Need for Fuzzy Control Mamdani Approach to . . . Logical (More Recent) . . . Both Approaches . . . Mamdani Approach Corresponding Crisp . . . to Fuzzy Control, Fuzzy Control: What . . . Definitions Logical Approach, Main Result Proof: Main Lemmas What Else? Home Page Samuel Bravo and Jaime Nava Title Page Department of Computer Science ◭◭ ◮◮ University of Texas at El Paso El Paso, TX 79968, USA ◭ ◮ sbravo09@gmail.com jenava@miners.utep.edu Page 1 of 13 Go Back Full Screen Close Quit
Need for Fuzzy Control Mamdani Approach to . . . 1. Need for Fuzzy Control Logical (More Recent) . . . • In many application areas, Both Approaches . . . Corresponding Crisp . . . – we do not have the exact control strategies, but Fuzzy Control: What . . . – we have human operators who are skilled in the Definitions corresponding control. Main Result • Human operators are often unable to describe their Proof: Main Lemmas knowledge in a precise quantitative form. Home Page • Instead, they describe their knowledge by using words Title Page from natural language. ◭◭ ◮◮ • These rules usually have the form “If A i ( x ) then B i ( u )”, ◭ ◮ where x is the input and u is the resulting control. Page 2 of 13 • For example, a rule may say “If a car in front is some- Go Back what too close, break a little bit”. Full Screen • Fuzzy control is a set of techniques for transforming these rules into a precise control strategy. Close Quit
Need for Fuzzy Control Mamdani Approach to . . . 2. Mamdani Approach to Fuzzy Control: Histor- Logical (More Recent) . . . ically the First Both Approaches . . . • For a given input x , a control value u is reasonable if: Corresponding Crisp . . . Fuzzy Control: What . . . – the 1st rule is applicable, i.e., its condition A 1 ( x ) Definitions is satisfied and its conclusion B 1 ( u ) is satisfied, Main Result – or the 2nd rule is applicable, i.e., its condition A 2 ( x ) Proof: Main Lemmas is satisfied and its conclusion B 2 ( u ) is satisfied, Home Page – etc. Title Page • Thus, the condition R ( x, u ) “the control u is reasonable ◭◭ ◮◮ for the input x ” takes the form ◭ ◮ ( A 1 ( x ) & B 1 ( u )) ∨ ( A 2 ( x ) & B 2 ( u )) ∨ . . . Page 3 of 13 • To get control value u ( x 0 ), we apply a defuzzification Go Back procedure to the corr. membership function R ( x 0 , u ). Full Screen Close Quit
Need for Fuzzy Control Mamdani Approach to . . . 3. Logical (More Recent) Approach to Fuzzy Con- Logical (More Recent) . . . trol Both Approaches . . . • Main idea: simply state that all the rules are valid, i.e., Corresponding Crisp . . . that the following statement holds: Fuzzy Control: What . . . Definitions ( A 1 ( x ) → B 1 ( u )) & ( A 2 ( x ) → B 2 ( u )) & . . . Main Result • For example, we can interpret A → B as ¬ A ∨ B , in Proof: Main Lemmas Home Page which case the above formula has the form Title Page ( ¬ A 1 ( x ) ∨ B 1 ( u )) & ( ¬ A 2 ( x ) ∨ B 2 ( u )) & . . . ◭◭ ◮◮ • Equivalently, we can use the form ◭ ◮ ( A ′ 1 ( x ) ∨ B 1 ( u )) & ( A ′ 2 ( x ) ∨ B 2 ( u )) & . . . , Page 4 of 13 where A ′ i ( x ) denotes ¬ A i ( x ). Go Back Full Screen Close Quit
Need for Fuzzy Control Mamdani Approach to . . . 4. Both Approaches Have a Universality Property Logical (More Recent) . . . • Fact: both Both Approaches . . . Corresponding Crisp . . . – Mamdani’s approach to fuzzy control and Fuzzy Control: What . . . – logical approach to fuzzy control Definitions have a universality (universal approximation) property. Main Result Proof: Main Lemmas • Meaning of universal approximation property: Home Page – an arbitrary control strategy can be, Title Page – with arbitrary accuracy, ◭◭ ◮◮ – approximated by controls generated by this approach. ◭ ◮ Page 5 of 13 Go Back Full Screen Close Quit
Need for Fuzzy Control Mamdani Approach to . . . 5. Corresponding Crisp Universality Property Logical (More Recent) . . . • Why do the corresponding fuzzy controls have the uni- Both Approaches . . . versal approximation property? Corresponding Crisp . . . Fuzzy Control: What . . . • Intuitive explanation: because the corresponding crisp Definitions formulas have the universal property. Main Result • In precise terms: for finite sets X and U , any relation Proof: Main Lemmas C ( x, u ) on X × U can be represented in both forms Home Page ( A 1 ( x ) & B 1 ( u )) ∨ ( A 2 ( x ) & B 2 ( u )) ∨ . . . ; Title Page ( A 1 ( x ) → B 1 ( u )) & ( A 2 ( x ) → B 2 ( u )) & . . . . ◭◭ ◮◮ • Proof: an arbitrary crisp property C ( x, u ) is described ◭ ◮ by the set C = { ( x, u ) : C ( x, u ) } , so: Page 6 of 13 C ( x, u ) ⇔ ∨ ( x 0 ,u 0 ) ∈ C (( x = x 0 ) & ( u = u 0 )); Go Back C ( x, u ) ⇔ & ( x 0 ,u 0 ) �∈ C (( x = x 0 ) → ( u � = u 0 )) . Full Screen • Fact: the corr. CNF & DNF representations are ac- tively used in digital design; e.g., in vending machines. Close Quit
Need for Fuzzy Control Mamdani Approach to . . . 6. Fuzzy Control: What Other Approaches Are Logical (More Recent) . . . Possible? Both Approaches . . . • Both Mamdani’s and logical approaches are actively Corresponding Crisp . . . used in fuzzy control. Fuzzy Control: What . . . Definitions • The fact that both approaches are actively used means Main Result that both have advantages and disadvantages. Proof: Main Lemmas • In other words, this means that none of these two ap- Home Page proaches is perfect. Title Page • Since both approaches are not perfect, it is reasonable ◭◭ ◮◮ to analyze what other approaches are possible. ◭ ◮ • In this paper, we start this analysis by analyzing what type of crisp forms like Page 7 of 13 ( A 1 ( x ) & B 1 ( u )) ∨ ( A 2 ( x ) & B 2 ( u )) ∨ . . . ; Go Back ( A 1 ( x ) → B 1 ( u )) & ( A 2 ( x ) → B 2 ( u )) & . . . . Full Screen are possible. Close Quit
Need for Fuzzy Control Mamdani Approach to . . . 7. Definitions Logical (More Recent) . . . • By a binary operation , we mean a function f : { 0 , 1 }× Both Approaches . . . { 0 , 1 } → { 0 , 1 } that transforms Boolean values. Corresponding Crisp . . . Fuzzy Control: What . . . • A pair of binary operations ( ⊙ , ⊖ ) s.t. ⊖ is commuta- Definitions tive and associative has a universality property if: Main Result – for every two finite sets X and Y, Proof: Main Lemmas – an arbitrary relation C ( x, u ) can be represented, for Home Page some A i ( x ) and B i ( u ) , as Title Page ( A 1 ( x ) ⊙ B 1 ( u )) ⊖ ( A 2 ( x ) ⊙ B 2 ( u )) ⊖ . . . ◭◭ ◮◮ • We say that pairs ( ⊙ , ⊖ ) and ( ⊙ ′ , ⊖ ) are similar if the ◭ ◮ relation ⊙ ′ has one of the following forms: Page 8 of 13 a ⊙ ′ b = ¬ a ⊙ b, a ⊙ ′ b or a ⊙ ′ b def def def Go Back = a ⊙¬ b, = ¬ a ⊙¬ b. Full Screen Close Quit
Need for Fuzzy Control Mamdani Approach to . . . 8. Main Result Logical (More Recent) . . . • Theorem. Every pair of operations with the univer- Both Approaches . . . sality property is similar to one of the following pairs: Corresponding Crisp . . . Fuzzy Control: What . . . ( ∨ , &) , (& , ∨ ) , ( ⊕ , ∨ ) , ( ⊕ , &) , ( ⊕ ′ , ∨ ) , ( ≡ , ∨ ) , ( ≡ , &) . Definitions Main Result • Thus, in addition to the Mamdani and logical approaches, we have 4 other pairs with the universality property. Proof: Main Lemmas Home Page • In essence, we have 2 new forms w/“exclusive or” ⊕ : Title Page ( A 1 ( x ) & B 1 ( u )) ⊕ ( A 2 ( x ) & B 2 ( u )) ⊕ . . . ; ◭◭ ◮◮ ( A 1 ( x ) ∨ B 1 ( u )) ⊕ ( A 2 ( x ) ∨ B 2 ( u )) ⊕ . . . ◭ ◮ • The meaning of the new forms: we restrict ourselves Page 9 of 13 to the cases when exactly one rule is applicable. Go Back Full Screen Close Quit
Need for Fuzzy Control Mamdani Approach to . . . 9. Proof: Main Lemmas Logical (More Recent) . . . • If the pairs ( ⊙ , ⊖ ) and ( ⊙ ′ , ⊖ ) are similar, then the Both Approaches . . . following two statements are equivalent to each other: Corresponding Crisp . . . Fuzzy Control: What . . . – the pair ( ⊙ , ⊖ ) has the universality property; Definitions – the pair ( ⊙ ′ , ⊖ ) has the universality property. Main Result • Out of all binary operations, only the following six are Proof: Main Lemmas commutative and associative: Home Page – the “zero” operation s.t. f ( a, b ) = 0 for all a and b ; Title Page – the “one” operation s.t. f ( a, b ) = 1 for all a and b ; ◭◭ ◮◮ – the “and” operation s.t. f ( a, b ) = a & b ; ◭ ◮ – the “or” operation s.t. f ( a, b ) = a ∨ b ; Page 10 of 13 – the “exclusive or” operation s.t. f ( a, b ) = a ⊕ b ; Go Back – the operation a ⊕ ′ b def = a ⊕ ¬ b . Full Screen Close Quit
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