Wireless Ad Hoc and Sensor Networks

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Distributed Fair Scheduling in Wireless Ad Hoc and Sensor Networks 311LEMMA 7.3.1If the weights are updated as in Equation 7.9 for a sufficiently long interval [ t1, t2],then the weight error ̃φ ij ( k + 1 ) is bounded, provided||α < 1PROOF Using Equation 7.9 and Equation 7.13, the weight estimation erroris expressed as̃ φ ( k+ 1) = α. ̃ φ ( k) + ( 1− α) φ + β.Eij ij ij ij(7.14)Choose a Lyapunov functionV = ̃φ 2 ( k)ij(7.15)The first difference of the Lyapunov equation can be obtained as∆V = V( k+ 1) −V( k)(7.16)or∆V = ̃ 2φ k+ − ̃2( 1) φ ( k)ijij(7.17)Substituting Equation 7.14 in Equation 7.17 to get∆V = [ αφ . ̃2( k) + ( − α) φ + β. E ] − ̃21φ ( k)ij ij ij ij(7.18)Equation 7.18 can be rewritten asEquation 7.19 can be simplified asThis further implies that2∆V = αφ̃2 2 2 2 2( k) + ( 1− α) φ + βE+ 2αφ̃( 1−α)φij ij ij ij2+ 21 ( − αφβ ) E + 2αφβ ̃ E − ̃ φ( k)ij ij ij ij ij2 2 2 2 2 2∆V =−( 1− α ) ̃ φ + ( 1− α) φ + β E + 2αφ̃ ( 1−αφ )ij ij ij ij+ 21 ( − αφβ ) E + 2αφβ̃ Eij ij ij ijijij(7.19)(7.20)2ij| ∆V| ≤−( 1− α 2 ) ̃ φ + 2α ̃ φ a+b,ij(7.21)
312 Wireless Ad Hoc and Sensor Networkswherea= |[( 1− αφ ) + βE]|ijij(7.22)andb= ( 1− α) 2 φ 2 + β 2 E 2 + 2( 1−α)φ βEij ij ij ij(7.23)| ∆V|≤ 0implies that⎡22 2b| ∆V| ≤−( 1−α)̃ αφij− ̃2 ij a−2. (7.24)⎣⎢( 1−α ) φ ⎤( 1−α ) ⎦⎥̃φij2 2 2α + α a + b( 1 −α)≥2( 1 − α )(7.25)LetB ij,φbe the bound on the weight estimation error, thenBij, φ2 2α + α + b( 1 −α)=2( 1 − α )(7.26)For ̃φ ij ≥ Bij, φ ∆V < 0 . Because ̃φ ij ≥ Bij,φ, from Equation 7.13 we getˆφ ≤σφijij(7.27)for some σ .LEMMA 7.3.2The actual weights ˆφ ij at each node using Equation 7.9 converge close to their targetvalues in a finite time.PROOF Because || α < 1, define ̃φ ij ( k) = x( k), then Equation 7.14 can beexpressed asxk ( + 1) = cxk ( ) + duk ( )(7.28)whereijc= d= − u k = ⎡ ⎣ ⎢ φ ⎤α, [( 1 α) β], ( )Eij⎦⎥(7.29)
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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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100 Wireless Ad Hoc and Sensor Netw
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4Admission Controller Design for Hi
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Admission Controller Design for Hig
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Page 326 and 327: 7Distributed Fair Scheduling in Wir
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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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