutions: Mamdanior Sugenoapproaches
Fuzzy Inference Techniques • Mamdani • The most commonly used fuzzy inference technique • He built one of the first fuzzy systems to control a steam engine • He applied a set of fuzzy rules supplied by experienced human operators. • E. Mamdani, “Application of fuzzy algorithms for control of simple dynamic plant” (Proc. IEE, Vol.121, No. 12, pp. 1585-1588, 1974) • E. Mamdani and S. Assilian, “An experiment in linguistic synthesis with a fuzzy logic controller”, (Int. J. of Man-Machine Studies, Vol.7, No.1, pp. 1- 13, 1975)
Fuzzy Inference Techniques • Sugeno • The ‘Zadeh of Japan’ • Sugeno, Michio. ”Industrial applications of fuzzy control,” Elsevier Science Inc., 1985.
Mamdani Fuzzy Inference • Four steps: • Fuzzification of the input variables • Rule evaluation (inference) • Aggregation of the rule outputs (composition) • Defuzzification.
Mamdani Fuzzy Inference We examine a simple two-input one-output problem that includes three rules: Rule: 1Rule: 1 IF x is A3 IF project_funding is adequate OR y is B1 OR project_staffing is small THEN z is C1 THEN risk is low Rule: 2Rule: 2 IF x is A2 IF project_funding is marginal AND y is B2 AND project_staffing is large THEN z is C2 THEN risk is normal Rule: 3Rule: 3 IF x is A1 IF project_funding is inadequate THEN z is C3 THEN risk is high
Take the crisp inputs, x1 and y1 (project funding and project staffing; e.g. x1=2million, y1:10 persons), and determine the degree to which these inputs belong to each of the appropriate fuzzy sets. Step 1: Fuzzification A1: Inadequate, A2: Marginal, A3: Adequate B1: Small, B2: Large
Step 2: Rule Evaluation • Take the fuzzified inputs, (x=A1) = 0.5, (x=A2) = 0.2, (y=B1) = 0.1 and (y=B2) = 0.7, and apply them to the antecedents of the fuzzy rules. • If a given fuzzy rule has multiple antecedents, the fuzzy operator (AND or OR) is used to obtain a single number that represents the result of the antecedent evaluation. • This number (the truth value) is then applied to the consequent membership function. (monotonic selection)
Step 2: Rule Evaluation • How the result of the antecedent evaluation can be applied to the membership function of the consequent? • Clipping(alpha-cut) • Cut the consequent membership function at the level of the antecedent truth. • losing some information. • it is often preferred because it involves less complex and faster mathematics • Scaling • offers a better approach for preserving the original shape of the fuzzy set. • Multiplying all its membership degrees by the truth value of the rule antecedent. • It loses less information
Step 2: Rule Evaluation scaling clipping
Step 3: Aggregation of the rule outputs • The process of unification of the outputs of all rules. • Combining with MAX operator
Input: the aggregate output fuzzy set Output: a single number The most popular method: Centroid technique. It finds the point where a vertical line would slice the aggregate set into two equal masses. Mathematically, it’s the center of gravity (COG) Step 4: Defuzzification
A reasonable estimate can be obtained by calculating it over a sample of points. Step 4: Defuzzification
Sugeno Fuzzy Inference • In Mamdani-style inference, to find the centroid, an integration across a continuously varying function is required. no computationally efficient! • MichioSugeno suggested to use a single spike, a singleton • Fuzzy Rules in zero-order Sugeno fuzzy model: IF x is A AND y is B THEN z is k where k is a constant.
Sugeno Aggregation of the Rule Outputs Rule 1: IF project_funding is adequate OR project_staffing is small, THEN risk is k1 Rule 2: IF project_funding is marginal AND project_staffing is large, THEN risk is k2 Rule 3: IF project_funding is inadequate, THEN risk is k3
Sugeno Defuzzification Weighted Average (WA) Suppose: k1=20, k2=50, k3=80
Mamdani or Sugeno? • Mamdani • widely accepted for capturing expert knowledge • more intuitive, more human-like manner • a substantial computational burden • Sugeno • computationally effective • works well with optimization and adaptive techniques • e.g. control problems, particularly for dynamic nonlinear systems.
Advantages and Problems of Fuzzy Logic • Advantages • general theory of uncertainty • wide applicability, many practical applications • natural use of vague and imprecise concepts • helpful for commonsense reasoning, explanation • Problems • membership functions can be difficult to find • multiple ways for combining evidence • problems with long inference chains