Wireless Ad Hoc and Sensor Networks

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Distributed Power Control of Wireless Cellular and Peer-to-Peer Networks 183Here P= ( p denotes the maximum transmission power level of eachmax)transmitter. The scheme assumes that there exists a unique power vectorP * , which would solve Equation 5.7. Thus, by a feasible system, the matrix* −1A is nonsingular, and 0 ≤ P = A µ ≤P.Iterative methods can be executedwith local measurement to find the power vector P * . Through somemanipulations, the following second-order iterative form of the algorithmis obtained as{ { }}( l )l l ll lp = min p , max 0, w γ p R + ( 1−w ) p+ 1 −iii1i ii(5.9)where w l = 1+1/ 1. 5n . The min and max operators in the Equation 5.9 areto guarantee that the transmitter power will be within the allowable range.The DPC schemes proposed in Bambos et al. (2000) and Jantti and Kim(2000) appear to result in unsatisfactory performance in terms of convergenceand during admission control as demonstrated in our simulations.The proposed work (Jagannathan et al. 2002, Dontula and Jagannathan2004) is aimed at addressing these limitations. A suite of closed-loop DPCschemes is presented next. Both state space (SSCD) and optimal schemesfor updating the transmitter power are described in detail. The convergenceproofs of these control schemes are also given, and they are demonstratedin simulation. Later, the DPC scheme is modified so that suitablepower can be selected during fading channels.5.2.3 State-Space-Based Controls DesignThe calculation of SIR can be expressed as a linear system asR( l+ 1) = R( l) + v()li i i(5.10)where Ri( l+ 1) , Ri( l)are the SIR values at the time instant l and l + 1,respectively, with vi( l)being the power value. Equation 5.10 is obtainedas follows: i.e., each user-to-user or user-to-base-station connection canbe considered as a separate subsystem as described by the equationR ( l+ 1)=ipi() l + ui()lI () li(5.11)p () l( )inGijκiwhere by definition Ri() l = , and interference Ii () l = ∑ j≠ipj()* l + , withGiiGiiIi() ln as the number of active links. The vi( l)in each system is defined
184 Wireless Ad Hoc and Sensor Networksu () las vi() il = , where u is the input to each subsystem, which dependsIi() li( l)only on the total interference produced by the other users. To maintainthe SIR of each link above a desired target and to eliminate any steadystateerrors, SIR error ei() l = Ri()l − γ i is defined as the difference betweenactual SIR and its target value γ i . Therefore, defining e() l = x(),l we haveiix( l+ 1) = J x( l) + Hv()li i i i i(5.12)where J i = 1, and H i = 1. According to the state space theory (Lewis1999), the equation in the preceding text represents a first-order linearstate space system. Our goal is to maintain a target SIR for each networklink while the transmitter power is adjusted so that the least possiblepower is consumed.THEOREM 5.2.1Given the SIR system as Equation 5.12, and if the feedback for the ith transmitteris chosen as vi() l =− kixi()l +ηiwith k i representing the feedback gains,η i representing the protection margin and the associated power update, P i( l+ 1) = ( vi( l) Ii( l) + pi( l),given in Table 5.1 is used, then the closed-loop systemis stable for each link, and the actual SIRs will converge to their correspondingtarget values.PROOF Applying the feedback into Equation 5.12 the closed-loop errorsystem in SIR Is got asx( l+ 1) = ( J − H k ) x( l)+ Hηi i i i i i i(5.13)TABLE 5.1State-Space-Based Distributed Power ControllerSIR system state equationSIR errorFeedback controllerPower updateR( l+ 1) = R( l) + v () li i ix () l = R()l −γi i iv () l =− k x () l +ηi i i ip( l+ 1) = ( v ( l) I ( l) + p( l))i i i iwherek i, γ i , andηi are design parameters
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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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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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