Lecture 3 Fuzzy Reasoning 1

inference engine core of every fuzzy controller the computational mechanism with which decisions can be inferred even though the knowledge may be incomplete. fuzzy inference engines perform an exhaustive search of the rules in the knowledge base to determine the degree of fit for each rule for a given set of causes. only one unique rule contributes to the final decision

fuzzy propositional implication defines the relationship between the linguistic variables of a fuzzy controller. Cartesian product Using the conjunctive operator (min) algebraic product the Cartesian product

fuzzy propositional implication example Two fuzzy sets ， Cartesian product ：

fuzzy propositional implication example

fuzzy propositional implication example relational matrix

fuzzy propositional implication example relational matrix The Cartesian product based on the conjunctive operator min is much simpler and more efficient to implement computationally and is therefore generally preferred in fuzzy controller inference engines. Most commercially available fuzzy controllers in fact use this method.

3.1 The Fuzzy Algorithm Assume that where ψ is some implication operator and

3.1 The Fuzzy Algorithm the membership function for N rules in a fuzzy algorithm is given by

3.2 Fuzzy Reasoning two fuzzy implication inference rules 1. Generalized Modus Ponens (or GMP) 广义取式（肯定前提）假言推理法 简称为广义前向推理法 For the special case Α’=Α and Β’=Β then GMP reduces to Modus Ponens. 肯定前提的假言推理 use in all fuzzy controllers.

3.2 Fuzzy Reasoning two fuzzy implication inference rules 2. Generalized Modus Tollens (or GMT) For the special case 广义拒式（否定结论）假言推理法 广义后向推理法 application in expert systems then GMT reduces to Modus Tollens 否定结论的假言推理

3.2 Fuzzy Reasoning Boolean implication Lukasiewicz implication Zadeh implication Mamdani implication Larsen implication

3.2 Fuzzy Reasoning Boolean implication For the case of Ν rules,

3.2 Fuzzy Reasoning Lukasiewicz implication For the case of Ν rules, Bounded sum

3.2 Fuzzy Reasoning Zadeh implication difficult to apply in practice

3.2 Fuzzy Reasoning Mamdani implication For a fuzzy algorithm comprising N rules

3.2 Fuzzy Reasoning Larsen implication For a fuzzy algorithm comprising N rules

3.3 The Compositional Rules of Inference Given, for instance The composition of these two rules into one can be expressed as: rule composition

3.2 The Compositional Rules of Inference the membership function of the resultant compositional rule of inference Mamdani implication Larsen implication

3.3 The Compositional Rules of Inference the procedure for determining the consequent (or effect), given the antecedent (or cause). Given and the compositional rule of inference: if the antecedent is consequent ？？

3.3 The Compositional Rules of Inference the max-min operators max-product operators:

3.3 The Compositional Rules of Inference example Slow Fast determine the outcome if A = ‘slightly Slow’ for which there no rule exists

3.3 The Compositional Rules of Inference The first step is to compute the Cartesian product and using the min operator

3.3 The Compositional Rules of Inference The second step using the fuzzy compositional inference rule: the Mamdani compositional rule

3.3 The Compositional Rules of Inference The final operation

3.3 The Compositional Rules of Inference using the max-product rule of compositional inference: The first step

3.3 The Compositional Rules of Inference using the max-product rule of compositional inference: The second step

3.3 The Compositional Rules of Inference using the max-product rule of compositional inference: The maximum elements of each column are therefore: the Mamdani compositional rule

3.3 The Compositional Rules of Inference Given If inputs ： then outputs ： if input Then output ? example

3.3 The Compositional Rules of Inference

The unit set