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Advances in PID Control Part 6

Tham khảo tài liệu 'advances in pid control part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 90 Advances in PID Control 2.4.1 Fuzzy logic theory background A fuzzy set A on a domain X is a set defined by the membership function ụa x which is a mapping from the domain X into the unit interval ụa - X 0 1 . 10 There are several ways to define a fuzzy set in particular we define it here using the analytic description of its membership function ụ A x f x . For instance see Fig. 3 the triangular membership function can be described as ụ x a b c max A . x a c x 0 min 1 b a c b J J 11 where a b and c are parameters that is related to the coordinates of the triangle s vertices whereas a Gaussian membership function can be described as ụ x n a exp 21 12 A static or dynamic system which makes use of fuzzy sets and the corresponding mathematical framework is called a fuzzy system. In order to derivate the FAPI controller Fig. 3. Example of Membership Functions Triangular a 1 b 0.5 c 0 and Gaussian n 0.5 a 0.4 updating law it is necessary to define the intersection of fuzzy sets connective AND obtained by considering a function t 0 1 X 0 1 0 1 that transforms the membership functions of fuzzy sets A and B into the membership function of the intersection of A and B that is t ụ A x ụB x ụ ahb x . 13 A function t can be qualified as an intersection function if it satisfies at least the following four requirements t 0 0 0 t a 1 t 1 a a boundary condition t a b t b a commutativity . . t a b t a b a a b b monotonicity . t t a b c t a t b c associativity From Basic to Advanced PI Controllers A Complexity vs. Performance Comparison 91 In the following analysis the probabilistic connective AND will be used ụ AnB x ụ A x ụB x . 15 The most common fuzzy systems are defined by means of if-then rules rule-based fuzzy systems. In the rule-based fuzzy systems the relationships between variables are represented in the following general form if antecedent proposition then consequent proposition. A fuzzy proposition is a statement like x is big where big is a linguistic label defined by