Wireless Ad Hoc and Sensor Networks

312 Wireless Ad Hoc and Sensor Networkswherea= |[( 1− αφ ) + βE]|ijij(7.22)andb= ( 1− α) 2 φ 2 + β 2 E 2 + 2( 1−α)φ βEij ij ij ij(7.23)| ∆V|≤ 0implies that⎡22 2b| ∆V| ≤−( 1−α)̃ αφij− ̃2 ij a−2. (7.24)⎣⎢( 1−α ) φ ⎤( 1−α ) ⎦⎥̃φij2 2 2α + α a + b( 1 −α)≥2( 1 − α )(7.25)LetB ij,φbe the bound on the weight estimation error, thenBij, φ2 2α + α + b( 1 −α)=2( 1 − α )(7.26)For ̃φ ij ≥ Bij, φ ∆V < 0 . Because ̃φ ij ≥ Bij,φ, from Equation 7.13 we getˆφ ≤σφijij(7.27)for some σ .LEMMA 7.3.2The actual weights ˆφ ij at each node using Equation 7.9 converge close to their targetvalues in a finite time.PROOF Because || α < 1, define ̃φ ij ( k) = x( k), then Equation 7.14 can beexpressed asxk ( + 1) = cxk ( ) + duk ( )(7.28)whereijc= d= − u k = ⎡ ⎣ ⎢ φ ⎤α, [( 1 α) β], ( )Eij⎦⎥(7.29)