Wireless Ad Hoc and Sensor Networks
Wireless Ad Hoc and Sensor Networks Wireless Ad Hoc and Sensor Networks
Distributed Power Control of Wireless Cellular and Peer-to-Peer Networks 185This is a stable linear system driven by a constant bounded input η i .Applying the well-known theory of linear systems (Lewis 1999), it is easyito show that xi( ∞ ) = η . In the absence of protection margin, the SIRkierror approaches to zero as t →∞.x iREMARK 1The transmission power is subject to the constraint pmin≤ pi≤ pmaxwherep min is the minimum value needed to transmit, p max is the maximumallowed power, and p i is the transmission power of the user i. Hence, fromEquation 5.11, pi( l+ 1)can be written as pi( l+ 1) = min( pmax, ( vi( l) Ii( l) + pi( l))).The power update presented in Theorem 5.2.1 does not use any optimizationfunction. Hence, it may not render optimal transmitter powervalues though it guarantees convergence of actual SIR of each link to itstarget. Therefore, in Table 5.2, an optimal DPC is proposed.THEOREM 5.2.2 (OPTIMAL CONTROL)Given the hypothesis presented in the previous Theorem 5.2.1, for DPC, with thefeedback selected as v() l =− k x() l +η , where the feedback gains are taken asi i i i( )−1k = H S H + T H S Ji i T ∞ i i i T ∞ i(5.14)TABLE 5.2Optimal Distributed Power ControllerSIR system state equationPerformance indexSIR errorAssumptionsFeedback controllerPower updateR( l+ 1) = R( l) + v () l∞∑i=1i i ixTx+vQvi T i i i T i ix () l = R()l −γi i iTi ≥ 0, Qi> 0 all are symmetricT−1TS = J [ S − SH ( H S H + T) H S ] J + Qiii i i iTi i i iT− Tk = ( H S H + T) 1 H S Jii∞v () l =− k x () l +ηi i i ii i ip( l+ 1) = ( v ( l) I ( l) + p( l))i i i i∞iiiwherek i, γ i , andηi are design parameters
186 Wireless Ad Hoc and Sensor Networkswhere is the unique positive definite solution of the algebraic Ricattiequation (ARE)S iS J S SH H S H T H S J Q( )⎡= − +⎣⎢⎤⎦⎥ +i i T i i i i T i i i i T i i−1(5.15)Then, the resulting time-invariant closed-loop system described byx( l+ 1) = ( J − HK ) x( l)+ Hηi i i i i i i(5.16)is asymptotically stable, if = 0.PROOF Follow the steps as in Lewis (1999).η iREMARK 2The proposed scheme minimizes the performance index∞∑ i=xTx i T i i + vQv i T i i where and are weighting matrices.The block diagram in Figure 5.1 explains the process of power controlusing SSCD/optimal schemes. The receiver as shown in the blockdiagram, after receiving the signal from the transmitter, measures theSIR value and compares it against the target SIR threshold. The differencebetween the desired SIR to the received signal SIR is sent to thepower update block, which then calculates the optimal power levelwith which the transmitter has to send the next packet to maintain therequired SIR. This power level is sent as feedback to the transmitter,which then uses the power level to transmit the packet in the next timeslot.1 T i Q iInterferenceTransmitter ∗ TPCP iFeedback∗ Transmitter power controlChannel with uncertaintiesTPC commandRadio channelReceiverDPCPower updateIEEE 802.11Target SIR (dB)Measured SIRFIGURE 5.1Block diagram representation of DPC.
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186 <strong>Wireless</strong> <strong>Ad</strong> <strong>Hoc</strong> <strong>and</strong> <strong>Sensor</strong> <strong>Networks</strong>where is the unique positive definite solution of the algebraic Ricattiequation (ARE)S iS J S SH H S H T H S J Q( )⎡= − +⎣⎢⎤⎦⎥ +i i T i i i i T i i i i T i i−1(5.15)Then, the resulting time-invariant closed-loop system described byx( l+ 1) = ( J − HK ) x( l)+ Hηi i i i i i i(5.16)is asymptotically stable, if = 0.PROOF Follow the steps as in Lewis (1999).η iREMARK 2The proposed scheme minimizes the performance index∞∑ i=xTx i T i i + vQv i T i i where <strong>and</strong> are weighting matrices.The block diagram in Figure 5.1 explains the process of power controlusing SSCD/optimal schemes. The receiver as shown in the blockdiagram, after receiving the signal from the transmitter, measures theSIR value <strong>and</strong> compares it against the target SIR threshold. The differencebetween the desired SIR to the received signal SIR is sent to thepower update block, which then calculates the optimal power levelwith which the transmitter has to send the next packet to maintain therequired SIR. This power level is sent as feedback to the transmitter,which then uses the power level to transmit the packet in the next timeslot.1 T i Q iInterferenceTransmitter ∗ TPCP iFeedback∗ Transmitter power controlChannel with uncertaintiesTPC comm<strong>and</strong>Radio channelReceiverDPCPower updateIEEE 802.11Target SIR (dB)Measured SIRFIGURE 5.1Block diagram representation of DPC.
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