Wireless Ad Hoc and Sensor Networks

444 Wireless Ad Hoc and Sensor NetworksConsider ∆J 1 from Equation 9.22 and substitute Equation 9.18 to get∆J 1 =ei ( k+ 1) − ei ( k) =( kvei( k ) + ̃α i( k) Ri( k)) − ei 2 ( k)2 2 2(9.23)Taking the second term of the first difference from Equation 9.22 andsubstituting Equation 9.19 yields−1 2 2∆J2= σ ⎡i k+ 1 − i k ⎤ 2 kveiki⎣α ̃ ( ) α̃ ( ) ( ) α̃( kR ) i( k)⎦ =− [ ]2 2− 2[ α̃i( kR ) i( k) ] + σRi ( k) [ ke v i( k)+ α̃i ( kR ) i ( k)] 2(9.24)Combine Equation 9.23 and Equation 9.24 to get∆J = ⎡2 2+ R kk ⎤ 2 2i v ei k + Ri k ke v⎣1 1 σ ( )⎦( ) 2σ( )[ i( k)][ α̃i( k) Ri( k)]− −( )( )− 1 −σR ( k) [ α̃( k) R ( k)] 2 ≤− 1 −δke ( k)2i i i( vmax)2 2i( )2−( 1 −σRi( k) )⋅ α̃ i( k)Ri( k) − σ Ri( k) 2 ⎡1−σRi( k) ⎤ kvei( k)⎣⎢2⎦⎥2(9.25)where d is given after (9.20). Now, taking expectations on both sides yields⎧⎪2 2E( ∆J) ≤ −E⎨( 1 −δkvmax) ei( k)⎩⎪222−( 1 −σ Ri( k)) ̃αi( kR ) i( k) + σ Ri( k) ⎡2−σRi( k) ⎤ ⎫1 ke v i( k)⎣⎢⎦⎥ ⎬ ⎪ ⎭⎪(9.26)Because EJ () > 0and E( ∆J)≤0 , this shows the stability in the mean, inthe sense of Lyapunov (Jagannathan 2006) provided the conditions inEquation 9.20 hold, so Ee [ i( k)] and E[ ̃α i( k)](and hence E[ αˆ i( k)]) arebounded in the mean if Ee [ i( k0)]and E[ ̃α i( k 0 )] are bounded. Summingboth sides of Equation 9.26 and taking limits lim E( ∆J), the errorl→∞E[ |()|] eik → 0 .Consider the closed-loop throughput error with estimation error, ε( k),ase ( k+ 1) = K e ( k) + α ( k) R( k) + ε(k). (9.27)i v i i i