Location based Power Control for Mobile Devices ina Cellular Network

Pubudu N. PathiranaSchool of Engineering and Technology, Deakin University

Victoria, Australia 3217, Email: pubudu@deakin.edu.au

Abstract— This paper provides location estimation basedpower control strategy for cellular radio systems via a locationbased interference management scheme. Our approach considersthe carrier-to-interference as dependent on the transmitter andreceiver separation distance and therefore an accurate estimationof the precise locations can provide the power critical mobileuser to control the transition power accordingly. In this fullydistributed algorithms, we propose using a Robust ExtendedKalman Filter (REKF) to derive an estimate of the mobile user’sclosest mobile base station from the user’s location, headingand altitude. Our analysis demonstrates that this algorithm cansuccessfully track the mobile users with less system complexity, asit requires measurements from only one or two closest mobile basestations and hence enable the user to transmit at the rate that issufficient for the interference management. Our power controlalgorithms based on this estimation converges to the desiredpower trajectory. Further, the technique is robust against systemuncertainties caused by the inherent deterministic nature of themobility model. Through simulation, we show the accuracy ofour prediction algorithm and the simplicity of its implementation.

Index Terms : Power Control, CarNet, Location tracking,mobility modelling, Robust Extended Kalman Filter, Wirelessnetworks.

I. INTRODUCTION

The transmitter power control has attracted much attentionduring recent times to achieve a desired carrier to interferenceratio at the receiver. The key objectives of power control isto achieve power saving mobile terminal as well as eliminat-ing unnecessary interferences[1], determining capacity and thequality of service[2]. The cochannel interference caused byfrequency reuse is the single most limiting factor on the systemscapacity[3]. Further, power control has also been shown toincrease the call carrying capacity of cellular systems for chan-nelized systems[4] and also for single channel systems.[1][5].Therefore the basic idea is to contiguously adjust the trans-mission power, thus interference in the receiver is minimized.Naturally this optimal power level that the transmitter needsto transmit is a dependent on the relative transmission distanceand the receiver conserves power by using only just enough forminimum interference reception.

Mobility management and location estimation of mobileterminals in a wireless network has been considered for smoothoperation of real-time applications. The location managementapproach has two components: location update and locationprediction. Some recent studies have focused mainly on theupdate method[6], [7], [8]. Accurate prediction of the mobil-ity of the mobile user can be used in a range of network

management scenarios including power control. Tabbane [9]proposed that a mobile terminal’s location can be derivedfrom its quasi-deterministic mobility behavior and can berepresented as a set of movements in a user profile. MobileMotion Prediction (MMP), which uses pattern matching andpattern recognition, has been proposed as an enhancementof Tabbane’s method [10]. Bhattacharya et al. [11] used aninformation-theoretic approach to characterize the complexityof the mobility tracking problem in a cellular network. Shan-non’s entropy measure is identified as a basis for comparinguser mobility models. By building and maintaining a dictionaryof individual users’ path updates, the proposed adaptive on-linealgorithm can learn subscribers’ profiles. These and severalother similar schemes do not perform well when randomfactors are re-introduced or assumptions such as those regardingrectilinear movement patterns are removed. Extended Kalmanfilter technique has been applied in [12], [13]. Yang et al. [14]proposed an application of sequential Monte Carlo (SMC)methodology to the problem of joint mobility tracking and hardhand-off detection. This is computationally expensive, althoughit is based on mobile user dynamic model assumptions. Asit has been demonstrated that further improvements can beachieved via efficient prediction, in this paper we propose usinga Robust Extended Kalman Filter (REKF) as a state estimatorin predicting a mobile user’s expected trajectory for efficientpower control. These robust state estimation ideas emergedfrom the work of Savkin and Petersen[15]. This approach notonly provides satisfactory results[16], but also eliminates therequirement of the knowledge or modelling of the user mobilitypattern and measurement noise as required by the extendedKalman filter implementation presented in [12]. Further, asthe unpredictable user mobility is modelled as state noise, therobustness of the estimator is of importance.

Mobility tracking based on Receiver Signal Strength Indica-tor(RSSI) measurements is solved by treating it as an on-lineestimation in a nonlinear dynamic system. For example, theextended Kalman filter has been used to solve this problemin [12] [17].

As the physical systems are limited in their maximum powertransmission capacity, as in [18], our algorithm provides asolution in the allowable transmission range. The initial startingpower can be arbitrary in the range and as in [2] and can set toa lower value in order to avoid disturbances when a new userjoins as well as saving the battery life.

As in [2] our approach is cooperative in the sense that

limited information is allowed to flow among adjacent basestations although the control mechanism follows from localmeasurements only. Even though the algorithm can easilybe implemented into asynchronous operation, as in [2], forsimplicity we use the synchronous mode. Traditionally, inGSM-type networks, the base stations are statically located.

II. MOBILE BASE STATION/MOBILE USER DYNAMIC MODEL

We use the terminology Car for a mobile base station in thispaper (other mobile vehicles or robots fitted with a base stationwould fit into the same category). The dynamic model for theith car, Cari and the mobile user to be used in this approachcan be given in two-dimensional Cartesian coordinates as[19]:

xi(t) = Axi(t) + B1ui(t) + B2w(t) (II.1)

where

A =[

Θ 00 Θ

], B1 = −B2 =

[Φ 00 Φ

]

with

Θ =[

0 10 0

], Φ =

[0−1

]. (II.2)

The dynamic state vector xi(t) =[Xi(t) Xi(t) Yi(t) Yi(t)

]′, where Xi(t) and Yi(t)

represent the position of the user with respect to the base station(the ith car) at time t, and their first-order derivatives X(t) andY(t) represent the relative speed along the X and Y directions.In other words, if xM (t) = [xM (t) xM (t) yM (t) yM (t)]′

represents the absolute state (position and velocity in orderin the X and Y directions respectively) of the mobileuser, and xi

C(t) =[xi

C(t) xiC(t) yi

C(t) yiC(t)

]′denotes

the absolute state of the ith car in the same order, thenxi(t) � xM (t) − xi

C(t). Furthermore, let ui(t) denote thetwo dimensional driving and acceleration commands of thecar from the respective accelerometer readings and let w(t)denote the unknown two-dimensional driving and accelerationcommands of the mobile user.

A. Measurement model

In cellular systems, the distance between the mobile and aknown base station is practically observable. Such informationis inherent in the forward link RSSI (received signal strengthindication) of a reachable base station. Measured in decibels atthe mobile station, RSSI can be modelled as having two com-ponents: one from path loss and one from shadow fading[12].Fast fading is neglected, assuming that a low-pass filter is usedto attenuate Rayleigh or Rician fade. Denoting the ith car asCari (Figure II.1), the uplink RSSI at the base station fromCari, yi(t), can be formulated as[20]

yi(t) = poi − 10ε log di(t) + vi(t), (II.3)

and the down link RSSI measurement at the ith mobile userfrom the base station is :

ymi (t) = pB − 10ε log di(t) + νi(t). (II.4)

( x ,y )jC C

j

( x ,y )iC C

i

( x ,y )kC C

k

di

dj

(x M M,y )

Fig. II.1. Network geometry

where poi (t) is the transition power function of the ith mobile.

ε is a slope index (typically two for highways and four formicrocells in the city), and vi(t) is the logarithm of the shad-owing component, which is considered as an uncertainty in themeasurement and as in [2], we assume there is no interferencebetween the uplink and the down link. di(t) represents thedistance between the mobile and base station of Cari, whichcan be further expressed in terms of the mobile’s position withrespect to the location of the ith car, i.e., (Xi(t), Yi(t))

di(t) =(

Xi(t)2 + Yi(t)2)1/2

(II.5)

In [12], three independent distance measurements are usedto locate a moving user in a two-dimensional domain, as GSMsystems sample the forward link signal levels of six neighboringcells. We can assume that the signal of mobile i will be receivedcorrectly if the CIR at base i is not less than a given value γt

i .i.e

γi =yi∑n

j=1,j �=i yj + vi≥ γtgt

i , i = 1, ...., n. (II.6)

with n denoting the number of mobiles in the network sharingthe same channel at a given instance, vi is the receiver noiseat the base station i.

[I − H] po − 10ε [I − H] D ≤ η

where

D =

log d1

...log dn

, ηi = γtgt

i vi

and [H]ij ={

0 i = j

γtgti i �= j

(II.7)

and the solution for II.7 in the equality gives the

popt(x, t) = 10εD + (I − H)−1η (II.8)

when x = [x1 · · ·xi · · ·xn]′. Due to the fact that mobiletransmission power in practical systems cannot be arbitrarilylarge, we have the constraint

0 ≤ poi ≤ pmax (II.9)

By taking a first order Taylor series expansion around x(tn) =xn, we can write the approximation

popt(x, t) = Ψ(xn)t + Φ(xn) ≈ popt, (II.10)

where, Ψ(xn) = ∇xDx|x=xnand Φ(xn) = D − tnΨ(xn) +

(I − H)−1η

III. FIRST ORDER APPROXIMATION

Consider the following simple first order differential equa-tion:

poi = f (po

i − h(t)) , t ∈ [0,Γn], i = 1, · · · , N (III.1)

where f(p) = −gp and h(t) = mt+c with m, c, and 0 < g < 1given constants. Clearly, f(·) is a contraction mapping.

Proposition 1. There exists a solution to III.1 in the intervalII.9.

Proof: Choosing g < 1/Γn, this can be easily seenusing the Picard’s local existence and uniqueness theorem. See[18][21]

Proposition 2. Equation III.1 converges to h(t) exponentially.

Proof: ∀t1, t2 ∈ [0, T ]

poi (t1) − h(t1) = (po

i (t2) − h(t2)) exp (−g(t1 − t2)).t1 > t2 ⇒ po

i (t1) − h(t1) < poi (t2) − h(t2)

In application of the equation III.1, as the mobile user usesthe estimated state(x is used, Ψ(x) and Φ(x) is used.

The measurement equation of a dynamic system can bewritten in the following form:

y(t) = C(x(t)) + v(t) (III.2)

This more general nonlinear equation is for the receiver signalstrength measuring two body dynamic system and is valid forthe general case of mobile sensor mobile base station case.where v(t) = [vi(t) vj(t)]′ with

C(x(t)) =

p1(t) − 10ε log(

X1(t)2 + Y1(t)2)

...pi(t) − 10ε log

(Xi(t)2 + Yi(t)2

)...

pn(t) − 10ε log(

Xn(t)2 + Yn(t)2)

(III.3)for n number of sensors with pi(t) is given by equation III.1.

IV. SET-VALUE STATE ESTIMATION WITH A NON-LINEAR

SIGNAL MODEL

We consider a nonlinear uncertain system of the form

x = A(x, u) +B2w

z = K(x, u) (IV.1)

y = C(x) +v,

as a general form of the system given by equation II.1 with itsmeasurement equation in the form of equation III.2, and definedon the finite time interval [0, s]. Here, x(t) ∈ R

n denotes thestate of the system, y(t) ∈ R

l is the measured output, andz(t) ∈ R

q is the uncertainty output. The uncertainty inputs arew(t) ∈ R

p and v(t) ∈ Rl. In addition, u(t) ∈ R

m is the known

control input. We assume that all of the functions appearingin (IV.1) have continuous and bounded partial derivatives.Additionally, we assume that K(x, u) is bounded. This wasassumed to simplify the mathematical derivations and can beremoved in practice[15]. The matrix B2 is assumed to beindependent of x, and is of full rank.

The uncertainty in the system is defined by the followingnonlinear integral constraint[15] :

Φ(x(0)) +∫ s

0

L1 (w(t), v(t)) dt ≤ d +∫ s

0

L2 (z(t)) dt,

(IV.2)where d ≥ 0 is a positive real number. Here, Φ, L1 and L2

are bounded non-negative functions with continuous partialderivatives satisfying growth conditions of the type

‖φ(x) − φ(x′)‖ ≤ β

(1 + ‖x‖ + ‖x′‖

)‖x − x

′‖, (IV.3)

where ‖ · ‖ is the Euclidean norm with β > 0, and φ =Φ, L1, L2. Uncertainty inputs w(· ), v(· ) satisfying this condi-tion are called admissible uncertainties.

We consider the problem of characterizing the set of allpossible states Xs of the system (IV.1) at time s ≥ 0 which areconsistent with a given control input u0(· ) and a given outputpath y0(· ); i.e., x ∈ Xs if and only if there exists admissibleuncertainties such that if u0(t) is the control input and x(· ) andy(· ) are resulting trajectories, then x(s) = x and y(t) = y0(t),for all 0 ≤ t ≤ s.

A. Robust Extended Kalman Filter

Petersen and Savkin in [15] presented an extended Kalmanfilter version of the solution to the set-value state estimationproblem for a linear plant with the uncertainty described by anIntegral Quadratic Constraint (IQC). This IQC is also presentedas a special case of equation IV.2. We consider the uncertainsystem described by (IV.1) and an IQC of the form

(x(0) − x0)′X0 (x(0) − x0)

+12

∫ s

0

(w(t)

′Q(t)w(t)

)+ v(t)

′R(t)v(t)dt

≤ d +12

∫ s

0

z(t)′z(t)dt, (IV.4)

where N > 0, Q > 0 and R > 0.

This amounts to the so called Robust Extended Kalman Filter(REKF) generalization presented in[15].

In the application of REKF to CarNet, the ith system(Cari and the mobile user) tracking the mobile user duringa corresponding time interval is represented by the nonlinearuncertain system in (IV.1) together with the following IQC(from equation IV.4):

(x(0) − x0)′Ni (x(0) − x0)

+12

∫ s

0

(w(t)

′Qi(t)w(t)

)+ v(t)

′Ri(t)v(t)dt

≤ d +12

∫ s

0

z(t)′z(t)dt. (IV.5)

Here Qi > 0, Ri > 0 and Ni > 0 with i ∈ {1, 2, 3}are the weighting matrices for system i. For each system i,these parameters represent the relative weightings among theunknowns: the mobile user’s acceleration, measurement noiseand the uncertainty in the initial condition. Therefore, therelative effect of these inputs on system performance can becontrolled by adjusting these parameters (i.e more dominantinputs can have higher weightings). The initial state (x0) isthe estimated state of the respective systems at acquisition orhandover time. This initial state is essentially derived from theterminal state of the previous system together with other dataavailable in the network (i.e., vehicle position and speed) tobe used as the initial state for the next system taking over thetracking. With an uncertainty relationship of the form of (IV.5),the inherent measurement noise (see equation III.2), the un-known mobile user acceleration and driving command, and theuncertainty in the initial condition are considered as boundeddeterministic uncertain inputs. In particular, the measurementequation with the standard norm bounded uncertainty can bewritten as (equation III.2)

y = C(x) + δC(x) + v0, |δ| ≤ ξ (IV.6)

where ξ > 0 is a constant indicating the upper bound of thenorm-bounded portion of the noise. By choosing z = ξC(x)and ν = δC(x),

∫ T

0|ν|dt ≤ ∫ T

0z′zdt. Considering v0 and the

uncertainty w satisfying the bound in the form of

Φ(x(0)) +∫ T

0

[w(t)′Qw(t) + v0(t)′Rv0(t)] dt ≤ d, (IV.7)

it is clear that this uncertain system leads to the satisfaction ofthe condition in inequality IV.2 and hence IV.4 (see [15]). Thismore realistic approach removes any noise model assumptionsin the development of the algorithm and guarantees its robust-ness.

B. Robust versus optimal state estimation

The Robust Extended Kalman Filter was introduced in theprevious subsection. It tends to increase the robustness of thestate estimation process and reduce the chance that a smalldeviation from the Gaussian process in the system noise causesa significant negative impact on the solution. However, welose optimality and our solution will be just sub-optimal.To explain the connection between REKF and the standardextended Kalman filter, consider the system IV.1 with

K(x, u) = νK0(x, u) (IV.8)

where K0(x, u) is some bounded function, and ν > 0 is aparameter. Then, the REKF estimate x(t) for the system IV.1,IV.8, IV.4 converges to x0(t) as ν tends to 0. Here x0(t) is theextended Kalman state estimate for the system IV.1 with theGaussian noise

[w(t)′ v(t)′

]satisfying

E

{[w(t)v(t)

] [w(t)′ v(t)′

]}=

[Q(t) 00 R(t)

].

See, e.g. [22]. The parameter ν in IV.8 describes the size ofuncertainty in the system and measurement noise. For small νour robust state estimate approaches the Kalman state estimatewith Gaussian noise; for larger ν we achieve more robustnessbut less optimality. Hence, we always have some trade-offbetween robustness and optimality. We will show below viacomputer simulations that with larger uncertainty (which isquite realistic) our robust filter still performs well whereas thestandard extended Kalman estimate diverges.

V. SIMULATION

To examine the performance of the Robust Extended KalmanFilter based power control algorithm, simulations were carriedout for two mobile users in a network of hexagonal basestations. The measurement from the closest base station is used- the connectivity is maintained by the closest base station.The bases station measures the RSSI and the unknown andunpredictable mobile user mobility is modelled by arbitrarytime functions.

The simulated service area contains two cars for illustrativepurposes and can obviously be scaled for as many mobile basestations and users as required. The parameters (Table V.1) wereused for our simulations.

Parameter Value Comments

poi 20 W Base station

transmission power

{N1, N2, N3} 0.1{1, 1, 0.95}I4 the initial

viscosity solution

{Q1, Q2, Q3} {5, 5, 1} × 10−3I2 uncertainty in

the user

driving command

{R1, R2, R3} {6, 7, 5} × 103

I1 for scenario 1 measurement noise

I2 for scenario 2 Weighting

T 70mins Simulation time

Amax 3.3m/s2 driving command

amplitude

Ts 0.6 s Sampling interval

x1(0) Initial state

30 km/hr @ 10o w.r.t the 1st car

x2(0) Initial state

50 km/hr @ 140o w.r.t 2nd car

x3(0) Initial state

10 km/hr @ 240o w.r.t the 3rd car

TABLE V.1

SIMULATION PARAMETERS.

In the simulation of the dynamic system, we chose thefunctions given in Table V.2 for arbitrary car accelerations(uis) and unknown mobile user acceleration (w), with φ1

and φ2 uniformly distributed random variables in the interval[0 0.2Amax]. Here Amax is simply a constant used in the simu-lation indicating the magnitude of the deterministic componentof the unknown mobile user acceleration (w(t)).

Car Acceleration

1 Amax [−3 sin(0.2t) + 0.2φ1 0.9 cos(0.05 ∗ t) + 0.2φ2]′

2 1.6Amax [3 sin(0.2t) + 0.2φ1 − 2.0 cos(0.09t) + 0.2φ1]′

TABLE V.2

ACCELERATIONS OF DYNAMIC ENTITIES

The equation for the state estimation and the correspondingRiccati differential equation obtained[15] asfollows:

˙x(t) = Ax(t) + B1ui(t)+X−1(t)[β1 (x(t))′ Ri (y(t) − β (x(t)))+ξ2β1 (x(t))′ β1 (x(t))], x(t) = x0. (V.1)

X + A′X + XA + XB2Q−1i B′

2X

−β1x(t)′Riβ1x(t) + ξ2β1 (x(t))′ β1 (x(t)) = 0

X(0) = Ni, (V.2)

where for single base station measurement β(xi) = pi withi corresponding to the closest base station, and for two basestation measurements β(x) = C(x)as shown in equation III.3.Here,

β1(x) = ∇xβ(x) =∂

∂xpo(x, t), (V.3)

x0 is the last state before the handover.

Time 0 4.2 6.9 10.8 26.7 33.7 40.11

period(mins) 4.2 6.9 10.8 26.7 33.7 40.11 47

Hexagonal 1 5 6 11 12 17 22

area

TABLE V.3

USER 1 MOBILITY

Time 0 10.4 14 31 35.6 41.8

period 10.4 14 31 35.6 41.8 47

Hexagonal area 4 3 7 11 10 14

TABLE V.4

USER 2 MOBILITY

A. Discussion of results

The simulation for the transmission power control of twointerested users purely based on signal strength measurementswas performed successfully in a 90× 90km suburban area. Wehave restricted the number of mobile agents to two to ensurethe clarity and simplicity in demonstration but the approach canobviously be scaled to as many agents as required. We choserealistic values for the simulation parameters (i.e., velocitiesand accelerations of vehicles) to depict a real application. Asingle base station measures the forward link signal in the GSMsystem, estimates the mobile user’s location and velocity, as

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

X direction distance(km)

Y di

rect

ion

dist

ance

(km

)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

21

22

20

User 2

User 1

Actual Trajectory

Estmated Trajectory

Fig. V.1. The trajectories(estimated and actual) of two typical mobile usersmoving in the network

0 5 10 15 20 25 30 35 40 45 50130

140

150

160

170

180

190

200

Time (minutes)

Powe

r (W

att)

Desired Transmition power Actual transmitted power

Fig. V.2. The power dissipation for user 1(Desired and actual) of the twouses

shown in Figures V.1 the two users are estimated to a highaccuracy. The respective times and corresponding hexagonalbases station area that the each users are in are given in TableV.3 and V.4 for user 1 and user 2 respectively. The actual anddesired power dissipation trajectories are shown in Figure V.2and V.3 for user 1 and user 2. Estimation error comparisonwith the standard Kalman filter is shown in Figures V.4 and V.5and shows the efficiency in using this algorithm as comparedwith the standard extended Kalman filter. Here we used noisiermeasurements (ξ = 0.5) to demonstrate that, while the standardKalman filter diverges, the REKF continues to track the actualtrajectory in spite of these large noise inputs.

VI. CONCLUSION

We have provided a fully distributed scheme for controllingthe power dissipation of a mobile user based on the locationestimations. To the best of our knowledge there have beenno other studies of such networks. The desire was to developan effective, robust and easily implementable algorithm withless burden on the system resources for the ever increasingconstraints on energy usage. We proposed using a RobustExtended Kalman Filter-based state estimation algorithm tocontrol the transition power of mobile user and hence removethe unnecessary levels of power dissipation.

0 5 10 15 20 25 30 35 40 45 50130

140

150

160

170

180

190

200

Time (minutes)

Powe

r (W

atts)

Desired transmtion powerActual transmition power

Fig. V.3. The power dissipation for user 2(Desired and actual) of the twouses

0 5 10 15 20 25 30 35 40 45 500

100

200

300

400

500

600

Time (minutes)

Error

in tra

nsmi

tion p

ower

REKFExtedned Kalman filter

Fig. V.4. Performance comparison between Extended Kalman filter and REKFfor user 1

It is evident from our research that our algorithms convergeto the desired power dissipation rapidly. Emerging from recenttheoretical developments, REKFs can successfully be used inthe prediction of a mobile user’s location in a wireless ad hocnetwork with trackers and measurers switching appropriately.As our implementation with a single base station uses onlythe measurement from the closest neighboring station, thecomputational efficiency of the overall network is significantlyimproved. This considerably reduces the network traffic whileimproving the computational efficiency in using this algorithm.This algorithm is clearly more computationally efficient thanthe extended Kalman filter implementation provided in many

0 5 10 15 20 25 30 35 40 45 500

5

10

15

Time (minutes)

Errro

r in tra

nsmi

tion p

ower

REKFExtended Kalman filter

Fig. V.5. Performance comparison between Extended Kalman filter and REKFfor user 2

PCS networks. It can also be implemented within the mobileuser, if necessary, rather than in the base station to reduce thesignaling traffic. Further, as no assumptions were made on themeasurement noise and uncertain user acceleration component,the robustness of this algorithm is ensured. In future, we willconsider extending this work to outdoor localization of smallsensor devices.

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