Proceedings of the 2006 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 2006
FrC20.4
Distributed Nonlinear Power Control in Cellular Mobile Networks with Each User Communicating with Several Base Stations Andrey V. Savkin Abstract— A simple distributed power control algorithm for communication networks with each user communicating to several base stations is proposed. We prove that the proposed algorithm is globally converging.
I. I NTRODUCTION In cellular wireless communication networks, mobile users avoid interference phenomenon by regulating transmission power. The main requirement is to provide each mobile user an acceptable connection by limiting the interference caused by other users transmitting via the same channel in different cells. This problem has attracted a lot of attention; see e.g. [1], [2], [3] and references therein. Several efficient distributed power control algorithms converging to a fixed point were proposed. All these papers on power control consider the situation where each user is allocated to one base station. A new important research field of Mobile Ad Hoc Networks (MANETs) has emerged in recent years. MANETs are self-organizing networks built dynamically in the presence of nodes equipped with radio interface devices. The nodes are capable of movement in an arbitrary fashion, and perform routing and switching functions. When two nodes are not within each other’s communicating range, they can still communicate with each other in a multihop fashion [4]. These networks are designed for temporary and special use, such as, on a battlefield or an emergency rescue operation where there may not be any established infrastructure for networking. Research in MANET has given rise to many new network architectures. The paper [5] introduces SpeedNet, an ad hoc wireless mobile network where base stations (possibly other vehicles in the network) monitor and control velocities of mobile users (cars) in a speed limit critical zone. One of distinctive features of such mobile ad hoc networks is that each user simultaneously communicates with several base stations at any time. The number of base stations for each user may vary from two or three (as in [6], [5]) to six (as in [7]). This feature is especially important in networks in which base stations are not statically located but are free to move randomly and organize themselves arbitrarily; thus, the network’s wireless topology may change rapidly and unpredictably [8], [5]. In the case of each user communicating to several base stations, the standard results
on power control are not applicable anymore and the power control problem becomes much harder. In this paper, we consider a quite general problem statement where each user communicates to an arbitrary number of base stations. The main goal is to minimize power expenditure under the constraint that each user is provided with an acceptable connection to all relevant base stations. We propose a new nonlinear distributed power control law. We show that the corresponding nonlinear closed-loop system is globally stable; i.e. all its trajectories converge to a unique equilibrium state which corresponds to the minimal power expenditure. The proofs of the results will be given in the full version of the paper. II. P ROBLEM S TATEMENT We consider a mobile radio system in which n mobiles share the same radio channel. Let N1 ≥ 1, N2 ≥ 1, . . . , Nn ≥ 1 be given integers. We assume that that each mobile i is assigned to Ni base stations B(i, 1), . . . , B(i, Ni ). Furthermore, we assume that at any time t ≥ 0, the radio signal of the mobile i is received correctly at the base station B(i, s), 1 ≤ s ≤ Ni if the carrier-to-interference ratio at the base station at time t is not less than a given constant γis,min : s pi (t) gii s,min . s p (t) + v s ≥ γi g i j=1,j=i ij j
γis (t) := n
Here pi (t) ≥ 0 is the transmission power of the mobile i at s ≥ 0 is the link gain from the mobile j to the base time t, gij station B(i, s), and vis > 0 is the receiver noise at the station B(i, s). Moreover, we will assume that s gii >0
∀i, s.
(2)
Furthermore, introduce the numbers hsii := 0, ∀i, s; s,min s gij γi hsij := ∀i = j, s, s gii bsi := the vectors lis :=
s hi1
and the matrices
This work was supported by the Australian Research Council. The author is with the School of Electrical Engineering and Telecommunications, the University of New South Wales, Sydney, NSW 2052, Australia. Email:
[email protected]
1-4244-0210-7/06/$20.00 ©2006 IEEE
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hsi2
γis,min vi s gii ...
hsin ⎛
⎜ ⎜ L(s1 , s2 , . . . , sn ) := ⎜ ⎝
∀i, s,
l1s1 l2s2 .. . lnsn
∀i, s,
(3)
(4)
⎞ ⎟ ⎟ ⎟ ⎠
(5)
for all 1 ≤ s1 ≤ N1 , 1 ≤ s2 ≤ N2 , . . . , 1 ≤ sn ≤ Nn . It is obvious that L(s1 , s2 , . . . , sn ) is a square matrix of order n. Furthermore, let L be the set of all such matrices L(s1 , s2 , . . . , sn ). In general, the set L consists of N1 N2 · · · Nn matrices. Now the requirement (1) can be re-written as ⎞ ⎛ n (6) pi (t) − ⎝ hsij pj (t) + bsi ⎠ ≥ 0 ∀i, s j=1
for any t ≥ 0. T , Furthermore, let p(t) := p1 (t) p2 (t) . . . pn (t) and introduce the following functions ⎛ ⎞ n hsij pj + bsi ⎠ ∀i = 1, . . . , n. (7) fi (p) := max ⎝ s=1,...,Ni
j=1
In our new notations, the requirement (6) can be re-written as pi (t) − fi (p(t)) ≥ 0
∀i = 1, . . . , n
for all t ≥ 0. Also, let f (p) := f1 (p) f2 (p) . . .
fn (p)
(8) T
.
Here L is the set of matrices (5), and · denotes the standard Euclidean matrix norm. Now we are in a position to present the main result of this paper. Theorem 3.1: Consider a radio system with gains satisfying (2) and (13). Then, the following statements hold: 1) The system (12) has a unique equilibrium state p0 . 2) This equilibrium state p0 is the unique vector of optimal values of transmission powers (see Definition 2.1). 3) Any trajectory of the system (12) converges to p0 . The proof of Theorem 3.1 will be given in the full version of the paper. Remark 3.1: Notice that in the case of each user communicating to one base station (N1 = N2 = · · · = Nn ) the system (12) is linear and the control law (11) is reduced to the well-known Foschini and Miljanic algorithm [2]. Remark 3.2: Another interesting problem is to extend the main result to the case of link gains with more complicated uncertain and nonlinear dynamics. We believe that the robust control and filtering techniques of [9], [10], [11], [12], [13] will be relevant to this problem.
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Definition 2.1: The vector popt = opt opt opt T is called a vector of optimal p2 . . . pn p1 values of transmission powers, if it satisfies the equations:
popt = fi (popt ) i
∀i = 1, . . . , n
(10)
popt i
and > 0 for any i. Our goal is to construct for all i = 1, . . . , n a rule for control of the power pi (t) using only γimin , pi (t), γi (t) and their historical values such that the power vector p(t) converges to popt for any initial condition p(0). III. D ISTRIBUTED P OWER C ONTROL We propose for each mobile i power control rule described by the equation: p˙i (t) = −(pi (t) − fi (p(t)))
∀i = 1, . . . , n.
(11)
It is obvious from (1), (7) that fi (p) =
max
s=1,...,Ni
γis,min pi (t) . γis (t)
Therefore, the distributed control rule (11) uses only γimin , pi (t), γi (t) and their historical values. Moreover, the equations (11) can be re-written as p(t) ˙ = −(p(t) − f (p(t))).
(12)
Notice that the system (12) is a nonlinear autonomous system of ordinary differential equations with a continuous righthand side. The main assumption for our result is as follows: m := max L < 1. L∈L
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