Wireless Ad Hoc and Sensor Networks
Wireless Ad Hoc and Sensor Networks Wireless Ad Hoc and Sensor Networks
Distributed Power Control of Wireless Cellular and Peer-to-Peer Networks 219THEOREM 5.4.1Given the DPC in the preceding text (with channel uncertainty), if the feedbackfrom the DPC scheme is selected as Equation 5.62, then the mean channelestimation error along with the mean SIR error converges to zero asymptotically,if the parameter updates are taken asˆ θ ( l+ 1 ) = ˆ θ () l + σψ () l e ( l+1 )i i i i T(5.64)provided2σ | ψ ( )| < 1i l(5.65)k vmax < 1 δ(5.66)where δand σ is the adaptation gain.1= ,−21 σ| ψi( l)| PROOF Define the Lyapunov function candidateJi ei T 1= () l ei() l + ⎡i T ⎣() l i()l ⎤σ κ θ ̃ θ ̃ ⎦(5.67)whose first difference is∆J ∆J ∆J ei T l ei l ei T 1= 1+ 2 = ( + 1) ( + 1) − ( l) ei( l)+σ κ ⎡⎣ ̃θi T ( l+ 1) θ̃ i( l+ 1) −θ̃ i T ( l) θ̃i( l)⎤⎦(5.68)Consider ∆J 1 from Equation 5.68 and substituting Equation 5.63 to get∆J1 = ei T ( l+ 1) ei( l+ 1) −ei T ( l) ei( l)TT= ( ke v i() l + ̃θ i() lψi() l ) ( kvei() l + θ̃i T lψi() l )− ei T () l ei()l(5.69)Taking the second term of the first difference from Equation 5.68 andsubstituting Equation 5.64 yields1∆J2= ⎡⎣i T l+ 1 i l+ 1 − i T l i l ⎤σ κ θ ̃ ( ) θ ̃( ) θ ̃ ( ) θ ̃( )⎦T T= −2[ ke v i( l)] θ̃i ( l) ψi() l −2⎡ ⎤ ⎣θ̃i T lψi()l ⎡ ⎦θ̃i Tlψi()l ⎤⎣ ⎦+ σψ i T () lψ i() l kvei()l + θ̃i T T⎡lψi() l ⎤ ⎡⎣⎦kvei() l + θ̃⎣i T lψi()l ⎤⎦T(5.70)
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
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Distributed Power Control of <strong>Wireless</strong> Cellular <strong>and</strong> Peer-to-Peer <strong>Networks</strong> 219THEOREM 5.4.1Given the DPC in the preceding text (with channel uncertainty), if the feedbackfrom the DPC scheme is selected as Equation 5.62, then the mean channelestimation error along with the mean SIR error converges to zero asymptotically,if the parameter updates are taken asˆ θ ( l+ 1 ) = ˆ θ () l + σψ () l e ( l+1 )i i i i T(5.64)provided2σ | ψ ( )| < 1i l(5.65)k vmax < 1 δ(5.66)where δ<strong>and</strong> σ is the adaptation gain.1= ,−21 σ| ψi( l)| PROOF Define the Lyapunov function c<strong>and</strong>idateJi ei T 1= () l ei() l + ⎡i T ⎣() l i()l ⎤σ κ θ ̃ θ ̃ ⎦(5.67)whose first difference is∆J ∆J ∆J ei T l ei l ei T 1= 1+ 2 = ( + 1) ( + 1) − ( l) ei( l)+σ κ ⎡⎣ ̃θi T ( l+ 1) θ̃ i( l+ 1) −θ̃ i T ( l) θ̃i( l)⎤⎦(5.68)Consider ∆J 1 from Equation 5.68 <strong>and</strong> substituting Equation 5.63 to get∆J1 = ei T ( l+ 1) ei( l+ 1) −ei T ( l) ei( l)TT= ( ke v i() l + ̃θ i() lψi() l ) ( kvei() l + θ̃i T lψi() l )− ei T () l ei()l(5.69)Taking the second term of the first difference from Equation 5.68 <strong>and</strong>substituting Equation 5.64 yields1∆J2= ⎡⎣i T l+ 1 i l+ 1 − i T l i l ⎤σ κ θ ̃ ( ) θ ̃( ) θ ̃ ( ) θ ̃( )⎦T T= −2[ ke v i( l)] θ̃i ( l) ψi() l −2⎡ ⎤ ⎣θ̃i T lψi()l ⎡ ⎦θ̃i Tlψi()l ⎤⎣ ⎦+ σψ i T () lψ i() l kvei()l + θ̃i T T⎡lψi() l ⎤ ⎡⎣⎦kvei() l + θ̃⎣i T lψi()l ⎤⎦T(5.70)
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