Wireless Ad Hoc and Sensor Networks

144 Wireless Ad Hoc and Sensor NetworksConsidering the input and hidden-layer weight updates, and usingthese in Equation 3.A.1 and combining with Equation 3.A.5, one obtains( vmax) +2 2∆J ≤− 1−σk |( e k)| 2γk |( e k)|vmax+ ρ−( − αϕˆ TT( k)ˆ ϕ( k))− −αϕˆT1 1 1 1 Γ 1 1 1( k)ˆ ϕ1( k) × ( V ϕˆ 1( k) + B1kve( k))Ṽk ( )ˆ ϕ1( k)ˆT2 − αϕ ( k)ˆϕ1( k)1 1×⎡⎣ˆ ( )ˆ ( k)⎤⎦2 − αϕ 1 1 T k ϕ1−ˆTT( k)ˆ ( k) + I− ˆTαϕ 2 2 ϕ2 Γ αϕ 2 2( k)ˆ ϕ2( k) × ( kek v ( ) + W ̃ϕ2( k) + ε( k) + dk ( ))ei( k)−ˆT1 − αϕ ( k)ˆ ϕ(k)2 2 22×⎡TT⎣I−αϕ k ϕ k ⎤⎦ + I −22ˆ 12( )ˆ2( ) αϕ 1ˆ TT1( k)ˆ ϕ1( k) tr[ Γ2Vˆ ( k )ˆ V ( k ) + 2 Γ Vˆ ( k )ˆ V ( k )]α112 2+ | I− αϕ 2ˆ 2 T ( )ˆ ϕ2( )| [ ˆ Tk k tr Γ W ( k) W ˆ ( k)+ 2Γ ˆ TW ( k) W ˆ ( k)]α2(3.A.6)where( )γ = β1 Wmaxϕ̃ 2max + εN+ dM+ Γ( 1−α2)ϕ2 2 max ϕ2maxWmax( ( ))+ k1 β2 + Γ 1−α1ϕ1 2 max ϕ1maxVmax(3.A.7)and( )ρ= ⎡β1( Wmaxϕ̃ max + εN+ dM)+ 2Γ 1−α2ϕ2 2 max ϕ2maxW⎤max⎣⎦( )× ( Wmax ̃ϕ2max + εN+ dM)⎡β2 α1ϕ1 2 max ϕ1 2 2+ Γ 1−maxV⎤max(3.A.8)⎣⎦