Wireless Ad Hoc and Sensor Networks

220 Wireless Ad Hoc and Sensor NetworksTABLE 5.3Distributed Power Controller during Fading Channels with Estimation ErrorBeing ZeroSIR system state equationwhereSIR errorFeedback controllerChannel parameter updatePower update⎡ yi()l ⎤yi( l+ 1) = [ αi( l) ri( l)]⎢ ⎥ + βi()lvi()l⎣ωi()l ⎦y () l = R()li= θ () lψ () l + β () l v () le () l = R()l −γi i iii T i i iv () l = − 1β ()[ l − ˆ θ () lψ () l + γ + k e ()] li i i i i v iˆ θ ( l+ 1 ) = ˆ θ () l + σψ () l e ( l+1 )i i i i TPl ( + 1) = ( v( lI ) ( l) + Pl ( ))i i i iwherek v, γ i , σ andηi are design parametersCombining Equation 5.69 and Equation 5.70 to get( )∆J =−ei T () l ⎡I− 1+σψ i T () lψi() l kv T k ⎤v ei()l⎣⎦( ) ⎡ ⎣+ 2σψ i T T() lψ i()[ l kvei()] l ⎡θ ̃ i T () lψi()l ⎤⎣ ⎦ − 1 −σψ i T () l ψ i() l θ ̃i T () l ψ i() l ⎤ ⎡ θ ̃⎦ ⎣i T () l ψ i(l)⎤⎦T( vmax) i − − i≤− 1−δk 2 |()| e l 2 1 σ | ψ ( l)|2( )̃θi Tlψi()l −1 −2σ | ψi( l)|ke2 v i()lσ | ψ ( l)|i2(5.71)where δ is given after Equation 5.66. Taking expectations on both sidesyieldsE( ∆J)≤−E⎛2× ( 1−kvmax) 2 |()| ei l 22σ | ψ ( )|δ −( 1−σ | ψ i( l)||) ̃θi Ti l⎜lψi()l + k2 vei( l)⎝⎜1 − σ | ψi( l)|⎞⎟⎠⎟(5.72)Because EJ ()> 0 and E( ∆J) ≤ 0,this shows the stability in the mean viasense of Lyapunov provided the conditions (Equation 5.65) and (Equation5.66) hold, so Ee [ i( l)] and E[ ̃θ i( l)](and, hence, E[ ˆ θi( l)])are bounded in themean if Ee [ i( l0)]and E[ ̃θ i( l )] are bounded in a mean. Summing both sides0of Equation 5.72 and taking limits lim E ( ∆J), the SIR error can be shownl→∞to converge E[ |()|] eil → 0.2