is implemented by a fuzzy relation R i and is defined as

Transcription

1 FS VI: Fuzzy reasoning schemes R 1 : ifx is A 1 and y is B 1 then z is C 1 R 2 : ifx is A 2 and y is B 2 then z is C 2... R n : ifx is A n and y is B n then z is C n x is x 0 and y is ȳ 0 z is C The i-th fuzzy rule from this rule-base R i : if x is A i and y is B i then z is C i is implemented by a fuzzy relation R i and is defined as R i (u, v, w) =(A i B i C i )(u, w) =[A i (u) B i (v)] C i (w) for i =1,...,n. 1

2 Find C from the input x 0 and from the rule base R = {R 1,...,R n }. Interpretation of logical connective and sentence connective also implication operator then compositional operator We first compose x 0 ȳ 0 with each R i producing intermediate result C i = x 0 ȳ 0 R i for i =1,...,n. Here C i is called the output of the i-th rule C i(w) =[A i (x 0 ) B i (y 0 )] C i (w), for each w. 2

3 Then combine the C i component wise into C by some aggregation operator: C = n C i = x 0 ȳ 0 R 1 x 0 ȳ 0 R n i=1 C(w) =A 1 (x 0 ) B 1 (y 0 ) C 1 (w) A n (x 0 ) B n (y 0 ) C n (w). input to the system is (x 0,y 0 ) fuzzified input is ( x 0, ȳ 0 ) firing strength of the i-th rule is A i (x 0 ) B i (y 0 ) the i-th individual rule output is C i(w) :=[A i (x 0 ) B i (y 0 )] C i (w) overall system output is C = C 1 C n. overall system output = union of the individual rule outputs 3

4 We present five well-known inference mechanisms in fuzzy rule-based systems. For simplicity we assume that we have two fuzzy IF-THEN rules of the form R 1 : if x is A 1 and y is B 1 then z is C 1 also R 2 : if x is A 2 and y is B 2 then z is C 2 fact : x is x 0 and y is ȳ 0 consequence : z is C Mamdani. The fuzzy implication is modelled by Mamdani s minimum operator and the sentence connective also is interpreted as oring the propositions and defined by max operator. The firing levels of the rules, denoted by α i,i=1, 2, are computed by α 1 = A 1 (x 0 ) B 1 (y 0 ), α 2 = A 2 (x 0 ) B 2 (y 0 ) The individual rule outputs are obtained by 4

5 C 1(w) =(α 1 C 1 (w)), C 2(w) =(α 2 C 2 (w)) Then the overall system output is computed by oring the individual rule outputs C(w) =C 1(w) C 2(w) =(α 1 C 1 (w)) (α 2 C 2 (w)) Finally, to obtain a deterministic control action, we employ any defuzzification strategy. 5

6 A1 B1 C1 u v w A2 B2 C2 xo u yo Inference with Mamdani s implication operator. v min Tsukamoto. All linguistic terms are supposed to have monotonic membership functions. The firing levels of the rules, denoted by α i,i=1, 2, are computed by α 1 = A 1 (x 0 ) B 1 (y 0 ),α 2 = A 2 (x 0 ) B 2 (y 0 ) In this mode of reasoning the individual crisp control actions z 1 and z 2 are computed from 6 w

7 the equations α 1 = C 1 (z 1 ), α 2 = C 2 (z 2 ) and the overall crisp control action is expressed as z 0 = α 1z 1 + α 2 z 2 α 1 + α 2 i.e. z 0 is computed by the discrete Centerof-Gravity method. If we have n rules in our rule-base then the crisp control action is computed as z 0 = n i=1 α iz i n i=1 α, i where α i is the firing level and z i is the (crisp) output of the i-th rule, i =1,...,n Example 1. We illustrate Tsukamoto s reasoning method by the following simple example 7

8 R 1 : if x is A 1 and y is B 1 then z is C 1 also R 2 : if x is A 2 and y is B 2 then z is C 2 fact : x is x 0 and y is ȳ 0 consequence : z is C Then according to the figure we see that A 1 (x 0 )=0.7, B 1 (y 0 )=0.3 therefore, the firing level of the first rule is α 1 = min{a 1 (x 0 ),B 1 (y 0 )} = min{0.7, 0.3} =0.3 and from A 2 (x 0 )=0.6, B 2 (y 0 )=0.8 it follows that the firing level of the second rule is α 2 = min{a 2 (x 0 ),B 2 (y 0 )} = min{0.6, 0.8} = 0.6, 8

9 the individual rule outputs z 1 =8and z 2 = 4 are derived from the equations C 1 (z 1 )=0.3, C 2 (z 2 )=0.6 and the crisp control action is z 0 =( )/( )=6. A1 B1 C u v z1 = 8 w A2 B2 C xo u yo v min z2 = 4 w Tsukamoto s inference mechanism. Sugeno. Sugeno and Takagi use the follow- 9

10 ing architecture R 1 : if x is A 1 and y is B 1 then z 1 = a 1 x + b 1 y also R 2 : if x is A 2 and y is B 2 then z 2 = a 2 x + b 2 y fact : x is x 0 and y is ȳ 0 cons. : z 0 The firing levels of the rules are computed by α 1 = A 1 (x 0 ) B 1 (y 0 ),α 2 = A 2 (x 0 ) B 2 (y 0 ) then the individual rule outputs are derived from the relationships z 1 = a 1 x 0 + b 1 y 0,z 2 = a 2 x 0 + b 2 y 0 and the crisp control action is expressed as z 0 = α 1z 1 + α 2 z 2 α 1 + α 2 10

11 A1 A2 u v α1 a1x + b 1y B1 B2 α2 x u y v min a2x + b 2y Sugeno s inference mechanism. If we have n rules in our rule-base then the crisp control action is computed as z 0 = n i=1 α iz i n i=1 α i, where α i denotes the firing level of the i-th rule, i =1,...,n Example 2. We illustrate Sugeno s reasoning method by the following simple example 11

12 R 1 : if x is BIG and y is SMALL then z 1 = x + y also R 2 : if x is MEDIUM and y is BIG then z 2 =2x y fact : x 0 is 3 and y 0 is 2 conseq : z 0 Then according to the figure we see that µ BIG (x 0 )=µ BIG (3)=0.8, µ SMALL (y 0 )=µ SMALL (2)=0.2 therefore, the firing level of the first rule is α 1 = min{µ BIG (x 0 ),µ SMALL (y 0 )} and from = min{0.8, 0.2} =0.2 µ MEDIUM (x 0 )=µ MEDIUM (3)=0.6, µ BIG (y 0 )=µ BIG (2)=0.9 it follows that the firing level of the second rule is α 2 = min{µ MEDIUM (x 0 ),µ BIG (y 0 )} 12

13 = min{0.6, 0.9} = 0.6. the individual rule outputs are computed as z1 = x 0 + y 0 =3+2=5,z2 =2x 0 y 0 =2 3 2=4 so the crisp control action is z 0 =( )/( )= α1= 0.2 u v x+y= α2=0.6 3 u 2 v min 2x-y=4 Example of Sugeno s inference mechanism. Larsen. The fuzzy implication is modelled 13

14 by Larsen s prduct operator and the sentence connective also is interpreted as oring the propositions and defined by max operator. Let us denote α i the firing level of the i-th rule, i =1, 2 α 1 = A 1 (x 0 ) B 1 (y 0 ), α 2 = A 2 (x 0 ) B 2 (y 0 ). Then membership function of the inferred consequence C is pointwise given by C(w) =(α 1 C 1 (w)) (α 2 C 2 (w)). To obtain a deterministic control action, we employ any defuzzification strategy. If we have n rules in our rule-base then the consequence C is computed as n C(w) = (α i C 1 (w)) i=1 where α i denotes the firing level of the i-th rule, i =1,...,n 14

15 A1 B1 C1 u v w A2 B2 C2 xo u yo v min Inference with Larsen s product operation rule. Simplified fuzzy reasoning R 1 : if x is A 1 and y is B 1 then z 1 = c 1 also R 2 : if x is A 2 and y is B 2 then z 2 = c 2 fact : x is x 0 and y is ȳ 0 consequence : z 0 The firing levels of the rules are computed by 15 w

16 α 1 = A 1 (x 0 ) B 1 (y 0 ), α 2 = A 2 (x 0 ) B 2 (y 0 ) then the individual rule outputs are c 1 and c 2, and the crisp control action is expressed as z 0 = α 1c 1 + α 2 c 2 α 1 + α 2 If we have n rules in our rule-base then the crisp control action is computed as z 0 = n i=1 α ic i n i=1 α, i where α i denotes the firing level of the i-th rule, i =1,...,n 16

17 L1 H2 L3 α1 c1 M1 M2 M3 α2 H1 H2 H3 c2 min z3 α3 17