Wireless Ad Hoc and Sensor Networks

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Distributed Power Control of Wireless Cellular and Peer-to-Peer Networks 217with the inclusion of noise, Equation 5.52 is written asR( l+ 1) = α ( l) R() l + β () l v() l + r() l ω () li i i i i i i(5.57)where ω( l)is the zero mean stationary stochastic channel noise with rl i( ) asits coefficient.The SIR of each link at time instant l is obtained using Equation 5.57.Carefully observing Equation 5.57, it is clear that the SIR at the timeinstant l + 1 is a function of channel variation from time instant l to l + 1.The channel variation is not known beforehand, and this makes the DPCscheme development difficult and challenging. Because α is not known,it has to be estimated for DPC development. As indicated earlier, all theavailable schemes so far ignore the channel variations during DPC development,and therefore, they render unsatisfactory performance.Now define y() l = R (), l then Equation 5.57 can be expressed asiiy( l+ 1) = α ( l) y() l + β () l v() l + r() l ω () li i i i i i i(5.58)The DPC development is given in two scenarios.CASE 1 α i , β i , and r i are known. In this scenario, one can select feedbackas−1v() l = β ()[ l −α () l y() l − r() l ω () l + γ + k ei i i i i i v i()] l(5.59)where the error in SIR is defined as e() l = R() l −γ . This implies thatiie( l+ 1) = k e( l)iv i(5.60)Appropriately selecting k v by 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 { i( l)} = 0.This rendersl→∞that yi( l) → γ .CASE 2 α i , β i , and r i are unknown. In this scenario, Equation 5.58 canbe expressed as⎡ yi()l ⎤yi( l+ 1) = [ αi( l) ri( l)]⎢ ⎥ + βi()lvi()l⎣ωi()l ⎦(5.61)= θ () lψ () l + β () l v()li T i i i
218 Wireless Ad Hoc and Sensor Networkswhere θi() l = [ αi() l ri()]l is a vector of unknown parameters, and ψ i () l= ⎡yi() l⎤ ⎣ ⎦ is the regression vector. Now selecting feedback for DPC asωi () lv() l = − 1β ()[ l − ˆ θ () lψ () l + γ + k e ()] li i i i v i(5.62)where θiˆ () l is the estimate of θ i (), l then the SIR error system is expressed ase( l + 1) = k e( l) + θ () lψ () l − ˆ θ () lψ() li v i i T i i T i= k e() l + ̃θ () lψ() lv i i T i(5.63)*TPC: Transmitter power control *DPC: Distributed power controlwhere ̃θi() l θi() l ˆ θi() l is the error in estimation. From Equation 5.63, it isclear that the closed-loop SIR error system is driven by the channel estimationerror. If the channel uncertainties are properly estimated, thenestimation error tends to zero. In this case, Equation 5.63 becomesEquation 5.60. In the presence of error in estimation, only the boundednessof SIR error can be shown. We can show that the actual SIR approaches thetarget provided the channel uncertainties are properly estimated. Figure 5.26illustrates the block diagram representation of the proposed DPC wherechannel estimation and power selection are part of the receiver. To proceedfurther, the Assumption 5.4.1 is required and therefore stated.= −Interference5.4.2.2 Adaptive Scheme DevelopmentASSUMPTION 5.4.1The channel changes slowly compared to the parameters’ updates.Radio channel with uncertainties (path loss,Rayleigh fading, shadowing)Transmitter∗ TPCTPC command∗ DPCTarget SIR (dB)Measured SIR (dB)Target SIR(dB)Feedback forpower selectionRadio channelwith uncertaintiesChannel parameterestimationPower selectionReceiverFIGURE 5.26Block diagram representation of distributed power control with channel uncertainties.
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Wireless Ad Hocand SensorNetworksPr
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18. Chaos in Automatic Control, edi
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CRC PressTaylor & Francis Group6000
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Table of Contents1 Background on Ne
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3 Congestion Control in ATM Network
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5.4.2 Distributed Power Controller
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7.3 Adaptive and Distributed Fair S
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9.2.2 Overview.....................
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the number of connections in each l
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y Maheswaran Thiagarajan, and tirel
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2 Wireless Ad Hoc and Sensor Networ
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10 Wireless Ad Hoc and Sensor Netwo
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2BackgroundIn this chapter, we prov
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Background 51u(k)x n (k + 1)x n (k)
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Background 53with x( 0)being the in
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Background 55with tr(.) the matrix
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Background 57nGiven a function ft (
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Background 592.3.2 Lyapunov Stabili
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Background 61as well as Jagannathan
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Background 63The subsequent example
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Background 65Evaluating this along
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Background 67Now, select the feedba
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Background 69is SISL if and only if
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Background 71L 1 (x)L(x, t)L 0 (x)0
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Background 73for some R > 0 such th
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Background 75Evaluating the first d
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Background 77Section 2.4Problem 2.4
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4Admission Controller Design for Hi
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7Distributed Fair Scheduling in Wir
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Distributed Fair Scheduling in Wire
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8Optimized Energy and Delay-Based R
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Optimized Energy and Delay-Based Ro
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9Predictive Congestion Control for
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Predictive Congestion Control for W
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10Adaptive and Probabilistic Power
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Adaptive and Probabilistic Power Co
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