Jacobian Matrix of a Fuzzy Control System - Fuzzy Modeling and Control: Theory and Applications - page 264

but ∂ ˜

x k

x q =

1if k = q and 0 in other case, therefore:

∂

⎛
⎝ ∂ ˜

⎞

n m m

x k

⎠ =

b ji c kj b ji c qj . (A.15)

∂ x q

k = 0 j = 1 j = 1

The derivative of the first term of ( A.14 ) can be written as:

m

j = 1

∂ b ji c kj

∂

b ji

n n m

b ji

c kj + ∂

c kj
∂

x k = ˜ x k . ˜ (A.16)

∂ x q ∂ x q x q

k = 0 k = 0 j = 1

Noting the previously calculated derived from a ki , it is easy to deduce for b ji that:

w i

M i

∂

w i ( b ji )

b l ji −

∂ x q

∂ b ji

∂

l , p = 1

x q =

. (A.17)

M i 2

l = 1

w i

Operating similarly:

N j N j

r = 1 ω r = 1 ω

r

j c kj

r
j

∂ ∂

⎛
⎝

⎞
⎠ =

j j c kj

N j N j N j

r = 1 ω r = 1 r = 1

r

j c kj

r r

ω − ω

∂ ∂ x q ∂ x q

c kj ∂

x q =

,

N j 2

∂ ∂ x q

N j

r = 1 ω

r = 1 ω

r
j

r
j

(A.18)

where

N j

r = 1 ω

r

j c kj

∂

N j

r
j

∂ω

c kj = , (A.19)

∂ x q ∂ x q

r = 1

and

N j

r = 1 ω

r
j

∂ω

∂

N j

r
j

= . (A.20)

∂ x q ∂ x q

r = 1