POWER CONTROL IN WIRELESS COMMUNICATIONS NETWORKS - FROM A CONTROL THEORY PERSPECTIVE Fredrik Gunnarsson *,**,1 Fredrik Gustafsson *,1 * Control & Communication, Dept. of Electrical Eng., Link¨ opings universitet, SE-581 83 LINK ¨ OPING, Sweden. Email: [email protected], [email protected]** Ericsson Research, Ericsson Radio Systems AB, P.O. Box 1248, SE-581 12 LINK ¨ OPING, SWEDEN. Email: [email protected]Abstract: The global communications system today (the telephone system yes- terday) is considered as the largest man-made system all categories. While the demand for increased bandwidth in such systems increases, an increased interest in utilizing the available resources efficiently can be observed. Here, the subset of wireless cellular communications systems will be in focus and methods for transmitter power control. Relevant aspects of power control are discussed with emphasis on practical issues, using an automatic control framework. Generally, power control of each connection is distributedly implemented as cascade control, with an inner loop to compensate for fast variations and an outer loop focusing on longer term statistics. Issues as capacity, load and stability are discussed and related to whether it is possible to accommodate the requirements of all users or not. The operation of such algorithms are illustrated by simulations. Copyright c 2002 IFAC Keywords: power control, wireless networks, distributed control, stability, smith predictor, time delays, disturbance rejection 1. INTRODUCTION The use of control theory applied to communica- tion systems is increasingly popular. More com- plex networks are being deployed and the critical resource management constitutes numerous con- trol problems. Wireless networks are for example pointed out as a new vistas for systems and con- trol in (Kumar, 2001). This paper surveys power 1 This work was supported by the Swedish Agency for Innovation Systems (VINNOVA) and in cooperation with Ericsson Research within the competence center ISIS, which all are acknowledged. control research and provides an extensive list of citations. It gives an overview from a control perspective of achievements in the area to date with pointers to interesting open issues. The power of each transmitter in a wireless net- work is related to the resource usage of the link. Since the links typically occupies the same fre- quency spectrum for efficiency reasons, they mu- tually interfere with each other. Proper resource management is thus needed to utilize the radio resource efficiently. Most methods discussed here are generally applicable. Some of the problems, however, are more emphasized in the 3G systems
Transcript
POWER CONTROL IN WIRELESSCOMMUNICATIONS NETWORKS - FROM A
CONTROL THEORY PERSPECTIVE
Fredrik Gunnarsson ∗,∗∗,1 Fredrik Gustafsson ∗,1
∗ Control & Communication, Dept. of Electrical Eng.,Linkopings universitet, SE-581 83 LINKOPING, Sweden.
Keywords: power control, wireless networks, distributed control, stability, smithpredictor, time delays, disturbance rejection
1. INTRODUCTION
The use of control theory applied to communica-tion systems is increasingly popular. More com-plex networks are being deployed and the criticalresource management constitutes numerous con-trol problems. Wireless networks are for examplepointed out as a new vistas for systems and con-trol in (Kumar, 2001). This paper surveys power
1 This work was supported by the Swedish Agency for
Innovation Systems (VINNOVA) and in cooperation with
Ericsson Research within the competence center ISIS,
which all are acknowledged.
control research and provides an extensive listof citations. It gives an overview from a controlperspective of achievements in the area to datewith pointers to interesting open issues.
The power of each transmitter in a wireless net-work is related to the resource usage of the link.Since the links typically occupies the same fre-quency spectrum for efficiency reasons, they mu-tually interfere with each other. Proper resourcemanagement is thus needed to utilize the radioresource efficiently. Most methods discussed hereare generally applicable. Some of the problems,however, are more emphasized in the 3G systems
based on DS-CDMA (Direct Sequence Code Di-vision Multiple Access) such as WCDMA. Moreon Radio Resource Management in general canfor example be found in (Holma and Toskala,2000; Zander, 1997), and with a power controlfocus in (Gunnarsson, 2000; Hanly and Tse, 1999;Rosberg and Zander, 1998).
A simplified radio link model is typically adoptedto emphasize the network dynamics of power con-trol. The transmitter is using the power p(t), andthe channel is characterized by the power gaing(t) (< 1). The “bar” notation indicates linearscale, while g(t) = 10 log10(g(t)) is in logarith-mic scale (dB). Hence, the receiver experiencesthe desired signal power C(t) = p(t)g(t) andinterference power I(t) from other connections.The perceived quality is related to the signal-to-interference ratio 2 (SIR) γ(t) = p(t)g(t)/I(t).For error-free transmission (and if the interferencecan be assumed Gaussian), the achievable datarate R(t) is given by (Shannon, 1956)
R(t) = W log2 (1 + γ(t)) [bits/s],
where W is the bandwidth in Hertz. From a linkperspective, power control can be seen as meansto compensate for channel variations in g(t). Thelink objective with power control can for examplebe
• to maintain constant SIR and thereby con-stant data rate
• to use constant power and variable coding toadapt the data rate to the channel variations
• to employ scheduling to transmit only whenthe channel conditions are favorable
This also depends on the data rate requirementsfrom the service in question.
1.1 Example: Power Control in CDMA Networks
Power control objectives are rather different whenconsidering networks and not only links. In aCDMA system, each user is allocated a code,and the signal space is essentially spanned bythe available orthogonal codes. The user’s datais recovered at the receiver by correlating thereceived signal with the allocated code. Due tochannels, nonlinearities etc, this orthogonality isnot preserved at the receivers. Instead, the codecorrelation (the scalar product) θij(t) ∈ [0, 1]between two codes of users i and j might benonzero.
Consider the simplistic uplink (mobile to basestation) situation in Figure 1. Assume that the
2 In dB: γ(t) = p(t) + g(t)− I(t)
mobiles are using the powers p1(t) and p2(t)respectively. The SIR of MS1 is given by
γ1(t) =p1(t)g1B
p2(t)g2B(t)θ12(t) + νB(t), (1)
where νB(t) represents thermal noise power atbase station B. The code correlation between mo-biles 1 and 2 θ12 is nonzero, since the signals havepassed through independent channels. Hence, theconnections are mutually interfering, and this factrestricts the achievable SIR’s to (see Theorem 4)
γ1γ2 <1
θ12θ21(2)
Limited transmission powers might further re-strict the achievable SIR’s.
PSfrag replacements
g1B
g2B
gB1gB2
MS1
MS2
a)b)
Figure 1. Simplistic uplink and downlink situationwith two mobiles connected to one base sta-tion to illustrate fundamental network limi-tations and objectives.
The downlink (base station to mobile) situation isslightly different. All the signals from a base sta-tion to a specific mobile have passed through thesame channel, and orthogonality can be assumedpreserved. (This is not true in reality, mainly dueto non-ideal receivers.) If the power of the signalsfrom other cells and thermal noise at mobile 1 isdenoted by ν1(t), the SIR is given by
γ1(t) =p1(t)gB1
ν1(t), (3)
The downlink power is limited to Pmax = PD+PC ,where PD is for user data and PC is for controlsignaling and pilot symbols used for channel esti-mation in the mobile. Hence PC ≥ p1(t) + p2(t)(vector addition due to different user codes). Tofully utilize the hardware investment, the basestation should use all the available power PDto provide services. The interesting question ishow this power, and thus resulting service quality,should be shared between the users.
Each connection-oriented service is typically reg-ularly reassigned a reference SIR, γti (t) (note theswitch to values in dB). Power control is used tomaintain this SIR based on feedback of the error
ei(t) = γti (t)−γi(t). The feedback communicationuses valuable bandwidth, and should be kept at aminimum. Let f(ei(t)) denote the feedback com-munication (essentially quantization). With pureintegrating control, this yields
pi(t+ 1) = pi(t) + βf (ei(t)) (4)
1.2 Aspects of Power Control
Being subjective, the following list constitutesimportant aspects of power control:
• Objectives. It is vital to clarify the aimof power control. As indicated in the exam-ple above, throughput maximation leads todifferent control strategies compared to fairobjectives where all users experience roughlythe same quality of service.
• Centralized/decentralized control. Cen-tralized power control is not practicallytractable. As discussed in Section 5, it mainlyserves as theoretical performance bounds tothe decentralized algorithms in Section 3.
• Feedback bandwidth. The feedback band-width should be stated as the number ofavailable bits per second for feedback com-munication. Then, this becomes a trade-offbetween error representation accuracy andfeedback rate as discussed in Section 3.1
• Power constraints. The transmission pow-ers are constrained due to hardware limita-tions such as quantization and saturation,which is in focus in Sections 3.3 and 3.5.
• Time delays. Measuring and control signal-ing take time, resulting in time delays in thedistributed feedback loops. The time delaysare typically fixed due to standardized sig-naling protocols, and are further treated inSection 3.3
• Disturbance rejection. The controller’sability to mitigate time varying power gainsand measurement noise is an important per-formance indicator further discussed in Sec-tion 3.3.
• Soft handover. One important coverage im-proving feature in DS-CDMA systems it thata mobile can be connected to a multitudeof base stations. This puts some specific re-quirements on power control which are brieflytouched upon in Section 4.4.
• Stability and convergence. Studying sta-bility and oscillatory behavior of the dis-tributed control loops as in section 3.4 is nec-essary, but not sufficient. The cross-couplingsbetween the loops also have to be considered.This is addressed in Section 5.2.
• Capacity and system load. As indicatedby the example above, the available radioresource is limited and have to be shared
among the users. An important distinction inSection 5.1 is therefore whether the networkcan accommodate all the users with associ-ated quality requirements.
2. SYSTEM MODEL
Most quantities will be expressed both in linearand logarithmic scale (dB). Linear scale is indi-cated by the bar notation g(t).
2.1 Power Gain
By neglecting data symbol level effects, the com-munication channel can be seen as a time vary-ing power gain made up of three componentsg(t) = gp(t) + gs(t) + gm(t) as illustrated byFigure 2. The signal power drops with distanced to the transmitter, and the path loss is mod-eled as gp = K − α log10(d). Terrain variationscause diffraction phenomenons and this shadowfading gs is modeled as AR(n)-filtered Gaussianwhite noise (n is typically 1-2, (Sørensen, 1998)).The multipath model considers scattering of radiowaves, yielding a rapidly varying gain gm (Sklar,1997).
0 5 10 15 20
−100
−95
−90
−85
−80
PSfrag replacements
gp
gs
gm
Travelled distance [m]
Pow
erG
ain
[dB
]
Figure 2. The power gain g(t) is modeled as thesum of three components: path loss gp(t),shadow fading gs(t) and multipath fadinggm(t). Here this is illustrated when movingfrom a reference point and away from thetransmitter.
2.2 Wireless Networks
Consider a general network with m transmittersusing the powers pi(t) and m connected receivers.For generality, the base stations are seen as multi-ple transmitters (downlink) and multiple receivers(uplink). The signal between transmitter i and re-ceiver j is attenuated by the power gain gij . Thusthe receiver connected to transmitter i will expe-rience a desired signal power Ci(t) = pi(t)gii(t)and an interference from other connections plus
noise Ii(t). The signal-to-interference ratio (SIR)at receiver i can be defined by
γi(t) =Ci(t)
Ii(t)=
gii(t)pi(t)∑j 6=i θij gij(t)pj(t) + νi(t)
, (5)
where θij is the normalized cross-correlation be-tween the waveforms of user i and j, and νi(t)is thermal noise. To keep the notation clear, wedefine θii = 1, and redefine the power gain toincorporate the cross-correlations:
gij := θij gij .
Depending on the receiver design, propagationconditions and the distance to the transmitter,the receiver is differently successful in utilizingthe available desired signal power pigii. Assumethat receiver i can utilize the fraction δi(t) ofthe desired signal power. Then the remainder(1− δi(t)
)pigii acts as interference, denoted auto-
interference (Godlewski and Nuaymi, 1999). Wewill assume that the receiver efficiency changesslowly, and therefore can be considered constant.Hence, the SIR expression in Equation (5) trans-forms to
γi(t) =δigii(t)pi(t)∑
j 6=i gij(t)pj(t) +(1− δi
)pi(t)gii(t) + νi(t)
.
(6)
From now on, this quantity will be referred toas SIR. For efficient receivers, δi = 1, and theexpressions (5) and (6) are equal. In logarithmicscale, the SIR expression becomes
γi(t) = pi(t) + δi + gii(t)− Ii(t). (7)
2.3 Power Control Algorithms
We adopt the loglinear power control modelin (Blom et al., 1998; Dietrich et al., 1996) toembrace central power control approaches. Thecascade control block diagram of a generic dis-tributed SIR-based power control algorithm is de-picted in Figure 3. The receiver computes the er-ror ei as the difference between the reference SIRγti and SIR (measured, subject to measurementnoise wi and possible filtered by the device Fi
3 ).The error is coded into power control commandsui by the device Ri, affected by command errorsxi on the feedback channel and decoded on thetransmitter side by Di. The control loop is subjectto power update delays of np samples and mea-surement delays nm samples. Typically, np = 1and nm. An outer loop adjusts the reference SIRto assure that the quality of service is maintained.Outer loop control is typically based on bit errorrates (BER) and block error rate (BLER) (Olofs-son et al., 1997; Wigard and Mogensen, 1996).
3 In practice, the desired signal power and the interference
are typically filtered separately with filters Fg,i and FI,i.
3. DISTRIBUTED POWER CONTROL
3.1 Feedback Bandwidth
The feedback signaling bandwidth is limited inreal systems. Typically, the communication is re-stricted to a fixed number k of bits per second.The evident trade-off is between error represen-tation accuracy and feedback command rate. Asingle bit error representation allow k feedbackcommands per second, while me bits error repre-sentation allow k/me commands per second. Thiscomparison is further explored in (Gunnarsson,2001).
Different error representations are proposed, forexample: single bit (the sign of the error) (Salmasiand Gilhousen, 1991), k-bit linear quantizer (Simet al., 1998) and k-bit logarithmic quantizer (Liet al., 2001). All these corresponds to quantizersand decoders in the devices Ri and Di in Figure 3.Note also that the error representation is relatedto the command error rate on the feedback chan-nel.
3.2 Power Control to Improve Link Performance
Power control algorithms aiming at optimizinglink performance mainly focus on link through-put and energy-effectiveness. One example (Gold-smith, 1997) actually meets the Shannon boundby transmitting more data when the channel isfavorable (instead of using only little power toequalize SIR). Such strategies primarily considersinformation theoretical aspects of power controlrather than network aspects.
3.3 Log-Linear Design
Early work such as (Foschini and Miljanic, 1993)addresses the problem in linear scale based on iter-ative methods for eigenvector computations (Fadeevand Fadeeva, 1963). Thereby, it is closely relatedto the global aspects in Section 5. In logarithmicscale, this is the special case β = 1 of an integrat-ing controller
pi(t+ 1) = pi(t) + βei(t) (8)
The algorithm proposed in (Yates, 1995) con-troller above with an arbitrary β. A compari-son to Figure 3 yields that this corresponds toRi{ei(t)} = ei(t) (the exact error signal is as-sumed), Di(q) = β
q−1 and no filtering. For reasonsthat will be evident later, the following inter-pretation is more natural when considering theacademic example of perfect error representation:
R(q) =β
q − 1, ui = pi, Di{pi(t−np)} = pi(t−np).
(9)
PSfrag replacements
Σ
Σ ΣΣ+−
q−nppi(t− np)ei(t)γti (t)
γi(t)γi(t)
Ri Di
δi + gii(t)− Ii(t)xi(t)
wi(t)
ui(t)
q−nmFi
Receiver
Outer
Loop
Decoding
Environ.Transmitter
QoSt
Figure 3. Block diagram of the receiver-transmitter pair i when employing a general SIR-based powercontrol algorithm. In operation, the controller result in a closed local loop.
This has motivated the alternative linear designsof R(q) as PI-controllers and more general linearcontrollers in (Gunnarsson et al., 1999). For exam-ple, it provides optimally fast I and PI controllerswhen subject to time delays. Perfect error rep-resentation results in a linear distributed controlloop with closed loop system Gll(q) and sensitivityS(q) given by
Gll(q) =R(q)
qnp+nm +R(q), S(q) =
qnp+nm
qnp+nm +R(q)(10)
With the single bit power control command asin (Salmasi and Gilhousen, 1991), the integratingcontroller becomes
pi(t+ 1) = pi(t) + β sign (ei(t)) (11)
The transmitter power is constrained in practice,typically quantized and bounded from above andbelow. Grandhi et al. (1995) proposes an algo-rithm to deal with powers bounded from above.In log-linear scale it is given by
pi(t+ 1) = min {pmax, pi(t) + βei(t)} (12)
This can be interpreted as one out of manypossible anti-reset windup implementations forPI-controllers (Astrom and Wittenmark, 1997),which thus can be employed to more general trans-mitter power constraints. The proposed powercontrol algorithm for limited and quantized trans-mitter powers in (Wu and Bertsekas, 1999) isnot linear and nor fully distributed, but is worthmentioning, due to its optimization approach.
Time delays are critically limiting the closed-loopperformance of any feedback system, and so alsowith power control. They therefore have to beconsidered in the design phase. The time delaysare known and fixed, since the signaling and mea-surement procedures are standardized, and prop-agation delays are neglectable (except possibly insatellite communications). For example, the typi-cal delay situation np = 1 and nm = 0 yields
γi(t) = pi(t− 1) + gii(t)− Ii(t) (13)
Since the delays are exactly known, time delayscan be compensated for using the Smith predic-tor (Astrom and Wittenmark, 1997) as describedin (Gunnarsson and Gustafsson, 2001b). Essen-tially, it is implemented as a measurement adjust-ment
γi(t) = γi(t) + pi(t)− pi(t− nm − np) (14)
The actual power levels might not be availablein the receiver, but rather the power controlcommands ui. These can, however, be used torecover the power level: pi = Di{ui}. With theSmith predictor, the closed loop system and thesensitivity becomes
Gll(q) =R(q)
qnp+nm(1 +R(q))(15)
S(q) =qnp+nm
qnp+nm(1 +R(q))(16)
The local behavior of the controllers above is il-lustrated in simplistic simulations in Figure 4. Wenote that the disturbance rejection is satisfactorywith most controllers. Furthermore, the benefitsof using the Smith predictor are more emphasizedwith single-bit error representation. Roughly thesame effect is obtained with linear design. Thisis in line with the results in (Kristiansson andLennartsson, 1999).
The Smith predictor might compensate for somedynamical effects, but the controllers still show de-layed reactions to changes in the power gains. Oneapproach to improve the reactions is to predict thepower gain. Considering only shadow fading, thefollowing model structure is relevant and fitted todata
gs(t) =C(q)
A(q)es(t), Var
{e2s(t)
}= σ2
e
Solve the Diophantine equation qm−1C(q) =A(q)F (q) +G(q) for F (q) and G(q) yields the op-
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14PSfrag replacements
a. b.
c. d.
γi(t)
[dB
]γi(t)
[dB
]
t [s]t [s]
Figure 4. Performance of different power controlalgorithms when subject to shadow fadingand the typical delay situation np = 1, nm =0. a. The PI-controller from pole placementsdesign with (solid) and without (dashed) pre-filter. b. The same controller at Ts = 0.015(solid) and Ts = 0.05 (dashed). c. An opti-mized PI-controller (solid) and an I-controllerwith Smith predictor (dashed). d. The I-controller with single-bit error representation(step-size 1 dB) with (dark gray) and without(light gray) Smith predictor.
timal m-step predictor (Astrom and Wittenmark,1997)
gs(t+m|t) =qG(q)
C(q)gs(t) (17)
Channel predictions are further studied basedon linear model structures (Choel et al., 1999;Ericsson and Millnert, 1996) and nonlinear modelstructures (Ekman and Kubin, 1999; Tanskanenet al., 1998; Zhang and Li, 1997)
When a disturbance model and an optimal predic-tor is available as above, the step to employ min-imum variance control is slightly short (Astromand Wittenmark, 1997; Gunnarsson, 2000):
R(q) =G(q)
q(q − 1)A(q)F (q), (18)
where F (q) and G(q) are obtained from the Dio-phantine equation above, and attention is payedto include integral action into the controller. Theperformance using predictive and minimum vari-ance controller is illustrated in Figure 5. Clearly,the error variance is significantly smaller than inFigure 4.
As indicated in previous sections, it is not alwaysjustifiable to let a user disturb other connectionssignificantly while aiming at a rather high SIRcompared to the propagation conditions. In theproposed algorithm by Almgren et al. (1994),users aiming at using a high power are forced touse a lower SIR. The algorithm expressed in thisframework is given by
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γi(t)
[dB
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[dB
]
t [s]
Figure 5. Predictive control improve the perfor-mance by predicting the disturbances. a. PI-controller with disturbance prediction (solid)and a minimum-variance controller (dashed).b. The single-bit error representation and I-control with the Smith predictor and distur-bance prediction.
R(q) =β
q − β , (19)
which does not include integral action. The algo-rithm is further explored in (Yates et al., 1997). Tokeep the inner control loop simple and intuitive,an alternative is to implement such priorities inthe outer control loop, see Section 3.5.
In real systems, SIR is typically not readily avail-able. One natural idea is to estimate SIR givenavailable measurements. Approaches to do so indifferent systems are discussed in (Andersin et al.,1998; Blom et al., 1999; Ramakrishna et al., 1997).A different approach is to base the power controlon the available measurements directly (Ulukusand Yates, 1998).
There have also been proposals which aim atequalizing the received power at the connectedbase station (Anderlind, 1997; Salmasi and Gilhousen,1991). The result by Ariyavisitakul (1994) con-cludes, however, that SIR based power controlprovides better control over the interference sit-uation.
All these algorithms are interpreted using the log-linear model. Uykan et al. (2000) proposes a PI-controller in linear scale which is hard to interpretin the log-linear model. Whether it is more effi-cient to pursue power control in logarithmic scalethan in linear scale is still unknown.
3.4 Analysis
Local stability analysis is straightforward whenthe error representation is ideal, since the localcontrol loop is all linear. E.g. root locus of the
poles to Gll(q) in (10) and (15) can be used toaddress local stability (Blom et al., 1998). It iseasy to see that time delays make the choice of βin the I-controller crucial to ensure local stability.For example the choice β = 1 and already a timedelay of one sample yield an unstable system,while it constitutes a dead-beat controller in theno delay situation. With the Smith predictor, thisdead-beat behavior is essentially recovered.
The single-bit error representation can be seen asrelay feedback in a linear system. The oscillatorybehavior can be approximated using discrete-timedescribing functions (Gunnarsson et al., 2001)yielding the oscillation period in samples as N =2 + 4(np + nm). Hence, the longer the delay, themore emphasized oscillatory behavior. This typeof controller is also locally analyzed in (Songet al., 2001), where the relay is approximated by aconstant and an additive disturbance yielding thesame relay output variance.
3.5 Outer Loop
SIR might be well correlated to perceived quality,but it is not possible to set a SIR reference offline,that result in a specific BER or BLER. Therefore,the inner control loop is operated in cascade withan outer loop, which adapts the SIR referencefor a specific connection. Reliable communicationcan be seen as low BLER requirements, whichin turn means that it is very hard to accuratelyestimate BLER. The errors appears rather seldomand it takes long time before the BLER estimateis stable. One implemented approach in severalsystems (Sampath et al., 1997) increases the SIRreference significantly when an erroneous block isdiscovered, and decreases the SIR reference whenan error-free block is received. Niida et al. (2000)provides experimental results of outer loop powercontrol using this method. One block comprisesmany bits. Therefore, it is easier to obtain a goodBER estimate. Then the relation between BERand BLER can be utilized to predict BLER basedon BER measurements (Kawai et al., 1999).
Another outer loop control mechanism is to adaptthe rate (Balachandran et al., 1999; Uvliden et al.,1998). In the uplink, rate adaption is supportedby the mobile in WCDMA (3GPP, 2001, docu-ment 25.321). Essentially, the optional feature al-lows the mobile to automatically increase its datarate and its SIR reference when the propagationconditions are favorable and vice versa.
3.6 Downlink Issues
As indicated by the introductory example, down-link power control objectives can be rather dif-ferent and specific. The power control objectives
depend on the policies of the network operator,and can be differentiated in terms of service re-quirements, resource utilization, subscription con-ditions etc. Revisit the introductory example inFigure 1. Some natural policies are:
• Aim at equal data rate (γ1(t) ≈ γ2(t)).• Aim at equal power usage (p1(t) ≈ p2(t)).• Use all power to transmit to the user with
highest power gain (i.e. mobile 1 as indicatedby the figure) to maximize throughput.
• First use power to meet the quality of servicerequirements of users with more expensivesubscriptions. Use the reminder to low-faresubscription users.
Various downlink power control and resource shar-ing issues are brought up in (De Bernardi et al.,2000; Lu and Brodersen, 1999; Song and Holtz-man, 1998; Vignali, 2001).
Another problem is uneven traffic distributionsthat are not supported by the cell layout. In sucha scenario, some cells might be overloaded, whileothers are under-utilized. The users select basestations based on measured pilot powers from thedifferent base stations. By controling the pilotpowers, the cells can be made larger or smaller.This is often referred to as cell breathing (Hwanget al., 1997).
4. STANDARDIZED POWER CONTROLALGORITHMS
Several power control algorithms are standardizedby 3GPP to be used in WCDMA (3GPP, 2001,document 25.214). In GSM, the situation wasdifferent, since the mobile serves as slaves, usingthe power determined by the system. Therefore,power control is not standardized except controland measurement signaling (GSM).
4.1 Fixed-Step Power Control
The power level is increased/decreased dependingwhether the measured SIR is below or abovetarget SIR, and implemented as:
Receiver : ei(t) = γti (t)− γi(t) (20a)
si(t) = sign (ei(t)) (20b)
Transmitter : pTPC,i(t) = ∆isi(t) (20c)
pi(t+ 1) = pi(t) + pTPC,i(t)(20d)
where si(t) are denoted the TPC (transmit-ter power control) commands. This is the de-fault choice both in the uplink and the downlinkclosed-loop power control. The uplink situationis slightly modified when the mobile is in softhandover. Then, the mobile receives power control
commands from every connected cell. To ensurethat the power is adapted to the best cell, themobile only increases the power if all commandsare equal to +1, otherwise the power is decreased.This algorithm is equal to the single-bit errorrepresentation with pure integration in (11).
4.2 Uplink Alternatives
This alternative algorithm is a different commanddecoding than above and is denoted ULAlt1. Itmakes it possible to emulate slower update rates,or to turn off uplink power control by transmittingan alternating series of TPC commands. In a5-slot cycle (j = 1, . . . , 5), the power updatepTPC,i(t) in (20c) is computed according to:
pTPC,i(t) =
∆i (j = 5)&(5∑
j=1
si(j) = 5)
−∆i (j = 5)&(
5∑
j=1
si(j) = −5)
0 otherwise
(21)
4.3 Downlink Alternatives
There are two downlink alternatives, both aimingat reducing the risk of using excessive powers. Inthe first one, here denoted by DLAlt1, the controlcommands are repeated over three consecutiveslots. The second one, denoted DLAlt2, reducesthe controllers ability to follow deep fades bylimiting the power raise. As with the ULAlt1,the commands are decoded differently than inSection 4.1, described as an alternative to (20c):
pTPC,i(t) =
−∆i si(t) < 0
∆i (si(t) > 0)&(psum,i(t) + ∆i < δsum)
0 otherwise
(22)
where psum,i(t) is the sum of the previous Npower updates and N and δsum are configurableparameters.
4.4 Soft Handover
One corecentral feature in DS-CDMA systemsis soft handover, where the mobile can connectto several base stations simultaneously. For bestperformance, the mobiles controls its power withrespect to the signal from the base station withthe most favorable propagation conditions. In-tuitively, the mobile only increases the power ifthe TPC commands from all the base stationsrequire it to do so. When command errors occur,
this might lead to unwanted effects. The mobilealgorithm of the TPC command combination isnot standardized, and the problem is addressedin Grandell and Salonaho (2001).
For downlink power control, the base stations ad-justs its powers according to the received TPCcommand from the mobile. Due to feedback er-rors, these powers might drift from the ideal powerlevels. To compensate for this drift, a centralizedpower balancing is proposed in the standards, see3GPP.
5. GLOBAL ANALYSIS
For practical reasons, power control algorithmsin cellular radio systems are implemented in adistributed fashion. However, the local loops areinter-connected via the interference between theloops, which affects the global dynamics as well asthe capacity of the system. An important globalissue is whether it is possible to accommodate allusers with their service requirements. The powergains reflect the situation from the transmittersto the receivers, and the results are thereforeapplicable to both the uplink and the downlink.
Sufficient conditions on global stability are de-rived, including the effect of time delays andgeneral log-linear power control algorithms andfilters. These conditions can be formulated asrequirements on the local loops. The interestingconclusion is thus that global stability can begranted by proper design of the local loops.
5.1 Performance Upper Bounds and Feasibility
The individual target SIR:s and the power gainsare considered constant in the global level analy-sis, where the latter is motivated by an assump-tion that the inner loops perfectly meets the pro-vided SIR reference, and thereby mitigates thefast channel variations. Note that values in linearscale are used in this section.
δigiipi∑j 6=i gij pj +
(1− δi
)pigii + νi
= γti , ∀i (23)
Introduce the matrices
Γt4= diag(γt1, . . . , γ
tm), Z = [zij ]
4=
[gijgii
],
∆4= diag
(δ1, . . . , δm
)
and vectors
p4= [pi] , η = [ηi]
4=
[νigii
].
The network itself put restrictions on the achiev-able SIR’s, and there exists an upper limit onthe balanced SIR (same SIR to every connection).
This is disclosed in the following theorem, neglect-ing auto-interference and assumes that the noisecan be considered zero
Theorem 1. (Zander, 1992). With probability one,there exists a unique maximum achievable SIR inthe noiseless case
γ∗ = max{γ0 | ∃ p ≥ 0 : γi ≥ γ0 , ∀i}.Furthermore, the maximum is given by
γ∗ =1
λ∗ − 1,
where λ∗ is the greatest real eigenvalue of Z .Note that λ∗ > 1 implies that γ∗ > 0. Moreover,the optimal power vector p∗ is the eigenvector ofλ∗ (i.e. kp∗ for any k ∈ R+ constitute an optimalpower vector.).
The effects of auto-interference is consideredin (Godlewski and Nuaymi, 1999)
The requirements in Equation (23) can be vector-ized to
p = Γt
((∆−1Z −E)p+ ∆
−1η), (24)
where E is the identity matrix. Solvability ofthe equation above is related to feasibility of therelated power control problem, defined as
Definition 2. (Feasibility). A set of target SIR:sΓt is said to be feasible with respect to a networkdescribed by Z, ∆ and η, if it is possible toassign transmitter powers p so that the require-ments in Equation (24) are met. Analogously, thepower control problem
(Z, η, ∆, Γt
)is said to be
feasible under the same condition. Otherwise, thetarget SIR:s and the power control problem aresaid to be infeasible.
The degree of feasibility is described by the fea-sibility margin, which is defined below. The con-cept has been adopted from Herdtner and Chong(2000), where similar proofs of similar and ad-ditional theorems covering related situations alsoare provided. Herdtner and Chong used the termfeasibility index RI and omitted auto-interference.
Definition 3. (Feasibility Margin). Given a powercontrol problem (Z, η, ∆, Γt), the feasibility margin
Γm ∈ R+ is defined by
Γm = sup{x ∈ R : xΓt is feasible
}
A motivation for introducing the name feasibilitymargin is to stress the similarity to the stabilitymargin of feedback loops. The following theoremcaptures the essentials regarding feasibility mar-gins.
Theorem 4. (Feasibility Margin). Given a powercontrol problem (Z, η, ∆, Γt), the feasibility mar-gin is obtained as
Γm = 1/µ∗
where µ∗ is
µ∗ = max eig{
Γt(∆−1Z −E)
}.
Moreover, if Γm > 1, the power control problemis feasible, and there exists an optimal powerassignment, given by
p =(E − Γt(∆
−1Z −E)
)−1
Γt∆−1η.
PROOF. See (Gunnarsson, 2000; Gunnarssonand Gustafsson, 2001a).
The power assignment above can of course beseen as a centralized strategy. However, since itrequires full information about the network it isnot plausible in practice, and the result mainlyserves as a performance bound.
The feasibility margin can also be related to theload of the system. When the feasibility marginis one, the system clearly is fully loaded (onlypossible when unlimited transmission powers areavailable). Conversely, when the feasibility marginis large, the load is low compared to a fullyloaded system. Thus the following load definitionis plausible.
Definition 5. (Relative Load). The relative load Lrof a system is defined by
Lr =1
Γm(= µ∗ in Theorem 4).
Feasibility of the power control problem is thusequal to a system load less than unity. For a moredetailed capacity discussion, see (Hanly, 1999;Zhang and Chong, 2000), where receiver and codesequence effects also are considered.
5.2 Convergence and Global Stability
As commented in Section 3.4, the integratingcontroller in (8) with β = 1 becomes unstablewhen subject to time delays. As concluded in thefollowing theorem, global stability is regained byemploying the Smith predictor.
Theorem 6. The algorithm (8) and βi = 1 subjectto a delay of n samples and employing the Smithpredictor converges to a unique equilibrium p∞that meets the requirements in (23) with equalityfor any initial power vector p0. The convergencerate is n + 1 times slower compared to the al-gorithm (8) applied to a delayless power controlproblem.
PROOF. See (Gunnarsson, 2000; Gunnarssonand Gustafsson, 2001a).
The WCDMA algorithm described in Section 4.1with single-bit error representation never con-verges to a fixed point, as disclosed in Section 3.4.Instead, it converges to a region characterized bythe following theorem.
Theorem 7. If the power control problem is fea-sible, the algorithm without and with the Smithpredictor (subject to a round-trip delay of totallynRT = 1 + np + nm samples, nRT = 1, 2, . . .)converges to a region where the SIR error for everyconnection is bounded (in dB) by
Without Smith: |γti − γi(t)| ≤ 2nRTβ
With Smith: |γti − γi(t)| ≤ (nRT + 1)β
and β is the step size. The result also holds whensubject to auto-interference.
PROOF. The result without the Smith pre-dictor is provided in (Herdtner and Chong,2000), while the result with the Smith predic-tor is from (Gunnarsson, 2000; Gunnarsson andGustafsson, 2001a).
The global system can be seen as a diagonalsystem of local control loops, inter-connected bya static nonlinearity via the interference. By lin-earizing the interference around the equilibriumcorresponding to the power vector that satis-fies (24), this picture is simplified to a diago-nal system with an unknown structured uncer-tainty. Full details are provided in (Gunnarsson,2000), and the remainder of the section providesa sketched road-map to the result. The existenceof such an equilibrium point is a consequence ofthe feasibility of the power control problem.
Introduce the MIMO system G(q)4= Gll(q)E,
where the closed local loop system Gll(q) is givenby (10) and (15). With the equilibrium powerdeflection p(t), the linearized global system canbe written as
p(t) = G(q)FI(q)∆ccp(t)+G(q)F g(q)g(t), (25)
where ∆cc is the Jacobian of the interference withrespect to the power vector. Figure 6 illustratesthe corresponding block diagram of the globalsystem (with a linear measurement filter).
The infinity norm of the interference Jacobian isgiven by
Lemma 8. The following relation holds for thematrix ∞-norm of the cross coupling matrix ∆cc
‖∆cc‖∞ = max1≤i≤m
(1− vi
Iti
)= δcc < 1
PSfrag replacements
∑
p(t)
∆ccFI(q)
G(q)Fg(q)g(t)
Figure 6. Block diagram of the global system,when approximating the interference by thecorresponding linearization with respect tothe equilibrium deflection p(t).
The value of the norm δcc will be referred to asthe degree of cross-coupling.
The degree of cross-coupling δcc allows a naturalinterpretation. Essentially, it reflects the influenceof the interference to the global system stability.Two cases are easily distinguished
• δcc = 0. Corresponds to the case when theinterference at every receiver is only noise, i.e.the local loops are operating independently.In such a situation, the local loop analysis inSection 3.4 is sufficient.
• δcc = 1. Impossible in practice, since it corre-sponds to the case when the thermal noise iszero, or the interference at one receiver is in-finite. However, in highly loaded interferencelimited systems, δcc is close to 1.
The Small Gain Theorem is directly applicable tothe block diagram in Figure 6 to address stabilityof the linearized system
Theorem 9. Let Gll(q) be the stable closed-looptransfer function of the local loop. Then the globalsystem in Figure 6 is stable if and only if the powercontrol problem is feasible and any of the followingproperties is satisfied
We note that a highly loaded system is muchharder to control (put narrower restrictions onlocal loop performance) than a slightly loadedsystem. For example rather aggressive strategiessuch as minimum-variance are only applicable insystems with little load to avoid violating theglobal stability.
6. CONCLUSIONS AND FUTURE WORK
The area of power control in wireless networksis surveyed, and the power control algorithmsare put into a control theory context to relateto the control nomenclature. With this commonframework, it is natural to address critical prop-erties such as stability and convergence. Still,many problems remains to be solved, primarily fordownlink power control, where the objectives aremore policy oriented and dependent on the net-work operator philosophy. Natural trade-offs arebetween fairness and throughput. The ambitionwith the extensive citation list is to provide, yetsubjective, an overview of central proposals, andpointers to interesting open problems.
7. BIOGRAPHY
Fredrik Gunnarsson received his PhD degreefrom Linkopings universitet, Sweden 2000. Cur-rently, he work on methods for power controlfield trial evaluation and with higher level radioresource management for WCDMA at EricssonResearch. He also holds a research position atLinkopings universitet, and is responsible for thetelecom projects within the competence centerISIS led by Prof. Lennart Ljung.
Fredrik Gustafsson is professor of the chairin Communication Systems at Linkopings univer-sitet. His research interests include adaptive signalprocessing with telecom, avionic, and automotiveapplications.
REFERENCES
3GPP. Technical specification group radio access network.
Standard Document Series 3G TS 25, Release 1999,
2001.
M. Almgren, H. Andersson, and K. Wallstedt. Power
control in a cellular system. In Proc. IEEE VehicularTechnology Conference, Stockholm, Sweden, June 1994.
E. Anderlind. Resource Allocation in Multi-Service Wire-
less Access Networks. PhD thesis, Radio Comm. Sys-
tems Lab., Royal Inst. Technology, Stockholm, Sweden,October 1997.
M. Andersin, N.B. Mandayam, and D.Y. Yates. Subspace
based estimation of the signal to interference ratio forTDMA cellular systems. Wireless Networks, 4(3), 1998.
S. Ariyavisitakul. Signal and interference statistics of
a CDMA system with feedback power control - part
II. IEEE Transactions on Communications, 42(2/3/4),
1994.
K. Astrom and B. Wittenmark. Computer Controlled Sys-
tems – Theory and Design. Prentice-Hall, EnglewoodCliffs, NJ, USA, third edition, 1997.
K. Balachandran, S.R. Kadaba, and S. Nanda. Channel
quality estimation and rate adaption for cellular mobile
radio. IEEE Journal on Selected Areas in Communica-
tions, 17(7), 1999.
J. Blom, F. Gunnarsson, and F. Gustafsson. Constrained
power control subject to time delays. In Proc. Interna-tional Conference on Telecommunications, Chalkidiki,
Greece, June 1998.
J. Blom, F. Gunnarsson, and F. Gustafsson. Estimationin cellular radio systems. In Proc. IEEE International
Conference on Acoustics, Speech, and Signal Process-
ing., Phoenix, AZ, USA., March 1999.
S. Choel, T. Chulajata, H. M. Kwon, B.-J. Koh, and S.-
C. Hong. Linear prediction at base station for closed
loop power control. In Proc. IEEE Vehicular Technology
Conference, Houston, TX, USA, May 1999.
R. De Bernardi, D. Imbeni, L. Vignali, and M Karls-
son. Load control strategies for mixed services in
WCDMA. In Proc. IEEE Vehicular Technology Con-
ference, Tokyo, Japan, May 2000.
P. Dietrich, R.R. Rao, A. Chockalingam, and L. Milstein.
A log-linear closed loop power control model. In Proc.
S. V. Hanly. Congestion measures in DS-CDMA. IEEETransactions on Communications, 47(3), 1999.
J.D. Herdtner and E.K.P. Chong. Analysis of a class ofdistributed asynchronous power control algorithms for
cellular wireless systems. IEEE Journal on Selected
Areas in Communications, 18(3), Mar 2000.
H. Holma and A. Toskala, editors. WCDMA for UMTS.
Radio Access for Third Generation Mobile Communi-
cations. Wiley, New York, NY, USA, 2000.
S.-H. Hwang, S.-L. Kim, H.-S. Oh, C.-E. Kang, and J.-Y
Son. Soft handoff algorithm with variable thresholds
in CDMA cellular systems. IEE Electronics Letters, 33
(19), 1997.
H. Kawai, H. Suda, and F. Adachi. Outer-loop control
of target SIR for fast transmit power control in turbo-
coded W-CDMA mobile radio. IEE Electronics Letters,
35(9), 1999.
B. Kristiansson and B. Lennartsson. Optimal PID con-
trollers including roll off and Schmidt predictor struc-
ture. In Proc. IFAC World Congress, Beijing, P. R.
China, July 1999.
P. R. Kumar. New technological vistas for systems and
control: The example of wireless networks. IEEE Con-
trol Systems Magazine, 21(1), 2001.
W. Li, V.K. Dubey, and C.L. Law. A new generic
multistep power control algorithm for the LEO satellite
channel with high dynamics. To appear in IEEE
Communications Letters, 2001.Y. Lu and R.W. Brodersen. Integrating power control,
error correcting coding, and scheduling for a CDMAdownlink system. IEEE Journal on Selected Areas in
Communications, 17(5), May 1999.S. Niida, T. Suzuki, and Y. Takeuchi. Experimental
results of outer-loop transmission power control usingwideband-CDMA for IMT-2000. In Proc. IEEE Vehic-
ular Technology Conference, Tokyo, Japan, May 2000.
H. Olofsson, M. Almgren, C. Johansson, M. Hook, andF. Kronestedt. Improved interface between link leveland system level simulations applied to GSM. In Proc.
IEEE International Conference on Universal PersonalCommunications, San Diego, CA, USA, October 1997.
D. Ramakrishna, N.B. Mandayam, and R.D. Yates. Sub-space based estimation of the signal to interference ratiofor CDMA cellular systems. In Proc. IEEE Vehicular
Technology Conference, Phoenix, AZ, USA, May 1997.Z. Rosberg and J. Zander. Toward a framework for power
control in cellular systems. Wireless Networks, 4(3),
1998.
A. Salmasi and S. Gilhousen. On the system design aspectsof code division multiple access (CDMA) applied todigital cellular and personal communications networks.
In Proc. IEEE Vehicular Technology Conference, New
York, NY, USA, May 1991.A. Sampath, P.S. Kumar, and Holtzman J.M. On setting
reverse link target SIR in a CDMA system. In Proc.
IEEE Vehicular Technology Conference, Phoenix, AZ,
USA, May 1997.C.E. Shannon. The zero error capacity of a noisy channel.
IRE Transactions On Information Theory, 2, 1956.
M.L. Sim, E. Gunawan, C.B. Soh, and B.H. Soong. Char-
acteristics of closed loop power control algorithms for a
cellular DS/CDMA system. IEE Proceedings - Commu-
nications, 147(5), October 1998.B. Sklar. Rayleigh fading channels in mobile digital com-
