Fuzzy Logic: An Introductory Course for Engineering Students (Studies in Fuzziness and Soft Computing)
Enric Trillas
Language: English
Pages: 204
ISBN: 331914202X
Format: PDF / Kindle (mobi) / ePub
This book introduces readers to fundamental concepts in fuzzy logic. It describes the necessary theoretical background and a number of basic mathematical models. Moreover, it makes them familiar with fuzzy control, an important topic in the engineering field. The book offers an unconventional introductory textbook on fuzzy logic, presenting theory together with examples and not always following the typical mathematical style of theorem-corollaries. Primarily intended to support engineers during their university studies, and to spark their curiosity about fuzzy logic and its applications, the book is also suitable for self-study, providing a valuable resource for engineers and professionals who deal with imprecision and non-random uncertainty in real-world applications.
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J= A j , that is μ j = μ A j . 1.5 Linguistic Variables 31 Example 1.5.2 In X = [0, 10], take μ0 (x) = 1 − x4 , 0, if if 0 4 ⎧x ⎨ 4, 4, , μ1 (x) = 1, 10 ⎩ 10−x x x 4 μ2 (x) = 0, x−6 4 , if if 0 6 x x , if if if 0 4 6 x x x 4, 6 , 10 6, 10. Graphically Since, obviously, μ0 (x) + μ1 (x) + μ2 (x) = 1 for all x ∈ [0, 10], {μ0 , μ1 , μ2 } is a fuzzy partition of the interval [0, 10]. Example 1.5.3 By proceeding in the same way that in last example, it is easy to prove that it is.
Algebras of Fuzzy Sets. . . . . . . . . . . . . 2.2.3 Non-contradiction and Excluded-Middle . 2.2.4 Decomposable Algebras . . . . . . . . . . . . 2.2.5 Standard Algebras of Fuzzy Sets . . . . . . 2.2.6 Strong Negations . . . . . . . . . . . . . . . . . 2.2.7 Continuous T-Norms and T-Conorms . . . 2.2.8 Laws of Fuzzy Sets . . . . . . . . . . . . . . . 2.2.9 Examples . . . . . . . . . . . . . . . . . . . . . . 2.3 On Aggregating Imprecise Information . . . . . . . 2.3.1 Aggregation Functions . . .
B) = N (G(N (a), N (b)), or G(a, b) = N (F(N (a), N (b)), for all a, b in [0, 1]. Notice that, in this case, it is enough to know N and F to have G, or N and G to have F. • If N is involutive (N (N (a)) = a, for all a ∈ [0, 1] or N 2 = N ), the triplet (min, max, N ) is a dual one. • All triplet (F, G, N ) verifies F(a, N (a)) G(b, N (b)), for all a, b in [0, 1], • It is F(a, N (a)) N (F(a, N (a))), for all a in [0, 1] • It is N (F(a, N (a))) N (N (F(a, N (a)))), for all a in [0, 1] • Given N.
Is just numerical, crisp in the form ‘x = x0 ’, that is, ‘x is P ∗ ’ is ‘x is x0 ’, or ‘x ∈ {x0 }’ and then μ P ∗ = μ{x0 } , with μ{x0 } (x) = 1, if x = x0 0, if x = x0 . In this case, μ Q ∗ (y) = sup T0 (μ{x0 } (x), J (μ P (x), μ Q (y))) = J (μ P (x0 ), μ Q (y)), ∀y ∈ Y, x∈X is a simpler expression that does not force to compute supx∈X . Sometimes, in addition to the input, the rule’s consequent μ Q is also numerical, that is ‘y is Q’ is ‘y = y0 ’, or ‘y is y0 ’ , or ‘y ∈ {y0 }’. In this case.
Premise variables that may be functions of the state variables and/or external disturbances. Each linear consequent equation represented by Ai x(t) + Bi u(t) is called a “subsystem”. x(t) ˙ = ri=1 h i (z(t)){Ai x(t) + Bi u(t)} y(t) = ri=1 h i (z(t)) Ci x(t) where, h i (z(t)) = wi (z(t)) , r i=1 wi (z(t)) p wi (z(t)) = Mi j (z(t)) j=1 for all t. The term Mi j (z(t)) is the grade of membership of z j (t) in Mi j . We have r 0, for i = 1, 2, . . . , r and all t. i=1 h i (z(t)) = 1, and h i.