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