II. Membership Functions Creation

52

APPENDIX

53

I. Operations On Type-2 Sets

Consider two fuzzy sets of type-2, ̃ and ̃ , in a universe . Let ̃ and _̃ be the membership grades (fuzzy sets in [ ]) of these two sets, represented, for each , as

̃ ∫ ( ) ⁄ and ̃ ∫ ( ) ⁄ , respectively, where indicate the primary memberships of and ( ) ( ) [ ]indicate the secondary memberships (grades) of . Using Zadeh’s Extension Principle [15,54,55], the membership grades for union, intersection and complement of type-2 fuzzy sets, ̃ and ̃ have been defined as follows [56]:

 Union

̃ ̃ _{̃ ̃}( ) _{̃ ̃} ∫ ∫ ( ( ) ( )) ( )⁄ ( )

 Intersection

̃ ̃ _{̃ ̃}( ) _{̃ ̃} ∫ ∫ ( ( ) ( )) ( )⁄ ( )

 Complement

̃̅ _̃̅ ̃ ∫ ( ) ( )⁄ ( )

Where represents the max t-conorm and represents a t-norm. The integrals indicate logical union. In the sequel, we adhere to these definitions, and, as in [56], we refer to the operations and ¬ as join, meet and negation, respectively.[57]

II. Membership Functions Creation

Triangular and trapezoidal membership functions retained in this work are defined using 4 linear functions, completely described by 5 points (a, b, c, d and e) as defined in Equations 1, 2 and 3, and illustrated in Figure 1. Triangular MFs are a particular case of trapezoidal MFs where b=c as illustrated in Figure 2.

( ) ( ( )) ( )

( ) ( )

( ) ( )

54

Fig 1 Trapezoidal MF example

a = -0:8; b = -0:4; c = 0:2; d = 0:6; e = 0:8

Fig 2 Triangular MF example a = -0:5; b = c = 0; d = 0:5; e = 1

0.5 y1

y2

Y=e y=0

y= Triangle

1

0

-0.5 0

x

Y2

y1

0 0.8

-0.8 -0.4 0.2 _0.6

y= ^Trapezoid

x

y=0 Y=e

55 KM Algorithm for Computing :

y^k ≤ y ≤ y^k+

y

𝑦

Calculate y^{′ y}

nfⁿ Nn=1

fⁿ

Nn=1 Replace

Find the value of k where ≤ 𝑘 ≤ 𝑁 and

𝑦 ^𝑁_𝑛= 𝑦^𝑛𝑓^𝑛 𝑓^𝑛

𝑁𝑛=

Cumpute :

Yes

y

𝑦 NO

𝑦 𝑦

Start

Inisialisation of 𝑓^𝑛:

𝑓^𝑛 𝑓^𝑛+ 𝑓^𝑛

𝑛 . . . 𝑁

;

𝑓^𝑛 𝑓^𝑛 𝑛 ≤ 𝑘 𝑎𝑛𝑑 𝑓^𝑛 𝑓^𝑛 𝑛 > 𝑘 ,

56 KM Algorithm for Computing :

𝑦^𝑘 ≤ 𝑦 ≤ 𝑦^𝑘+

y

𝑦

Calculate y^{′ 𝑦}

nfⁿ Nn=1

fⁿ

Nn=1 Replace

Find the value of k where ≤ 𝑘 ≤ 𝑁 and 𝑦 ^𝑁_𝑛= 𝑦^𝑛𝑓^𝑛

𝑓^𝑛

𝑁𝑛=

Cumpute :

Yes

y

𝑦 NO

𝑦 𝑦

Start

Inisialisation of 𝑓^𝑛:

𝑓^𝑛 𝑓^𝑛+ 𝑓^𝑛

𝑛 . . . 𝑁

𝑓^𝑛 𝑓^𝑛 𝑛 ≤ 𝑘 𝑎𝑛𝑑 𝑓^𝑛 𝑓^𝑛 𝑛 > 𝑘 ,

57

FUZZY PID STRUCTURES

we present some different FUZZY PID structures.[24][21] :

where is error , is change of error and rate of change of error.

Fig 3 Three-input fuzzy PID (coupled rules)

Fig 4 Three-input fuzzy PID (decoupled rules)

𝑒

𝑒

𝑒

𝑒

𝐺_𝑒

𝐺_𝑒 𝐺 _𝑒

𝐺 _𝑒 𝑒

𝑒 ^{𝐺 2𝑒}

𝐹𝐿𝐶

𝑓(𝑒) . 𝐹𝐿𝐶 𝑓( 𝑒)

𝐹𝐿𝐶 𝑓₃( 𝑒)

𝐹𝐿𝐶 𝑓(𝑒 𝑒 𝑒)

𝑢𝑃𝐼𝐷

𝑢_𝑃𝐼𝐷

𝑢_𝐼

𝑢_𝑝

𝑢_𝐷

𝑢_𝑃𝐼𝐷 𝐺_𝑢

𝐺_𝑢 +

+

+ + + + +

𝑍 𝑍 𝐺 ²_𝑒

58

Fig 5 Two-input fuzzy PID (coupled rules)

Fig 6 Two-input fuzzy PID (decoupled rules)

𝑒

𝑒

𝑒

𝑒

𝑒 𝐺_𝑒

𝐺_𝑒

𝐺_𝑒 𝐺 _𝑒

𝐺 _𝑒

𝐹𝐿𝐶 𝑓(𝑒 𝑒)

𝐹𝐿𝐶 𝑓(𝑒)

𝐹𝐿𝐶 𝑓(𝑒)

𝐹𝐿𝐶 𝑓(𝑒)

+ +

+ +

𝑢_𝑃𝐼𝐷 𝐺_𝑃𝐷

𝐺_𝑃𝐼 𝑢_𝑃𝐷

𝑢_𝑃𝐼𝐷 𝐺_𝑃

𝐺_𝐼 𝑢_𝑃 +

+

+ + 𝑍

𝑍

𝑢_𝑃 𝑢_𝑃𝐼𝐷

𝐺_𝐷 𝐺_𝐼 𝐺_𝑝

𝑍 𝐺_𝐷

+ +

+ + +

+

59

Fig 8 One-input fuzzy PID (decoupled rules) 𝐺_𝑒

𝐹𝐿𝐶 𝑓(𝑒)

𝐹𝐿𝐶 𝑓(𝑒)

𝐹𝐿𝐶 𝑓₃(𝑒)

𝐺_𝑝

𝐺_𝐼

𝐺_𝐷 𝑢_𝑃𝐼

𝑢_𝑃

𝑢_𝑃3

𝑢_𝑃𝐼𝐷

𝑍

𝑍

+

+ +

+ +

𝑒