FS VI: Fuzzy reasoning schemes
1: if x is A1 and y is B1 then z is C1
2: if x is A2 and y is B2 then z is C2
............
n: if x is An and y is Bn then z is Cn
x is x̄0 and y is ȳ0
z is C
The i-th fuzzy rule from this rule-base
i : if x is Ai and y is Bi then z is Ci
is implemented by a fuzzy relation Ri and is
defined as
Ri(u, v, w) = (Ai × Bi → Ci)(u, w)
= [Ai(u) ∧ Bi(v)] → Ci(w)
for i = 1, . . . , n.
1
Find C from the input x0 and from the rule
base
= {1, . . . , n}.
Interpretation of
• logical connective ”and”
• sentence connective ”also”
• implication operator ”then”
• compositional operator ”◦”
We first compose x̄0 × ȳ0 with each Ri producing intermediate result
Ci = x̄0 × ȳ0 ◦ Ri
for i = 1, . . . , n. Here Ci is called the output of the i-th rule
Ci(w) = [Ai(x0) ∧ Bi(y0)] → Ci(w),
for each w.
2
Then combine the Ci component wise into
C by some aggregation operator:
C=
n
Ci = x̄0 × ȳ0 ◦ R1 ∪ · · · ∪ x̄0 × ȳ0 ◦ Rn
i=1
C(w) = A1(x0) × B1(y0) → C1(w) ∨ · · · ∨
An(x0) × Bn(y0) → Cn(w).
• input to the system is (x0, y0)
• fuzzified input is (x̄0, ȳ0)
• firing strength of the i-th rule is
Ai(x0) ∧ Bi(y0)
• the i-th individual rule output is
Ci(w) := [Ai(x0) ∧ Bi(y0)] → Ci(w)
• overall system output is
C = C1 ∪ · · · ∪ Cn .
overall system output = union of the individual rule outputs
3
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
1 :
also
2 :
if x is A1 and y is B1 then z is C1
fact :
x is x̄0 and y is ȳ0
if x is A2 and y is B2 then z is C2
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
αi, i = 1, 2, are computed by
α1 = A1(x0) ∧ B1(y0),
α2 = A2(x0) ∧ B2(y0)
The individual rule outputs are obtained by
4
C1 (w) = (α1 ∧ C1(w)),
C2 (w) = (α2 ∧ C2(w))
Then the overall system output is computed
by oring the individual rule outputs
C(w) = C1 (w) ∨ C2 (w)
= (α1 ∧ C1(w)) ∨ (α2 ∧ C2(w))
Finally, to obtain a deterministic control action, we employ any defuzzification strategy.
5
A1
v
u
w
B2
A2
xo
C1
B1
u
C2
v
yo
w
min
Inference with Mamdani’s implication operator.
Tsukamoto. All linguistic terms are supposed to have monotonic membership functions.
The firing levels of the rules, denoted by
αi, i = 1, 2, are computed by
α1 = A1(x0)∧B1(y0), α2 = A2(x0)∧B2(y0)
In this mode of reasoning the individual crisp
control actions z1 and z2 are computed from
6
the equations
α1 = C1(z1),
α2 = C2(z2)
and the overall crisp control action is expressed as
α1z1 + α2z2
z0 =
α1 + α2
i.e. z0 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
n
i=1 αi zi
,
z0 = n
i=1 αi
where αi is the firing level and zi 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
1 :
also
2 :
if x is A1 and y is B1 then z is C1
fact :
x is x̄0 and y is ȳ0
if x is A2 and y is B2 then z is C2
consequence :
z is C
Then according to the figure we see that
A1(x0) = 0.7, B1(y0) = 0.3
therefore, the firing level of the first rule is
α1 = min{A1(x0), B1(y0)}
= min{0.7, 0.3} = 0.3
and from
A2(x0) = 0.6, B2(y0) = 0.8
it follows that the firing level of the second
rule is
α2 = min{A2(x0), B2(y0)}
= min{0.6, 0.8} = 0.6,
8
the individual rule outputs z1 = 8 and z2 =
4 are derived from the equations
C1(z1) = 0.3,
C2(z2) = 0.6
and the crisp control action is
z0 = (8 × 0.3 + 4 × 0.6)/(0.3 + 0.6) = 6.
C1
B1
A1
0.7
0.3
0.3
u
v
B2
A2
u
w
C2
0.6
0.8
0.6
xo
z1 = 8
yo
v
min
z2 = 4
Tsukamoto’s inference mechanism.
Sugeno. Sugeno and Takagi use the follow9
w
ing architecture
1 :
if x is A1 and y is B1 then z1 = a1x + b1y
also
2 :
if x is A2 and y is B2 then z2 = a2x + b2y
fact : x is x̄0 and y is ȳ0
cons. :
z0
The firing levels of the rules are computed
by
α1 = A1(x0)∧B1(y0), α2 = A2(x0)∧B2(y0)
then the individual rule outputs are derived
from the relationships
z1∗ = a1x0 + b1y0, z2∗ = a2x0 + b2y0
and the crisp control action is expressed as
α1z1∗ + α2z2∗
z0 =
α1 + α2
10
A2
A1
α1
u
v
a1 x + b 1 y
B2
B1
α2
x
y
u
v
min
a2 x + b 2 y
Sugeno’s inference mechanism.
If we have n rules in our rule-base then the
crisp control action is computed as
n
∗
α
z
i
i
,
z0 = i=1
n
α
i=1 i
where αi 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
1 :
if x is BIG and y is SMALL then
z1 = x + y
also
2 :
if x is MEDIUM and y is BIG then z2 = 2x − y
fact :
x0 is 3 and y0 is 2
conseq :
z0
Then according to the figure we see that
µBIG(x0) = µBIG(3) = 0.8,
µSM ALL(y0) = µSM ALL(2) = 0.2
therefore, the firing level of the first rule is
α1 = min{µBIG(x0), µSM ALL(y0)}
= min{0.8, 0.2} = 0.2
and from
µM EDIU M (x0) = µM EDIU M (3) = 0.6,
µBIG(y0) = µBIG(2) = 0.9
it follows that the firing level of the second
rule is
α2 = min{µM EDIU M (x0), µBIG(y0)}
12
= min{0.6, 0.9} = 0.6.
the individual rule outputs are computed as
z1∗ = x0 + y0 = 3 + 2 = 5, z2∗
= 2x0 − y0 = 2 × 3 − 2 = 4
so the crisp control action is
z0 = (5 × 0.2 + 4 × 0.6)/(0.2 + 0.6) = 4.25.
1
0.8
0.2
α1= 0.2
u
1
v
x+y=5
0.9
0.6
3
α2=0.6
u
2
v
min
2x-y=4
Example of Sugeno’s inference mechanism.
Larsen. The fuzzy implication is modelled
13
by Larsen’s prduct operator and the sentence
connective also is interpreted as oring the
propositions and defined by max operator.
Let us denote αi the firing level of the i-th
rule, i = 1, 2
α1 = A1(x0) ∧ B1(y0),
α2 = A2(x0) ∧ B2(y0).
Then membership function of the inferred
consequence C is pointwise given by
C(w) = (α1C1(w)) ∨ (α2C2(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) = (αiC1(w))
i=1
where αi denotes the firing level of the i-th
rule, i = 1, . . . , n
14
A1
u
v
w
B2
A2
xo
C1
B1
u
C2
yo
v
w
min
Inference with Larsen’s product operation rule.
Simplified fuzzy reasoning
1 :
if x is A1 and y is B1 then z1 = c1
also
if x is A2 and y is B2 then z2 = c2
2 :
fact :
x is x̄0 and y is ȳ0
consequence :
z0
The firing levels of the rules are computed
by
15
α1 = A1(x0) ∧ B1(y0),
α2 = A2(x0) ∧ B2(y0)
then the individual rule outputs are c1 and
c2, and the crisp control action is expressed
as
α1c1 + α2c2
α1 + α2
If we have n rules in our rule-base then the
crisp control action is computed as
n
αici
,
z0 = i=1
n
i=1 αi
where αi denotes the firing level of the i-th
rule, i = 1, . . . , n
z0 =
16
L1
H2
L3
α1
c1
M1
M2
M3
α2
c2
H1
H3
H2
α3
min
17
z3