Wireless Ad Hoc and Sensor Networks

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Distributed Power Control and Rate Adaptation 271rate, and the required power changes nonlinearly with change of modulationrate. The exact cost is expressed as:C u k ,r k =P k SNR ( r i( k )) ui(k)TX( i( ) i( )) 0( ) ⋅SNR( 0)Rr ( ( k))(6.45)where ui( k) is a burst size, ri( k) is a modulation, Rr () is a transmission ratefor modulation r, SNR() r and SNR( 0)are SNRs for modulation r, and 0(the lowest, available modulation), respectively, P0( k)is the power calculatedfor the modulation 0 (zero), and k is a time instant.REMARK 4The radio access standards have a limitation on how long a channel canbe used by a given node. Hence, the burst size is limited by the modulationrate used as u( k)/ Rr ( ( k )) ≤ max_tx_duration.iiREMARK 5The DP is minimizing cost functions over the u i and r i . Thus the optimalvalue of r i for a given u i can be inferred a priori, as a lowest rate thatsupports a given burst size. Hence, the transmission cost can be expressedas function of only u i .*C ( u( k )) =u( k) ⋅P( k) ⋅CM( u( k))TX i i 0i(6.46)* *where CM( ui( k )) = SNR( ri ( k))/[ SNR( 0) ⋅R( ri( k))]is the cost of transmittinga burst ui( k) using optimal modulation r i* ( k)for the burst size.6.9.2.3 Approximating Cost FunctionOverall, the cost function for the dynamic programming optimizationproblem is expressed as*J ( x( k )) =B( x( k )) +C ( u( k )) +J 1( x( k+ 1) )k i i TX i k+ i(6.47)where Jk( xi( k))is a cost function from time k to N (last step of algorithm)*with initial state xi( k) , Bx ( i( k)) is the cost function of queuing, CTX( xi( k))isa function of transmitting burst of size ui( k) , and Jk+ 1( xi( k+1))is a to-gocost from time k + 1.REMARK 6The cost function as in Equation 6.38, when used in DP will yield an optimalcontrol law. However, the calculation of such a law is computationally intensiveand highly sensitive to a number of possible values of ui( k).Instead, an approximated quadratic function is proposed in form ofCQ( u ( k )) = α ⋅P ( k) ⋅[ u ( k)]i(6.48)where parameter a is selected such that the least-square error for approximationis minimized.0i2i
272 Wireless Ad Hoc and Sensor NetworksEnergy consumed (mW ∗ burst time)160014001200100080060040020000 1 2 3 4 5 6 7Burst size (packets)FIGURE 6.17Rate selection as a function of queue utilization.Actual vs estimated energy spent on TXActual power costEstimated (2.4 ∗ u^2)REMARK 7Because the cost of transmission CQ( ui( k))is not exact, the calculated controllaw will be suboptimal.Figure 6.17 presents an example for the set of modulations presentedin Table 6.1. In this case, a is calculated to be equal to 2.4.Now, the final cost function is expressed as2 2J ( x( k )) =Q ( x( k )) +R ( u( k )) +J 1 ( x(k+ 1))k i k i k i k+ i(6.49)where Q k = γ and Rk = αP 0( k)are parameters from Equation 6.41 andEquation 6.45. The cost function in Equation 6.49 is in a quadratic form;hence, we can calculate an optimal control law using the standard Riccattiequation (Bertsekas 1987) because of the linear nature of the buffer dynamics.Next, the proposed solution is presented.6.9.3 The Riccatti EquationFirst, we notice that the rate adaptation problem should match closely thestate of the system, x i , to the outgoing flow, w i , rather than the queueutilization, because keeping an adequate flow of the data is more importantthan keeping the queue at a certain level. Hence, we consider a newstate variable that is equal to the sum of state x i and outgoing flow w i(negative value).z( k) = x( k) + w ( k)i i i(6.50)
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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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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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