Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

and the weight adaptation equation is given by
Wk+1 = Wk + ∆Wk . (16.24)
The feedback loop in fig. 16.3 first generates Y0 arbitrarily and later makes
correction through a change in Wk, which subsequently helps to determine
controlled Xk+1 and hence Yk+1. Wk -1 denotes the fuzzy compositional inverse
(strictly speaking, pre-inverse) for matrix Wk. The stability of the learning
model is guaranteed, vide theorem 16.3.
Theorem 16.3: The error vector Ek in the learning model converges to a
stable point for 0< α < 2 and the steady-state value of error is inversely
proportional to α.
Proof: We have
Yk = Wk+1 o Xk+1
= (Wk + ∆Wk ) o Xk+1
= {Wk + α Ek o Xk+1 T / ( ( Xk+1 T ) o ( Xk+1)) } o Xk+1
= {Wk + α Ek o Xk+1 T / ( (Xk+1 T )o (Xk+1) )} o ( Wk -1 oYk)
≤ Yk + {α Ek o Xk+1 T / ( (Xk+1 T ) o ( Xk+1)} o (Wk -1 o Yk)
= Yk + {α Ek o Xk+1 T / ( (Xk+1 T )o ( Xk+1))}o Xk+1
≈ Yk + α Ek.
Thus, Yk+1 ≤ Yk + α Ek (16.25)
= Yk + α (D- Yk)
⇒ Yk+1 ≤ αD + (I - αI) Yk (16.26)
⇒ [ E I - (I - α I)] Yk = (α - α ′ )D, for 0 < α ′≤ α (16.27)
where E is the extended difference operator and I is the identity matrix.