Wireless Ad Hoc and Sensor Networks

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