Wireless Ad Hoc and Sensor Networks

238 Wireless Ad Hoc and Sensor NetworksBy appropriately selecting k v via placing the eigenvalues within a unitcircle, it is easy to show that the closed-loop SIR system is asymptoticallystable in the mean or asymptotically stable, lim Ee { ( )} . This rendersli l = 0→∞that yi( l)→ γ .CASE 2 a i , b i , and r i are unknown. In this scenario, SIR error equation(see Chapter 5) can be expressed asyi()lyi( l+ 1 ) = [ α ⎡ ⎤i( l) ri( l)]⎢ ⎥ i(l(6.6)⎣⎢i()l ⎦⎥ +Tβ ) vi( l) = θi( l) ψi( l) + βi( l) vi( l)ωTywhere θi() l = [ αi() l ri()]l is a vector of unknown parameters, andi () lψ i() l = [ ωi() l ]is the regression vector. Now, selecting feedback control for DPC asv() l = − 1β ()[ l − ˆ θ () l ψ () l + γ + k e ()] li i i i v i(6.7)where θiˆ () l is the estimate of θ i () l , then the SIR error system is expressed asTTe ( l + 1 ) = k e ( l) + θ () l ψ () l − ˆ θ () l ψ () l = k e lT() + ̃θ () l ψ () li v i iiiiv iii(6.8)where ̃θi() l θi() l ˆ θi() l is the error in estimation. From Equation 6.8, it isclear that the closed-loop SIR error system between a transmitter–receiverpair in an ad hoc wireless network is driven by channel estimation error.If the channel uncertainties are properly estimated, then estimation errortends to be zero. In this case, Equation 6.8 becomes Equation 6.5. In thepresence of error in estimation, only boundedness of error in SIR can beshown. We can show that the actual SIR approaches the target providedthe channel uncertainties are properly estimated. To proceed further,Assumption 6.3.1 is used in Jagannathan et al. (2004).= −e l k e l l l lASSUMPTION 6.3.1The channel should change slowly in comparison to parameter updates.REMARK 1The channel estimation scheme works well when the channel changesslowly compared to estimation. However, to accommodate the suddenchanges in channel conditions, an additional safety factor has to be introducedin this work.Consider now the closed-loop SIR error system with channel estimationerror, (l), as( + 1) = ( ) + ̃θ () ⋅ ψ () + ε ()Ti v i i(6.9)where e(l) is the error in estimation which is considered bounded above|()| ε l ≤ ε , with e N a known constant.Ni