Delay-Stability of Power Control in Wireless Networks

Delay-Stability of Power Control in

Wireless Networks

BENJAMIN C. HEINRICH

Master’s Degree Project

Stockholm, Sweden May 2011

XR-EE-RT 2011:012

Abstract

In this thesis, we investigate the stability of uplink power control algorithms in wireless

networks. We derive an abstract block-diagram model of the power-control loop similar

to the model in [6]. The power control loop regulates the energy output of the mobile

devices based on measurements of the incoming signal strengths, background noise and

interference. The goal of the implemented algorithm is to maintain a certain Signal-to-

Interference Ratio (SIR) for all users.

Our analysis is done locally by linearizing the system around a steady state. There,

we can use a system-specific multivariate Nyquist criterion to analyze stability. In this

framework, we also find bounds on the rate of convergence as a performance measure.

A focus in this work lies on the influence of time delays and how one can compensate

for them. Consequently, we investigate Time-Delay Compensation (TDC, see [6]) and find

an extended version of it.

We also extend our model to incorporate binary control feedback to match the real-

world system. The emerging oscillatory behavior is then predicted and investigated by

multivariate describing-function methods.

The work is concluded by evaluating our findings with simulations using a Mat-

lab/Simulink model.

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ii

Acknowledgments

I guess this work involved the most persons a Master’s thesis has ever involved. Conse-

quently, there is a huge number of people I would like to thank. Furthermore, I want to

apologize to all the people I did forget in this section.

First of all, I want to thank my parents and grandparents for making my stay in Sweden

possible with their generous support - amongst others monetarily. Also, I would like to

thank ERASMUS for supporting me within their possibilities. For being very patient with

my various requests and supporting my non-standard exchange, I want to thank Simone

Schuler and Manja Schubert, the exchange coordinators of their respective institutes at the

University of Stuttgart and the KTH. The former was recently replaced by Georg Seyboth;

thanks for managing the remainder of the exchange bureaucracy. Mr. Seyboth has also

taken over the role of my exchange supervisor for Stuttgart since Marcus Reble, who was

originally supposed to do that, left Stuttgart during my thesis. This brings me to my

examiner at KTH, Stockholm: Professor Elling Jacobsen. A special thanks to you sir, for

spontaneously filling in for Professor Mikael Johansson, who originally agreed on being my

examiner, but was not reachable when I arrived in Stockholm. In addition to an examiner,

I had two supervisor. Now, why is that? Officially, I did my exchange with the department

of Electrical Engineering (EE), but my work as actually made and supervised at the math

department, Institute for Optimization and Systems Theory.

Hence, I would like to thank the two people who dedicated a lot of their time - especially

in the last months - to supervising me, discussing with me and correcting my many many

drafts. A huge shoutout for Professor Ulf Jonsson for sharing his mathematical insight

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with me. Thanks for all those valuable suggestions along the way. Last, but certainly not

least, I want to thank Anders Moller who single-handedly supervised the last 20% of the

work and 50% of the writing. Thank you for all those hours of discussion and being so

picky about every line I wrote and also doubting every line I calculated. This is meant

absolutely positively! You really contributed to quality of this thesis.

In memory of

Ulf Jonsson

Contents

1 Introduction 1

1.1 Wireless networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Attenuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Time Shift and Z Transform . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.3 List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.4 List of used Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Modeling 11

2.1 Controller Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.3 Auto-Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.4 Signal-to-Interference Ratio . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 One-Mobile Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Multiple-Mobile Case . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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vi Contents

2.2.3 Delay Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.4 Complete Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.5 Rearranging the Block Diagram . . . . . . . . . . . . . . . . . . . . 17

2.3 Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Transfer Function of our Block Diagram . . . . . . . . . . . . . . . 20

2.3.2 Block implementations . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Steady-State Analysis 25

3.1 Computing the Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Connection between the Linearized Interference and Feasibility Matrix . . 31

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Nyquist Analysis 33

4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1.1 One-Dimensional Nyquist Criterion . . . . . . . . . . . . . . . . . . 36

4.1.2 Multi-Dimensional Nyquist Criterion . . . . . . . . . . . . . . . . . 37

4.2 Analytical Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 General Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.2 Problem-Specific Usage . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Rate of Convergence 47

5.1 Scaling in the Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2 Scaling in the Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . 49

5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Contents vii

6 Time-Delay Compensation 53

6.1 Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.2 Block Diagram and Transfer Function . . . . . . . . . . . . . . . . . . . . . 54

6.3 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.4 Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.4.1 Traditional TDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.4.2 Extended TDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7 Describing Functions 67

7.1 Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.2 Problem Specific Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.3 Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

8 Simulations 79

8.1 Influence of the Gain Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.2 Influence of the Dynamics Matrix . . . . . . . . . . . . . . . . . . . . . . . 83

8.2.1 Changing the Integrator Gain . . . . . . . . . . . . . . . . . . . . . 84

8.2.2 Changing the Overall Delay . . . . . . . . . . . . . . . . . . . . . . 86

8.2.3 Adding a Proportional Part . . . . . . . . . . . . . . . . . . . . . . 88

8.2.4 Adding Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8.2.5 An Interesting Observation . . . . . . . . . . . . . . . . . . . . . . . 91

8.3 Influence of Time-Delay Compensation . . . . . . . . . . . . . . . . . . . . 92

8.3.1 Under- and Overcompensating the Delay . . . . . . . . . . . . . . . 92

8.3.2 Under- and Overestimating the Gain . . . . . . . . . . . . . . . . . 94

8.3.3 Rule of Thumb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8.4 Influence of Binary Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8.5 Influence of Varying the Gain Matrix . . . . . . . . . . . . . . . . . . . . . 99

viii Contents

8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

9 Conclusions 105

9.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

9.2 Possible Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Bibliography 110

List of Figures

1.1 Typical Depiction of a Cellular Wireless Network . . . . . . . . . . . . . . 3

1.2 Definition of the Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Toy Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Step-by-Step Derivation of our Block Diagram (pt.1) . . . . . . . . . . . . 16

2.3 Step-by-Step Derivation of our Block Diagram (pt.2) . . . . . . . . . . . . 18

2.4 Rearranged Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Simplified Rearranged Block Diagram . . . . . . . . . . . . . . . . . . . . . 20

3.1 Simplified Rearranged and Linearized Block Diagram . . . . . . . . . . . . 30

4.1 Standard Loop Layouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Nyquist Contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Stability Region of the Dynamics Matrix . . . . . . . . . . . . . . . . . . . 41

4.4 Stability Investigation of G1 under DPC Control . . . . . . . . . . . . . . . 42

4.5 Stability Investigation of G1 with System h1 . . . . . . . . . . . . . . . . . 43

4.6 Stability Investigation of the Infeasible Network G2 . . . . . . . . . . . . . 44

4.7 Time-Domain Plots of the Considered Examples . . . . . . . . . . . . . . . 44

5.1 Stability Region of the Scaled Dynamics Matrix hI for δ from 2–6 . . . . . 50

5.2 Rates of Convergence for Integral Control of G1 . . . . . . . . . . . . . . . 51

5.3 Time-Domain Plot of Figure 5.2c and 5.2d . . . . . . . . . . . . . . . . . . 52

6.1 Block Diagram with Additional Feedback Paths to Cancel Delays . . . . . 55

ix

x List of Figures

6.2 The Base Station’s TDC Controller . . . . . . . . . . . . . . . . . . . . . . 56

6.3 Reduced Block Diagram of the Time-Delay-Compensated System . . . . . 57

6.4 Stability Region of the BS-TDC Dynamics Matrix . . . . . . . . . . . . . . 59

6.5 Stability Region of the Under-/Overcompensated BS-TDC Dynamics Matrix 60

6.6 Stability Region of the under-/overcompensated extended TDC Dynamics

Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.1 Split System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.2 Block Diagram with Binary Control . . . . . . . . . . . . . . . . . . . . . . 70

7.3 Check of Low-Pass Behavior of g . . . . . . . . . . . . . . . . . . . . . . . 71

7.4 Motivation for Introducing ε . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.5 Test of Rule of Thumb 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.6 Time-Domain Plots of System with Binary Feedback . . . . . . . . . . . . 76

8.1 Feasible Reciprocal Eigenvalues of Γ†Fi . . . . . . . . . . . . . . . . . . . . 82

8.2 Investigation of Increasing Integrator Gain κI . . . . . . . . . . . . . . . . 84

8.3 Time-Domain Plot of Figure 8.2c . . . . . . . . . . . . . . . . . . . . . . . 85

8.4 Investigation of Increasing Overall Delay δ . . . . . . . . . . . . . . . . . . 86


8.6 Investigation of Additional Proportional Part κP . . . . . . . . . . . . . . . 88


8.8 Investigation of the Influence of Filtering . . . . . . . . . . . . . . . . . . . 90

8.9 Nyquist plot of Under and Overcompensation by TDC . . . . . . . . . . . 93

8.10 Time-Domain Plot of Figure 8.11f . . . . . . . . . . . . . . . . . . . . . . . 94

8.11 Nyquist plot of Under and Overestimation by TDC . . . . . . . . . . . . . 95

8.12 Additional TDC examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8.13 Information versus Binary Feedback . . . . . . . . . . . . . . . . . . . . . . 98

8.14 Investigating TDC for a Changing Gain Matrix (low delay) . . . . . . . . . 100

8.15 Investigating TDC for a Changing Gain Matrix (high delay) . . . . . . . . 101

8.16 Investigating Binary Feedback for a Changing Gain Matrix . . . . . . . . . 102

List of Tables

1.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Transfer functions of typical implementations of system components . . . . 22

8.1 Comparison between the Approximated and the Actual Feasibility Region . 81

8.2 Maximal κI for different Systems . . . . . . . . . . . . . . . . . . . . . . . 91

xi

xii List of Tables

Chapter 1

Introduction

This Diploma Thesis discusses the stability and performance of power control in wireless

networks. A special focus lies on the influence of delays, which virtually always occur in

these systems. The point of origin for this work was Fredrik Gunnarsson’s PhD thesis on

the power control in cellular radio systems [6]. Our results are presented in a manner such

that readers from different backgrounds should be able to follow.

The investigated system is introduced in Chapter 1. First, we give a rough overview of

key features of wireless communication networks and then clarify special terminology. The

last section comments on the notation we use throughout this thesis. Chapter 2 covers

the modeling of the introduced system. The main result of the second chapter will be the

system’s transfer function, which is derived via log-linear block diagrams. The following

chapters – Chapter 3 to 5 – treat the analysis of the linearized system. This includes

finding the linearization and investigating its stability as well as its rate of convergence,

both utilizing Nyquist’s criterion. In Chapter 6 we take a look into the so-called Time-Delay

Compensation (TDC) to assess its applicability. We compare two different compensators

and discuss their respective advantages and disadvantages. In the 7th chapter, we will

then expand our model to have binary feedback, which is indeed the case in the real-world

system. Harmonic-balance techniques will be used to predict the amplitude and period

length of the emerging steady-state oscillations. Chapter 8 is intended to give insight into

the system’s dynamics as well as to back up our earlier investigations, both by simulation.

1

2 1 Introduction

We wrap up our results in the final 9th chapter. There, we also discuss the contributions

and possible extensions of this work.

1.1 Wireless networks

In the information age, there are two things which are most important: being informed

and being so whenever, wherever. The only means to achieve this goal to date are wireless

communication networks. The term ‘wireless communication networks’ encompasses a

variety of networks, ranging from short distance Bluetooth interconnections to long distance

WiMax networks, and from two participants to hundreds.

This work will focus on the well-established Third Generation (3G) Network. Although

the fourth generation is already on its way, the 3G system will still play a huge role in

this decade. The reason for this is that its infrastructure is prevalent and thus means

of enhancing the performance of the existing hardware are in demand. Note that other

wireless networks are also covered to a certain extent due to the similarities between all

types of wireless communication.

In the upcoming sections, we will elaborate on the considered framework and the ter-

minology used in this thesis in order to circumvent ambiguities. At the end of this chapter,

a legend of our notation will be given.

1.2 Framework

In the framework of this thesis, wireless communication connects base stations (BS) and

mobile stations (MS). The latter are in most cases mobile phones. One BS provides for a

number of associated MS:s in its so-called cell, see Section 1.2.1. In most wireless networks,

communication is bilateral, i.e. both the BS and the MS:s are transmitters as well as

receivers. The signal’s passage between transmitter and receiver – sometimes including

their respective antennas – is the so-called channel, which is treated in the homonymous

1.2 Framework 3

Section 1.2.2. The channel, however, is prone to diverse disturbances. A short introduction

on our disturbance modeling is given in Section 1.2.3.

1.2.1 Cell

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

Figure 1.1: Typical Depiction of a Cellular Wireless Network

Hexaeder are traditionally used to depict network cells although they can be found in every size and shape.Each cell consists of one base station (BS), depicted as e, and a number of mobiles (MS), depicted asdots. Note, though, that especially when near a border of a cell, a MS might receive signals from multipleBS.

Due to their limited range and capacities, BS:s have only a limited effective area of

service. This area is called a cell. Cells vary strongly in size and form, dependent on the

expected density of users and geographic conditions. They are, nevertheless, often depicted

as homogeneous hexaeders (see Figure 1.1).

A mobile is assigned to a cell, and consequently to a base station. Ideally, a MS

interacts only with its associated BS. In reality, however, there is a number of phenomena

one has to consider. On the one hand, there will be interference among the users sharing

the cell’s frequency band; and not only amongst those, but also cross-cell interference when

frequency reuse has to be considered. On the other hand, MS:s are by their very nature

mobile, i.e. they will most likely not stay in one cell but traverse from one to another.

When in between two or more cells, multiple connections will occur. In order to provide

continuous connectivity in such cases, means of handing over an MS from one BS to another

4 1 Introduction

have to be considered. Note, though, that hand overs are not considered in this thesis.

Furthermore, we will soon go from a cell point of view to a channel point of view. The

channel is the topic of the next section.

1.2.2 Channel

signal Encoding Modulation Physical Propagation Demodulation Decodingperceivedsignal

Attenuation and Interference

Transmitter Receiver

Channel

Figure 1.2: Definition of the Channel

In this thesis, the Channel comprises everything that happens between sending the signal and perceivingit. This includes the dynamics of the en-/decoding and modulation. In particular, it includes the radiowave phenomena such as attenuation and interference.

We call the complete signal path from the transmitter to the receiver the channel.

The communication medium is air, or, to be more precise, an electro-magnetic carrier

wave transmitted through it. The carrier wave is superposed by the digitally-encoded ac-

tual signal. Typically, the carrier wave is within the Ultra High Frequency (UHF) Band,

i.e. between 300 and 3000 MHz. This bandwidth allows for high data-transmission capac-

ities while weather effects (moisture, rain) are only slight. Unfortunately, mountains and

buildings shield and reflect those radio waves. [6]

We can differentiate between the downlink channel – from BS to MS – and the uplink

channel – vice versa. It is much easier to decouple the downlink signals, i.e. make them

orthogonal in the signal-space, because they originate from a single source. In contrast,

the uplink signals are almost impossible to decouple, especially when considering mobiles

traversing between cells and being connected to two or more BS at the same time. For this

reason, the uplink channel is of special interest for this work. Next, we discuss the channel

attenuation in some detail.

1.3 Notation 5

1.2.3 Attenuations

The channel is cumbered by various wave propagation phenomena. We use the term

attenuation for all channel distortions which do not originate from other signals. This

section follows along the lines of Section 3.3 of [6].

One of the main phenomena of wave propagation is the signal loss due to the distance

between the transmitter and the receiver. It is often modeled as a factor inverse propor-

tional to the distance r to the power of some constant a. The factor a was in early studies

said to be four, while later investigations assume it to be somewhere between two and five,

dependent on topography. Naturally, open spaces correspond to less attenuation, and thus

smaller a, while urban regions hold greater a.

Another important attenuation happens due to shielding by the environment. In con-

trast to the path loss, this effect can lead to fast drops in the channel quality even though

the MS moves slowly. Analogous to the phenomenon when stepping from the light into a

shadow and back, this is called shadowing.

The last phenomenon we want to cover in this section is the so-called multi-path fading.

It occurs when the signal from transmitter to receiver traverses different paths due to

reflections from the environment. As a result of the different path lengths positive or

negative interference can occur at the receiver.

Both shadowing and fading are correlated to the movement of the MS as well as to the

movement of the environment. In this work, we will model all mentioned effects in one

single attenuation factor, see Chapter 2.

1.3 Notation

In this section, we do not only list the used symbols and conventions, but especially want

to point out the subtle difference between describing delays with the z transform and

the time-shift operator q. Although, from a practical point of view, they might be used

interchangeably, one has to stress the difference in order to be rigorous.

6 1 Introduction

1.3.1 Time Shift and Z Transform

We will frequently use two different notations for time delays. The first notation z−n, where

n is the number of samples delay, directly originates from the well-established z trans-

form [1]. Consequently, this is a time-discrete frequency-domain notation and works well

in the framework of time-discrete transfer functions.

The second notation involves the so-called time-shift operator q. This notation is fairly

common in the communication literature [6, 7, 13]. The time-shift operator is defined in

the time domain. We will illustrate its application with the help of a simple example.

Consider the following time-dependent function

f(t) = a(t) + b(t− 1)− c(t− 2),

where a, b and c are arbitrary chosen time dependent functions. Utilizing q yields

f(t) = a(t) + q−1b(t)− q−2c(t).

Accordingly, one can shift the time n steps by employing qn, where a negative n corresponds

to back-shifting while a positive n, naturally, corresponds to a forward shift. With the help

of q we can thus depict time-discrete algorithms like transfer functions. Consider e.g. a

time-discrete Euler-backward integrator

y(t) = y(t− 1) + u(t).

Rewritten with the time-shift operator q it reads:

y = q−1y + u

=q

q − 1u,

which maps directly to the z-transform of an integrator Z(y) = zz−1

Z(u).[1]

1.3 Notation 7

1.3.2 Conventions

The notation in the mathematical formulas of this work employs the conventions from

Table 1.1. Note especially the use of overlined symbols for variables measured in linear

scale in contrast to normal symbols, which are measured in dB.

Table 1.1: Conventions

Font Description

normal symbol measured in logarithmic scale

overlined symbol measured in linear scalebold letters vector (n× 1)

Capital Letters matrix (n×m)vector[i], Matrix[i,j] entry at ith row, jth columnCAPITAL old letters transfer functions/operator (MIMO)

old letters transfer functions/operator (SISO)

1.3.3 List of Abbreviations

The following more or less common abbreviations are used throughout this thesis.

Table 1.2: Abbreviations

Abbreviation Description

3G Third GenerationBS Base StationDPC Distributed Power ControlLTI Linear Time-Invariant

MIMO Multiple-Input Multiple-OutputMS Mobile StationSIR Signal-to-Interference RatioSISO Single-Input Single-OutputUHF Ultra High Frequency

8 1 Introduction

1.3.4 List of used Symbols

Here, we list all used symbols. Note that linear and logarithmic versions are not shown

explicitly.

Table 1.3: Notation

Symbol Description

α, α auto-interference

γ, γ Signal-to-Interference Ratio

γ†, γ† target Signal-to-Interference Ratio

γ††, γ†† new target Signal-to-Interference Ratio (γ† + F−1ι ζ)

δ number of samples delay (often: total,i.e. δb + δm + 2δp)

δb number of samples delay in the base station

δm number of samples delay in the mobile station

δp number of samples propagational delay

δTDC number of samples delay compensated for with TDC (δm + 2δp)

δ∗TDC number of samples delay compensated for with extended TDC (δb+δm+

2δp)

δε number of samples error in the TDC

δ∗ε number of samples error in the extended TDC

ζ, ζ channel attenuation

ι, ι interference signal

. . . continued on next page

1.3 Notation 9

Table 1.3 – continued from previous page

Symbol Description

κ controller parameter

κI controller parameter, integrator gain

κP controller parameter, proportional part

λ an eigenvalue

λ(·) the eigenvalues of ‘·’

σ, σ noise term

ς, ς usable signal

φ filter parameter

Φ(·(z)) counter-clockwise phase rotation of ‘·’ about the origin when z counter-

clockwise traverses the unit circle

Γ attenuated Signal-to-Interference matrix

B, b base-station-controller transfer function

D, d delay transfer function

F, f filter transfer function

G, g generic transfer function

G0, g0 open-loop transfer function

H, h pooled lower-loop transfer function, also: dynamics function

K, k controller transfer function

M,m mobile-station-controller transfer function

F, f feasibility gain matrix

. . . continued on next page

10 1 Introduction

Table 1.3 – continued from previous page

Symbol Description

G, g gain matrix

I identity matrix

p, p output power of the mobile stations

pss, pss output power of the mobile stations in steady state

Γ†F feasibility matrix

I,∇I interference function, linearized interference function

Chapter 2

Modeling

We will begin the modeling of the uplink power control of the 3G network by defining a

quality measure for radio-based communication networks and thus our controller reference.

Taking this as our point of departure, we will derive a block diagram of our system, step-

by-step, going from a simplistic one-mobile case to a full block diagram with various delays.

Finally, this block diagram will be used to deduce the transfer function of the considered

wireless network. The block diagrams and transfer functions will aid in the investigation

of the performance of different networks and controller designs throughout this work.

2.1 Controller Objective

In order to model our system, we, first of all, define a controller objective. In the case

of wireless networks the main objective is to provide a certain quality of service. Here,

it is most convenient to measure this reference in terms of the well-established Signal-to-

Interference Ratio, short SIR. It is frequently defined in the literature, see e.g. [14]; we

begin our definition as follows:

11

12 2 Modeling

Definition 1 (Signal-to-Interference Ratio). The SIR for mobile i is defined as

γi∆=

signaliinterferencei

∆=ςiιi

or γi∆= signali − interferencei

∆= ςi − ιi,

dependent on whether we measure in [mW ] (linear) or [dB] (logarithmic).

Moving on, we define the signal ς and the interference ι in more detail.

2.1.1 Signal

First, let us have a look at the signal ςr picked up at receiver r. It is generated by a mobile i

with power output pi. We assume the signal strength to be directly proportional to this

power output. The sent signal, however, will never entirely reach the target receiver r.

Consequently, we multiply it by a factor gri ∈ [0, 1], the transmission gain. This gain

– or attenuation, as it is always less than one – includes effects such as shadowing and

multi-path fading amongst others, which were discussed in Section 1.2.3. The model for

the signal follows as

ςr∆= gripi or ςr

∆= gri + pi,

again, dependent on whether we measure in linear or in logarithmic scale.

2.1.2 Interference

Our interference model consists of three parts. First, we have the interference from other

users; it occurs due to unorthogonal transmission from other mobiles j affecting the re-

ceiver r and is modeled as a factor grj ∈ [0, 1] times the interfering transmission power pj .

Second, we have background noise which is modeled by the additive noise term σr.

For notational convenience, we introduce the following convention: The target receiver r

of mobile i is also called receiver i. Thus, we go from a transmitter-receiver point of view

to a channel point of view. Note that in this framework we cannot explicitly see which

2.1 Controller Objective 13

physical BS a MS is connected to; what we very well can differentiate are the different

channels at one base station. We can thus tackle in- as well as inter-cell interference very

conveniently.

Using the above convention, we introduce the so-called gain matrix G[ij] = gij , where the

diagonal elements are the channel-gains while the off-diagonal elements are cross-channel

gains. In an ideal case, where there is no interference and no channel attenuation, the gain

matrix would thus be the identity matrix. This matrix is naturally time-dependent since

the connectivity of the mobiles changes with time. However, we will in most parts of this

work consider the gain matrix fixed in the time scale of the analysis. Figure 2.1 represents

a small toy network with two mobiles and two BS. We want to stress that those ‘Base

Stations’ may in fact be different channels of the same physical BS.

The interfering signal ιi at BS i is given by the non-linear interference function Ii,

which is, making use of the above, defined as

Ii∆= 10 log10(Ii), where Ii

∆= σi +

∑

j 6=i

gij pj .

Here, i, j ∈ {1, . . . , n} and n is the number of mobiles.

e e

H H

g11

g21 g12

g22

Figure 2.1: Toy Network

Here, two base stations (e) and two mobile phones (H) are depicted. The channel gains are given by g11and g22. The cross-channel gains, which depict the strength of the interference, are given by g12 and g21.

14 2 Modeling

2.1.3 Auto-Interference

A further source of interference can actually be the sent signal itself. This so-called auto-

interference is the third part of our interference modeling and happens when the receiver

cannot pick up the entire signal. The unused part of the signal then effectively acts as

additional noise. In order to model auto-interference, we hence add another term αigiipi to

the interference part. Here, 0 ≤ αi < 1 is the positive auto-interference factor. Moreover,

the interfering part of the signal has to be subtracted from ςi. Considering both terms, we

get the overall signal and interference as

ςi∆= (1− αi)giipi and (2.1)

Ii∆= σi + αigiipi +

∑

j 6=i

gij pj (2.2)

2.1.4 Signal-to-Interference Ratio

Plugging the above discussed model for the signal (2.1) and the interference (2.2) into

Definition 1 yields

γi =(1− αi)giipi

σi + αigiipi +∑

j 6=i gij pjor (2.3)

γi = gii + pi + 10 log10(1− αi)− 10 log10(σi + αigiipi +∑

j 6=i

gij pj), (2.4)

where, again, i, j ∈ {1, . . . , n} and n is the number of mobiles. Keep in mind that overlined

symbols are linearly measured values while non-overlined are values measured in logarith-

mic scale. Having defined the objective for our system, we move on to the derivation of its

block diagram.

2.2 Block Diagram 15

2.2 Block Diagram

Block diagrams can give valuable insight into systems dynamics. One of their many ad-

vantages is that one does not have to model each part of the system in every detail, but

instead uses abstract blocks, which can be defined in more detail later. Consequently, block

diagrams provide a convenient framework to derive general transfer functions for a whole

class of systems.

In our class of systems, radio-based networks, measurements are predominantly done

in dB; it is therefore natural to build up a block-diagram model of the system using these

logarithmically-scaled signals. The main advantage of doing so is that we get a linear

relation for the SIR which allow for easier analysis.

Note that block diagrams are most commonly used in the frequency domain, where

every block depicts a transfer function. They, however, do also work as a description

of a control algorithm in the time domain when utilizing the time-shift operator q, see

Section 1.3.1. In this section, we will solely use the time-domain interpretation, as it is

predominantly used in communications literature.

2.2.1 One-Mobile Case

We begin the derivation of the block diagram by looking at the simple one-mobile case,

which is illustrated in Figure 2.2a. It occurs when there is only one mobile in the considered

network or if we model the disturbances from other mobiles solely as noise. The latter might

be appropriate for a very large number of mobiles interfering with each other.

The controller within the base station b compares a target SIR γ† to the actual SIR γ.

The Signal-to-Interference Ratio in logarithmic scale is just the difference between the

signal ς and the interference ι. The difference between γ† and γ is then used as the

logarithmic control error e, which is used to compute the output u of the BS’ controller b;

the output is subsequently transmitted to the mobile m, which adjusts its power output p

accordingly. The power output is then fed back, impaired by the channel gain g, and

together with ι determines the next γ and thus the next e.

16 2 Modeling

Note that we split the SIR into two parts. The lower feedback path contains the damped

signal (ς = p+ g) while the upper path involves the interference ι. In this special case, the

latter is pure noise σ.

b mγ†

g

σ

e u p

ς

−

ι

(a) One-mobile case

I

B Mγ†

ζ

σ

e u p

ι

ς

−

(b) Multiple-mobiles case

Figure 2.2: Step-by-Step Derivation of our Block Diagram (pt.1)

(2.2a) The controller in the BS b compares the target SIR γ† to the actual SIR (γ = ς − ι), which is splitinto the an upper noise path (ι = σ) and a lower signal path (ς = p+ g), resulting in the control error e.The computed control signal u is sent to the mobile m, which has the power output p.(2.2b) Other mobiles and the consequently emerging interference I (see (2.2)) are considered. From nowon all signals are ∈ Rn, where n is the number of mobiles, and the channel attenuation g becomes ζ[i] = gii,where i ∈ {1, . . . , n}.

2.2.2 Multiple-Mobile Case

The second considered case, depicted in Figure 2.2b, is the multiple-mobile case. Here,

the influence of other mobiles is explicitly modeled by the interference function I, which

was defined in (2.2). Now, both paths accommodate feedback; the upper interference

path, however, has a nonlinearity in it. This nonlinearity will be treated in more detail

in Chapter 3. The attenuation in the lower signal path is henceforth denoted by ζ ∈ Rn,

where ζ[i] = gii.

Note that all signals from now on are vectors in Rn, where n is the number of considered

mobiles. Accordingly, all blocks represent multivariate operators/transfer functions. Due

to the distributed nature of our system, only local informations are available. Thus, the

controller blocks are assumed to be diagonal, i.e. B = diag{bi} and M = diag{mi}.


2.2.3 Delay Case

The next addition to our block diagram are time delays, which are an issue in the real-

world system. They occur mainly because of computation time in the signal processing/the

controller of the base station and/or of the mobile and may also occur due to the fact that

the propagation velocity of radio waves is limited. The latter, however, is commonly not an

issue in mobile networks since the cells are sufficiently small in size and thus propagational

delays can be neglected in most cases.

In Figure 2.3a, time delays are depicted by Dj . In particular, DB, DM and Dp are

the delays that occur in the base station, the mobile and due to propagation, respectively.

Generally, Dj is defined as diag(

q−δj,ii

)

, where the δj,i are the number of samples delay

of block j affecting channel i and q is time-shift operator. Keep in mind that the delay

might vary for each channel. Note that Dj corresponds to diag(

z−δj,ii

)

when considering

the frequency domain.

2.2.4 Complete Block Diagram

In order to complete our block diagram, we add an interference- and a signal-filter block,

Fι and Fς , which leads to Figure 2.3b. These filter blocks are composed of possibly differing

channel filters fi and thus defined as F = diag{fi}. Filters are implemented in real-world

systems in order to reduce measurement noise. The reason for explicitly modeling them

here is that they might generate additional dynamics and possibly delays which may not

be neglectable.

2.2.5 Rearranging the Block Diagram

There is only one block in Figure 2.3 which is assumed to be non-linear and widely unknown.

This block is the interference I. To isolate the interference, we pool all of the systems

dynamics into the dynamics block H. We do so by rearranging Figure 2.3b to Figure 2.4,

18 2 Modeling

I Dp

DBB Dp DMM

Dp

γ†

ζ

σ

e u p

ι

ς

−

(a) Delay case

Fι I Dp

DBB Dp DMM

Fς Dp

γ†

ζ

σ

e u p

ι

ς

−

Base Station ChannelMobile Station

(b) Complete diagram

Figure 2.3: Step-by-Step Derivation of our Block Diagram (pt.2)

(2.3a) Time delay blocks for base station DB, the mobile DM and the propagation Dp are added toFigure 2.2b; computational delays are combined with their respective controller transfer function to stresstheir inseparability.(2.3b) Additionally, filters F for the signal ς and the interference ι are introduced; The dashed lines indicatethe base station, the channel and the mobile station.


I

Fι DBB Dp DMM Dp

Fς

F−1ι Fς Dp

γ†

ζ

σ

e u

pι

−

γ††

H

Figure 2.4: Rearranged Block Diagram

The attenuation ζ is pulled out and a new input γ†† is defined. Note that here, for notational convenience,p are the delayed output powers p[t− δp]. The dashed line indicates the blocks which are pooled into theso-called dynamics block H.

which, introducing the new input

γ††∆= F−1

ι γ† + FςDpζ,

ultimately leads to Figure 2.5. Note that the definition of the new input γ†† obviously

requires the interference filter Fι to be invertible and its inverse to be stable. One can,

however, argue that this restriction is simply a modeling issue and thus does not affect the

real system. Another point of view is that unstable F−1ι would immediately be stabilized

by its own inverse, see Figure 2.4. Having completed the derivation of the block diagram

for the considered system, we move on to the derivation of its transfer function.

20 2 Modeling

I

Hγ††

pι

Figure 2.5: Simplified Rearranged Block Diagram

Wireless network in a standard layout from the new γ†† to p. This structure facilitates the investigationof the influence of the interference.

2.3 Transfer Function

Transfer functions are the very basis for investigating linear time-invariant (LTI) systems.

They give a description of the system’s input-output behavior in the frequency domain.

The transformation from the time domain into the frequency domain is commonly done

by Fourier transformation for continuous-time signals and by the z transformation for

discrete-time systems. More information on this topic can be found in every good control

theory book.

In this short section, we will derive the transfer function of the final block diagram,

Figure 2.5. Additionally, we will state some transfer functions for the so-far abstract

controller- and filter-blocks at the end of this section. The introduced controller- and

filter-implementations will be used in the subsequent analysis.

2.3.1 Transfer Function of our Block Diagram

The simplified block diagram, depicted in Figure 2.5, yields the simple transfer function

p = H(γ†† + ι).

2.3 Transfer Function 21

The transfer function of the dynamics block H can be derived by considering Figure 2.4 as

follows:

p = DpDMMDpDBB(Fι(γ†† + ι)− Fςp)

⇔ p = (I +DMBFς)−1DMBFι(γ

†† + ι)

where D = DBDMD2p contains the overall delay and I is the n-dimensional identity matrix.

Consequently,

H = (I +DMBFς)−1DMBFι. (2.5)

This transfer function will find great use in the Nyquist analysis in Chapter 4. All blocks –

except, of course, the interference – are assumed to be diagonal. This depicts that only local

information is available. Assuming, furthermore, that our system is homogeneous, i.e. all

channels share the same controller, filter and delay we get hi = h for all i. Hence, H = hI,

with

h =bmfι

zδ + bmfς, (2.6)

where b = B[i,i], m = M[i,i], f = F[i,i] and δ = δb + δm + 2δp is the number of samples

delay in total. We end this chapter by introducing some typical implementations of the

still-abstract sub-blocks.

2.3.2 Block implementations

A strength of the block diagram approach is that we did not need to specify the single

blocks in more detail. If we, however, want to analyze the system dynamics, they have to

be defined at some point. Table 2.1 depicts some possible realizations for the controllers in

the base station and the mobile station as well as two possible filters. Note that the local

average filter fLA has no stable inverse (see Section 2.2.5).

22 2 Modeling

Table 2.1: Transfer functions of typical implementations of system components

Transfer Function Description

dj = 1

zδj

Delay of one, often used as additional factor

kP = κP P-controller with gain κPkI = κI

zz−1

I-controller with integrator gain κIkPI = κP + κI

zz−1

PI-controller with gain κP and integrator gain κI

fLA =∑ν−1

i=0 z−i

νlocal average filter, which takes ν ∈ N steps into account

fEF = z(1−φ)z−φ

, φ ∈ (0, 1) exponential forgetting filter; ≈ 21−φ

contributing measurements [12]

We will frequently consider the following standard case

bi = dm,ikP,i =κP,izδm,i

, mi = db,ikI,i =zκI,i

zδb,i(z − 1), fι,i = fς,i = 1,

that means we have an integrator in the mobile which is driven by the proportional con-

troller in the base station. Note, however, that in the current diagonal LTI layout it

actually does not matter where which controller is situated1. Thus, it is sufficient to model

the homogeneous unfiltered system as

h(z) =k(z)

zδ + k(z), (2.7)

where k is the joined controller of base and mobile station and δ is the overall delay δb +

δm + 2δp.

For the given standard case we can without loss of generality assume the proportional

base-station gain to be one. This leads to the following transfer function for our lower-loop

dynamics:

hI =κI

zδ−1(z − 1) + κI. (2.8)

1The whereabouts of the specific controllers will play a role later on, when we model the binary feedbackbetween base station and mobile in Chapter 7.

2.4 Conclusion 23

2.4 Conclusion

In order to model our system, we first defined our controller objective, the so-called SIR γ.

This required us to have a look at different wave-propagation phenomena which disturb

wireless communication. During the course of this discussion, we defined the so-called gain

matrix G which will be used from now on to depict the networks connectivity. Connec-

tivity in this context does not only stand for ‘which mobile interacts with which other

mobile’; since we assume every mobile to have an influence on all other mobiles in the

same network/model, connectivity here means ‘how strong are those interactions’.

The outcome of this chapter is not only that we have a transfer function for our system,

in the course of its derivation we also built the system’s block diagram. This block diagram

can aid greatly in understanding system dynamics and especially helps to explicitly see

what happens where. A drawback of the system as-is is that we still have a nonlinearity in

it: the interference function I. We will tackle this problem with a linearization approach

in the next chapter.

24 2 Modeling

Chapter 3

Steady-State Analysis

In this chapter, we will first give an analytical expression for the steady-state vector of

power outputs – henceforth only referred to as the steady state. Then, we will study its

feasibility, i.e. the existence of a feasible steady state, which is, of course, indispensable if

one wants to stabilize the system. Finally, we will linearize our system around this steady

state in order to gain further insight by using tools from linear systems analysis such as

the Nyquist criterion.

3.1 Computing the Steady State

A steady state of our system is a point in state space where the power-output level of every

mobile remains constant for all times if it is unperturbed. This state is defined as follows:

Definition 2 (Steady State). The system is said to be in a steady state if

p[t+ 1]− p[t] = 0, ∀t

where pi[t] is the power output level of mobile i at sample t, and i ranges from 1 to n, the

total number of considered mobiles. The steady-state power-output level is denoted pss.

We search for a steady state where the network provides every user with a satisfying quality

of service, i.e. γi ≥ γ†i , where the latter is the minimal SIR to provide sufficient service,

25

26 3 Steady-State Analysis

the target SIR. It has been shown that the steady state where every mobile i has a SIR of

exactly γ†i is optimal in the sense that the vector of power outputs p is indeed minimal [10].

A minimal power output is desirable because it leads to maximal battery life. Hence, using

(2.3), we search for the output powers where

γ†i!= γi =

(1− αi)giipi,ssσi + αigiipi,ss +

∑

j 6=i gij pj,ss. (3.1)

In the following derivation we drop the additional index ‘ss’ for improved readability.

Rearranging (3.1) to

pi =γ†i

(1− αi)gii

(

σi + αigiipi +∑

j 6=i

gij pj

)

,

and bringing it in matrix-vector form,

p1...

pn

= diag

(

γ†i(1− αi)gii

)

σ1...

σn

+

α1g11 g1n. . .

gn1 αngnn

p1...

pn

,

yields

p = Γ†(σ + F p)

⇔ p = (I − Γ†F )−1Γ†σ, (3.2)

where

Γ† ∆= diag

(

γ†i(1− αi)gii

)

, σ[i]∆= σi and F[i,j]

∆=

αigii for i = j

gij for i 6= j.

Note, though, that various definitions for what we call F , Γ† and σ can be found in

the literature [6, 10]. Most interestingly, the spectrum of Γ†F can differ dependent on

3.2 Feasibility 27

the notation1. The spectrum plays a role in the investigation of the system’s feasibility,

stability and rate of convergence.

Furthermore, note that the derivation of the steady state was independent of b and m.

As long as we assume our system to have integral behavior, i.e. a pole at 1, this steady

state will be preserved. Note that this is the same steady state as for the famous DPC

algorithm [3], where

pi[t+ 1] =γ†iγipi[t].

It translates to a simple integrator in our framework. A proof can be found in [9].

We will now focus on the feasibility of the steady state pss. The investigation of its

stability can be found in Chapter 4, an estimation of the rate of convergence in Chapter 5.

3.2 Feasibility

The power output vector in the steady state pss must be non-negative in order to be feasible.

This is simply due to the fact that one cannot send out negative energy. Translated into

an equation this condition gives, with (3.2),

(I − Γ†F )−1Γ†σ!≥ 0. (3.3)

The latter part, Γ†σ, is always non-negative as all γ†i , gii as well as σi are non-negative and

consequently Γ† as well as σ is non-negative. Hence, the feasibility depends solely on the

first part of (3.3). In order to prove its non-negativity we utilize the following theorem:

Theorem 1 (from [15], abbreviated). The following statements are equivalent

(I − A)−1 =

∞∑

i=0

Ai and ρ(A) < 1,

where A is a n× n-matrix and ρ(A) is its spectral radius.

1Not further investigated in this thesis.


In our case that means if and only if ρ(Γ†F ) < 1 we can rewrite

(I − Γ†F )−1 as

∞∑

i=0

(Γ†F )i

and thus show the non-negativity and consequently feasibility of pss by proving Γ†F to be

non-negative. We show this by simply writing out

Γ†F =

α1

(1−α1)γ†1

g1n(1−α1)g11

γ†1. . .

gn1

(1−αn)gnnγ†n

αn

(1−αn)γ†n

.

Keep in mind that αi < 1, see Section 2.1.3. Thus, Γ†F is always non-negative since all

of its components are non-negative. Therefore, the feasibility condition condenses to the

spectral radius condition

ρ(Γ†F ) < 1. (3.4)

Accordingly, the matrix product Γ†F will from now on be called feasibility matrix.

For positive matrices the Perron-Frobenius theorem states that the biggest eigenvalue,

i.e. the spectral radius, is always simple and real. This holds also for non-negative matrices,

but only if they are irreducible [4]. Irreducibility for matrices is closely related to graph

theory and it basically means that every node has to be reachable from every other node,

no matter how many steps it takes. In our framework, a reducible matrix would mean that

there exists a mobile or a group of mobiles which does/do not influence the other mobiles

and hence it would make only limited sense to model them in the same network.

We have just shown that the system is always feasible when ρ(Γ†F ) < 1. A first

3.2 Feasibility 29

approximation of the feasibility directly from the gain matrix G can be derived as follows:

ρ(Γ†F ) < 1

⇔ maxk

|λk(Γ†F )| < 1

and thus

∣∣∣∣∣∣∣∣∣

λk

α1

(1−α1)γ†1

g1n(1−α1)g11

γ†1. . .

gn1

(1−αn)gnnγ†n

αn

(1−αn)γ†n

∣∣∣∣∣∣∣∣∣

< 1 ∀k, (3.5)

where k ∈ {1, · · · , n} is the arbitrary chosen numbering of the eigenvalues. Using Gersch-

gorin Disks [5] we know that the eigenvalues of a matrix A lie within the union of circles

around its diagonal elements aii. The circles’ radii are their respective off-diagonal row-

or column-sum, where, of course, the smaller radius gives the better bound. Hence, the

following condition is sufficient for feasibility:

αi(1− αi)

γ†i +

n∑

j = 1

j 6= i

1

1− αj

gjigjjγ†j < 1, ∀j

or

αi(1− αi)

γ†i +

n∑

j = 1

j 6= i

1

1− αi

gijgiiγ†i < 1, ∀i

.

We conclude the subsequent rule of thumb:

Rule of Thumb 1. Assuming the channel gains gii to be one, no auto-interference and

all target SIR γ†i to be equal, the system’s feasibility can be guaranteed if all row- or all

column-sums of the cross-channel gains are smaller than the reciprocal SIR.


3.3 Linearization

Now that we have found the system’s feasible steady state, we can linearize the interfer-

ence I around it and by doing so linearize the whole system. The linearized interference

will henceforth be denoted ∇I. Keep in mind that I = 10 log10(I), see Section 2.1.2. Due

to this nonlinearity we must apply the chain rule properly, which yields

∇I∆=∂I

∂p

∣∣∣∣p=pss

= diag(Ii)−1 ∂I

∂p

∣∣∣∣p=pss

and thus, using (2.2),

∇I[i,j] =

αigiipi,ss/Ii for i = j

gij pj,ss/Ii for i 6= j, (3.6)

where Ii is defined by (2.2) and evaluated at p = pss.

Employing the linearized interference gives Figure 3.1, which, using (2.5), yields

∆p = H(γ†† +∇I∆p)

⇔ ∆p = (I − H∇I)−1Hγ††

= (I +DMB(Fς − Fι∇I))−1DMBFιγ††,

where ∆p is the vector of output powers, linearized around the steady state pss.

∇I

Hγ††

∆pι

Figure 3.1: Simplified Rearranged and Linearized Block Diagram

The linearized interference function ∇I allows for an expression of ι as a function of ∆p and thus for linearanalysis of the now completely linearized system.

3.4 Connection between the Linearized Interference and Feasibility Matrix 31

3.4 Connection between the Linearized Interference

and Feasibility Matrix

Most interestingly, there exists a close connection between the linearized interference ∇I

and the feasibility matrix Γ†F .

Theorem 2. The linearized interference ∇I and the feasibility matrix Γ†F , both evaluated

at p = pss, have the same eigenvalues.

Proof. Using (2.2) and (2.3), we can write

Ii =(1− αi)giipi,ss

γ†i,

which, plugged into (3.6), yields

∇I[i,j] =

αi

(1−αi)γ†i for i = j

gij(1−αi)gii

pj,sspi,ss

γ†i for i 6= j. (3.7)

Keeping in mind the structure of Γ†F

Γ†F[i,j] =

αi

(1−αi)γ†i for i = j

gij(1−αi)gii

γ†i for i 6= j

we rewrite (3.7) into

∇I = diag(pss)−1Γ†Fdiag(pss).

Using the fact that λ(A) = λ(M−1AM), where M is a non-singular matrix, completes the

proof.


3.5 Conclusion

In this short chapter, we found the system’s steady state. We also gave a condition for its

feasibility as well as a rule of thumb which provides a sufficient condition for the system’s

feasibility solely based on the gain matrix G and γ†. Most importantly, we brought the

whole system into a linear form by linearizing the interference function I around the steady

state. Based on the linearization, we will investigate the system’s stability locally in the

next chapter.

Note that we could show a direct connection between the linearized interference ∇I

and the feasibility matrix Γ†F . This facilitates the analysis and also shows how the target

SIR and the gain matrix affect stability.

Chapter 4

Nyquist Analysis

The Nyquist stability criterion is a classical easy-to-use tool in linear systems analysis

which was postulated 1932 by Harry Nyquist [16]. Despite of its age, Nyquist’s criterion

is still widely used up to date. One of its great advantages is that one can illustrate the

stability of even multi-dimensional systems in one figure, the so-called Nyquist plot. With

the linearization from Section 3.3 and some minor modifications, Nyquist’s criterion is

applicable to our system.

In this chapter, we will first state the general Nyquist criterion for the one- and the

multi-dimensional case, both in their discrete-time version. Then, we will utilize the prop-

erties of our system to derive a problem-specific Nyquist criterion. This criterion will then

be used to investigate some example networks.

33

34 4 Nyquist Analysis

K G

(a) Open Loop

G

K

−

(b) Parallel Closed Loop

K G−

(c) Serial Closed Loop

h

∇I

+

(d) Our closed loop

Figure 4.1: Standard Loop Layouts

Loop layouts (a)–(c) are considered standard. Both, the serial and the parallel loop can be used to deriveNyquist’s criterion since their poles are the same. The difference in our case is not only the exchangedposition of controller and plant but also the positive, instead of the negative, feedback. Note that thepositioning of the feedback as upper loop is solely for the purpose of generating resemblance with theformerly-derived block diagrams.

4.1 Preliminaries

Nyquist’s criterion allows stability analysis of a closed loop (Figure 4.1b, 4.1c) by only

considering the open loop (Figure 4.1a). For its statement we use the open-loop transfer

function

G0 = GK, (4.1)

where G is the transfer function of the plant and K is the transfer function of the controller.

Note that our system is not entirely in one of the standard layouts (Figure 4.1b, 4.1c) which

are normally used to state the Nyquist criterion. Their transfer functions are

Gserial = (I +GK)−1GK = (I +G0)−1G0

Gparallel = (I +GK)−1G = (I +G0)−1G,

4.1 Preliminaries 35

-2 -1 0 1 2-2

-1

0

1

2

Im

Re-2 -1 0 1 2

-2

-1

0

1

2

Im

Re

g0=⇒

Figure 4.2: Nyquist Contour

On the left, we have the plot of z = ejφ when φ ∈ [0, 2π), the well-known unit circle. On the right, wehave the so-called Nyquist contour, which is the unit circle under the mapping of g0.

where I is the n-dimensional identity matrix and (4.1) holds. We, nevertheless, state the

criterion in a classical way but want to point out the swapped position of ‘plant’ ∇I and

‘controller’ H. The problem’s layout results in

Ghere = (I − H∇I)−1H = (I −G∗0)

−1H,

where G∗0 = H∇I. The positive – instead of the more common negative – feedback is

already accounted for.

When talking about Nyquist’s criterion the statement of the so-called Nyquist contour

is hard to avoid. Basically, the Nyquist contour is the image of the unit circle under the

mapping of the considered function. So, for example, an open-loop transfer function of

g0(z) =0.5

z(z − 1) + 0.5

would lead to Figure 4.2.

Keep in mind that we are analyzing the linearized system, not the system itself. Thus,

the results of this chapter will only hold in a neighborhood around the steady state.


4.1.1 One-Dimensional Nyquist Criterion

Here, we state the time-discrete one-dimensional Nyquist criterion without proof. Proofs

can frequently be found in the literature.

Theorem 3 (Nyquist Criterion). Assume G0(z) has no poles on the unit circle. Let

Φ(1 +G0(z)) be the counter-clockwise phase rotation of 1 +G0(z) about the origin when z

counter-clockwise traverses the unit circle and let #p be the number of poles of G0(z) inside

the unit circle. Then, the closed loop is stable if and only if

Φ(1 +G0(z))!= 2π#p.

Note that 1 +G0(z) must not be 0 for all z on the unit circle for this theorem to hold.

The more classical version of the Nyquist criterion is formulated in the continuous fre-

quency domain. There, the poles in the left s half plane are of interest. Furthermore,

often Φ(G0(z)) instead of Φ(1 +G0(z)) is considered and consequently the phase rotation

about the so-called Nyquist point (−1, 0) instead of the origin is measured. One may also

find definitions where z only traverses half the unit circle and therefore half the rotation

has to occur.

4.1 Preliminaries 37

4.1.2 Multi-Dimensional Nyquist Criterion

The multi-dimensional extension of the classical formulation reads very similar and is,

again, stated without proof. An insightful proof can e.g. be found in [11].

Theorem 4 (Multivariate Nyquist Criterion). Assume G0(z) has no poles on the unit

circle. Let Φ (det(I +G0(z))) be the counter-clockwise phase rotation of det(I + G0(z))

about the origin when z counter-clockwise traverses the unit circle and let #p be the number

of poles of G0 inside the unit circle. Then, the closed loop is stable if and only if

Φ (det(I +G0(z)))!= 2π#p.

Note that det(I + G0(z)) must not be 0 for any z on the unit circle for this theorem to

hold. Furthermore, G0(z) is n-dimensional and accordingly I is the n-dimensional identity

matrix.

Nyquist’s Criterion is also capable of handling poles at the border of stability, i.e. poles

that lie exactly on the unit circle. In order to include those, one has to add diminishing

indentations to the Nyquist contour – in the time-discrete case the unit circle – enclosing

these poles. Then each conjugate-complex pair of poles corresponds to an additional #p.


4.2 Analytical Usage

So far, the multivariate Nyquist criterion is of little practical use as we would have to plot

the Nyquist locus for every single controller K. In this section, we utilize the structure of

our system to get a practicable test for stability. A similar derivation was done in [2].

4.2.1 General Derivation

Assume the controller K to be frequency-independent and the plant G to be diagonal with

only identical sub-systems g, i.e. G = gI. Using the facts that

det(·) =∏

i

λi(·) and λi(I + cM) = 1 + cλi(M),

where λi(·) denotes the eigenvalues of ‘·’, M is a matrix and c is a constant, we get

det(I +G(z)K) =∏

i

(1 + g(z)λi(K)).

Keep in mind that we are ultimately interested in the phase rotation. In order to isolate

the phase, we can write any complex number in the form

z = z ejφ,

where z ∈ C; z is its amplitude and φ its phase. Since exponents add up when multiplied,

we can write

Φ (det(I +G(z)K)) =∑

i

Φ (1 + g(z)λi(K)) ,

where Φ(·) is the phase rotation of ‘·’. Pulling out the eigenvalues yields


i

Φ((λi(K)

−1 + g(z))λi(K))

=∑

i

Φ(λi(K)

−1 + g(z))+ Φ(λi(K))︸︷︷︸

=0

4.2 Analytical Usage 39

and hence


i

Φ(λi(K)

−1 + g(z)). (4.2)

Thus, we have shown that the stability of the closed loop can be investigated by counting

the number of counter-clockwise encirclements of −λi(K)−1 by the Nyquist contour of G(z)

when z traverses the unit circle counter-clockwise. This means we only have to study one

Nyquist plot of g(z) in order to infer stability for all K.

4.2.2 Problem-Specific Usage

As mentioned in Section 4.1, the plant G(z) corresponds in our case to the dynamics

block H(z), which includes the actual controllers, while the usual controller K(z) here only

includes the linearized interference ∇I. Assuming the system to be homogeneous, see

Section 2.3.2, we have H(z) = h(z)I. Taking also the positive feedback into account, we

get from (4.2) that

Φ (det(I +∇Ih(z))) =∑

i

Φ(−λi(∇I)−1 + h(z)

).

Hence, we can conclude the following theorem:

Theorem 5. Let h(z) be the stable transfer function of the dynamics block (see Figure 2.4).

For i from 1 to n, where n is the number eigenvalues, let Φ (−λi(∇I)−1 + h(z)) be the

counter-clockwise phase rotation of h(z) about λi(∇I)−1 when z traverses the unit circle in

counter-clockwise direction. Then the linearized system is stable if and only if

∑

i

Φ(−λi(∇I)−1 + h(z)

) != 0.

Since cancellations of clockwise and counter-clockwise encirclements are rare for the con-


sidered system, we will frequently use the sufficient condition

Φ(−λi(∇I)−1 + h(z)

) != 0 ∀i.

This theorem allows for an in-depth investigation of the influence of the linearized inter-

ference ∇I on the system’s performance. We want to point out that h(z) has to be stable

for Theorem 5 to hold.

4.3 Application

In this section, we show how to apply the problem-specific stability analysis. As a simple

example, we will consider two three-mobile networks G1 and G2. We will check feasibil-

ity and then move on to checking the performance of different controllers with the just

introduced Nyquist analysis. Our results will be verified by time-domain simulations.

First of all, we have to make sure to use only stable h(z). A check of our standard

case (2.7) with a PI-controller gives Figure 4.3. There, we plot the maximum κP and κI

leading to a stable lower loop. We begin with an overall delay δ of 2 in order to have

strictly proper update rules in both, the base station and the mobile.

For the sake of this discussion, consider the following target SIR and noise terms:

γ†i = 10 and σi = 10 ∀i (4.3)

These terms are arbitrarily chosen since they basically only scale the results. Now, consider

two networks with their respective gain matrices G1 and G2.

G1 =

1.000 0.012 0.013

0.021 1.000 0.023

0.031 0.032 1.000

G2 =

1.000 0.120 0.130

0.210 1.000 0.230

0.310 0.320 1.000

Applying Rule of Thumb 1 we expect, on the one hand, that the second network is not

4.3 Application 41

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

κP

κI

δ = 2

δ = 3

δ = 4

δ = 5

δ = 6

Figure 4.3: Stability Region of the Dynamics Matrix

Considering (2.7) with a PI-controller and an overall delay δ = δb + δm + 2δp from two to six we getthe shaded regions of stability for the parameters κI and κP . The stability of the dynamics matrix is anecessary condition for Theorem 5 to hold. Note that this figure, naturally, also captures the case of soleproportional or integral control on its axes.

feasible and know, on the other hand, that the first case is indeed feasible; both, of course,

regarding a target SIR of ten. Checking the eigenvalues of the respective feasibility matrices

reveals

λ(Γ†F1) = {0.503282,−0.321858,−0.181424}

λ(Γ†F2) = {5.03282,−3.21858,−1.81424}

and thus that the spectral radius condition (3.4) is indeed violated for G2. Accordingly,

we move on to the analysis of G1.

As a first control law, we investigate the so-called Distributed Power Control (DPC)

Algorithm. It was first postulated by Foschini and Miljanic in 1993 [3] and had great

impact on the communication community [14]. In our framework, DPC corresponds to


æææ

-6 -4 -2 0 2-2

-1

0

1

2

Im

Re

Figure 4.4: Stability Investigation of G1 under DPC Control

Here, the red dots depict the reciprocal eigenvalues of the feasibility matrix Γ†F1 of network G1. The blackcurve represents the Nyquist plot of hDPC . Since there are no encirclements of the reciprocal eigenvaluesby the Nyquist plot, the DPC algorithm stabilizes network 1, pursuant to Theorem 5.

unfiltered integral control with a gain κI of one; a computational delay of one, but no

further delays are considered. Hence, using (2.8), hDPC equates to

hDPC(z) =κI

z − 1 + κI=

1

z.

The examined system is stable when the Nyquist plot of hDPC(z) does not encircle any

eigenvalues of the feasibility matrix Γ†F . The latter is a function of the gain matrix G

while the former is not. Plotting the Nyquist curve of the DPC algorithm gives the unit

circle, see Figure 4.4. Keeping in mind the spectral radius feasibility condition (3.3), we

see that for DPC stability and feasibility coincide. Thus, any feasible system with no more

than one delay is stabilized – this naturally includes G1.

As a further example, consider a slightly more realistic case where there is an overall

delay δ of two. For the sake of this discussion, we choose an I-controller with gain κI = 0.5

resulting, with (2.8), in

h1(z) =0.5

z2 − z + 0.5.

This is a stable transfer function, which can be shown by checking its poles or simply

looking up Figure 4.3. Plotting the reciprocal eigenvalues of Γ†F1 and the Nyquist plot

of h1(z) yield Figure 4.5. For the chosen parameters there are no encirclements of the

4.3 Application 43

æææ

-6 -4 -2 0 2-2

-1

0

1

2

Im

Re

Figure 4.5: Stability Investigation of G1 with System h1

The same definitions as in Figure 4.4 hold. Hence, since there are no encirclements, the overall system isstable. Note that the increased delay involves a further encirclement of the origin and, even though theintegrator gain is smaller, a larger curve.

inverse poles of Γ†F1. Thus, according to Theorem 5, the system is stable.

Now, reconsider the second network, G2, and a two-delay PI-controller with κP = 1

and the same integral part as before, κI = 0.5, i.e.

h2(z) =1.5z − 1

z3 − z2 + 1.5z − 1.

Plotting this controller in conjunction with the reciprocal eigenvalues of Γ†F2 gives Fig-

ure 4.6. This plot suggests that G2 is stabilized by h2(z). That, however, is a fallacy

since there exists no feasible steady state which could be stabilized for G2. Furthermore,

the parameters are chosen such that h2(z) is unstable. Hence, the requirements for Theo-

rem 5 were not met in the first place. With this daunting example we want to stress the

importance of checking the required conditions before applying Theorem 5.

The time-domain plots in Figure 4.7 confirm the anticipated behavior. The powers for

the individual mobiles approach their respective steady states for our first example while

in the second case, the system is unstable.


æææ

-2 0 2 4 6-2

-1

0

1

2Im

Re

Figure 4.6: Stability Investigation of the Infeasible Network G2

This figure is an example for the misuse of Theorem 5. It contains no encirclements of the reciprocaleigenvalues of Γ†F2 (red) by the Nyquist plot of h2 (black). Accordingly, the plot suggests stability of theconsidered system. This, of course, is wrong since the requirements for Theorem 5 were not met.

0 10 20 30

18

20

22

24

26

t

pi(t)

(a) Time-Domain Plot of Figure 4.5

0 10 20 300

100

200

300

400

t

pi(t)

(b) Time-Domain Plot of Figure 4.6

Figure 4.7: Time-Domain Plots of the Considered Examples

These plots show the course of the output powers pi of the considered three mobiles over the time t in red.Their respective steady-state power pss are depicted by a dashed black line. Note that the network G2,which is considered in the right plot, has no feasible steady state for the considered parameters.

4.4 Conclusion 45

4.4 Conclusion

In this chapter, we first presented some basics of Nyquist analysis. Then, we moved

on to utilize the system’s characteristics to derive a problem-specific Nyquist criterion.

Furthermore, we showed how to apply this criterion on a simple example and pointed out

that special attention should be paid to the theorem’s requirements.

We now have a powerful tool to investigate the system’s stability. Keep in mind,

however, that Nyquist analysis is based on the system’s linearization and thus only holds

in a neighborhood of the steady state. We will see in Chapter 8, though, that typically

this region is adequately large for our considered system.

So far, the Nyquist analysis gives only information whether the system is stable or not.

The subsequent chapter will deal with analyzing the system from a performance point of

view, relying heavily on the results from this chapter.


Chapter 5

Rate of Convergence

We have just investigated the system’s stability. Another very important system property

is the time which our controller needs to stabilize the system. This chapter is dedicated to

finding a measure for this rate of convergence.

First, we will introduce a scaling for the system states in the time domain. With

this scaling we are able to give exponential bounds on its rate of convergence. In the

next section, we will translate the time-domain scaling into the frequency domain via the

z transform. Finally, we will investigate the rate of convergence for the example of the

previous chapter.

47

48 5 Rate of Convergence

5.1 Scaling in the Time Domain

In order to find a bound on the rate of convergence of our system, first consider the following

scaled version of the system’s states

∆p(t)∆= at∆p(t), (5.1)

where a > 1 is the scaling factor. Suppose the scaled system is stable, i.e.

limt→∞

∆p(t) = ∆pss (5.2)

has to hold. Since we have linearized our system around the steady state, ∆pss equals 0.

Furthermore, it follows directly from (5.1) that

∆p(t) = a−t∆p(t),

and with (5.2) that our nominal system in ∆p converges factor a faster than the scaled

system in ∆p does. The analysis of the system’s stability was done in the frequency domain,

not the time domain. Hence, we move on to the z transform of the just introduced scaling.

5.2 Scaling in the Frequency Domain 49

5.2 Scaling in the Frequency Domain

For the translation of the time-domain scaling into the frequency domain we use

atf(t) ◦–• F (z/a) (5.3)

from [1], where a is a constant, f(t) is a function in the time domain and F (z) is its

z transform. This scaling has, of course, a negative effect on the systems stability. To

investigate the stability of the scaled system, we once more utilize Theorem 5. Accordingly,

we have to check the stability of H(z) = H(z/a). This includes the check of the poles of

the scaled transfer function itself as well as the Nyquist contour of the scaled system.

Note that the poles of the feasibility matrix Γ†F are not frequency dependent and thus

do not change with the scaling; the Nyquist plot of h(z), however, will. Unfortunately,

this happens in a non-linear fashion. Some intuition on the nature of the changes can be

obtained by the examples in the next section.


5.3 Examples

Consider the standard case (2.8) with integral control and an overall delay of two where

hI(z) =κI

z2 − z + κI.

Scaling this system with (5.3) yields

hI(z) =a2κI

z2 − az + a2κI.

Now, let us first investigate the set of stable a for all stable κI . Figure 5.1 was obtained

by simply checking the poles of hI(z) and shows the region of stability.

0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.2

1.4

1.6

1.8

2.0

κI

a

δ = 2

δ = 3

δ = 4δ = 5δ = 6

Figure 5.1: Stability Region of the Scaled Dynamics Matrix hI for δ from 2–6

Considering h with an I-controller and an overall delay δ = δb + δm + 2δp, we get the shaded regions ofstability for the parameters κI and a. Increasing delay leads to decreased maximal possible a. Note thatthe stability of hI is just a necessary condition for the system’s stability, no sufficient one.

The next step, the Nyquist stability analysis, cannot be done so easily, though. We

5.3 Examples 51

æææ

-6 -4 -2 0 2-2

-1

0

1

2

Re

Im

a = 1.0

a = 1.1a = 1.2

(a) G1, κI = 0.1, δ = 2

æææ

-6 -4 -2 0 2-2

-1

0

1

2

Re

Im

a = 1.0a = 1.1

a = 1.2

(b) G1, κI = 0.2, δ = 2

æææ

-6 -4 -2 0 2-2

-1

0

1

2

Re

Im

a = 1.0a = 1.1

a = 1.3a = 1.2

(c) G1, κI = 0.4, δ = 2

æææ

-6 -4 -2 0 2-2

-1

0

1

2

Re

Im

a = 1.0a = 1.1

a = 1.3a = 1.2

(d) G1, κI = 0.5, δ = 2

Figure 5.2: Rates of Convergence for Integral Control of G1

Each subfigure holds plots of the same system, differently scaled. Dashed lines mark unstable systems,dotted lines mark unstable h and thus invalid plots. Most interestingly, smaller gain may lead to fasterrate of convergence – compare Figure 5.2c and 5.2d. Keep in mind, that for Theorem 5 to hold, h has tobe stable. This is indeed not given for a ≥ 1.2 in Figure 5.2a.

consider the example network G1 from Section 4.3 and pick four gains κ1 = 0.1, κ2 = 0.2,

κ4 = 0.4 and κ5 = 0.5. Figure 5.2 illustrates the influence of growing a, beginning at one,

the unscaled case, up to the respective borders of stability. It could be expected that the

biggest stable gain yields the best rate of convergence. This is, however, not the case here,

see Figure 5.2.

Remember that the scaling only gives an upper bound on the rate of convergence. A

time-domain plot of an interesting example can be found in Figure 5.3. Here, it is shown

that a higher gain can lead to a slower rate of convergence, just as it was predicted by our

frequency-domain analysis.


5.4 Conclusion

In this short chapter, we could find bounds on the systems rate of convergence by simply

introducing a scaling factor a. This was motivated by a time-domain consideration but

could easily be adopted into the frequency domain, where we could use the results from

Chapter 4. We also investigated the rate of convergence with a small example, where

time-domain simulations could verify our predictions.

Note, however, that we just found bounds on the rate of convergence, not the rate

itself. Those bounds might be conservative but qualitatively they often still give valuable

information. Furthermore, note that our results are based on a linearization around the

system’s steady state and are thus only valid locally. After finding a performance measure,

we now move on to improving the system’s performance with respect to time delays.

0 5 10 15 20 25 30 3515

20

25

30

t

pi(t)

Figure 5.3: Time-Domain Plot of Figure 5.2c and 5.2d

We compare the time-domain plots for network G1 under hbase with κI = 0.4 (red) and 0.5 (blue). Thebounds derived in Figure 5.2c and 5.2d suggest that the smaller gain should converge faster. This is indeedthe case. The dash-dotted pink line depicts the bound with a = 1.3.

Chapter 6

Time-Delay Compensation

The investigations in the previous chapters have shown that time delays are critical to

the stability of our system. In this chapter, we will therefore present the so-called Time-

Delay Compensation (TDC), see e.g. [6]. First, we will state the general idea behind TDC.

Then, we will visualize how TDC is implemented in the framework of the block diagrams

from Section 2.2. This will lead us directly to the derivation of the transfer function of the

delay-compensated system. Furthermore, we will derive an extended TDC controller for our

system that guarantees stability for any delay. Finally, we will investigate to which extent

TDC can compensate for time delays and what happens if one over- or undercompensates.

6.1 Idea

The goal of our controller is to guarantee a sufficient Signal-to-Interference Ratio γ. In

order to achieve this, it compares the target SIR γ† to the measured SIR γm resulting in

the control error

e[t] = γ†[t]− γm[t].

Problems occur when the measured data is outdated and thus

γm[t] = ς[t− δς ]− ι[t− δι],

53

54 6 Time-Delay Compensation

where δς and δι are the number of samples delay for the signal and the interference, re-

spectively. Using the time-shift operator q (see Section 1.3.1), the measured SIR follows

as

γm[t] = q−δς ς[t]− q−διι[t].

As a matter of fact, we know the current base station controller output u[t]. Thus, we can

approximate the actual signal ς[t] and consequently compensate for the delay simply by

subtracting the approximated delayed signal and adding the approximated current signal.

γmTDC [t] = q−δς ς[t]− q−διι[t]−q−δς ς[t] + ς[t]︸︷︷︸

TDC

⇔ γmTDC [t] = ς[t]− q−διι[t]

This results in the control error

e[t] = γ†[t]− ς[t]− q−διι[t],

which now only involves the interference delay δι. We will see later on that the delay in

the lower loop has a much larger effect on the system’s stability. Thus, the TDC will prove

very useful in our case.

6.2 Block Diagram and Transfer Function

Adding additional feedback paths to Figure 2.3b in order to cancel the lower delayed signal

path and to feed back the undelayed signal leads to Figure 6.1. The outer TDC path ideally

cancels the lower loop completely, leaving only the undelayed feedback of the ideal mobile.

Thus, theoretically all delays except the delay DB in the base station are canceled from

the lower loop.1

For notational convenience, let us condense the just introduced additional paths into a

1Note that the classical TDC is basically a Smith predictor

6.2 Block Diagram and Transfer Function 55

block T = diag(ti) (see Figure 6.2). This leads to a new base-station controller BTDC with

an inner loop. Component-wise it reads

bTDC =dbb

1 + dbbt, (6.1)

where

t = m− dpdmmdpfς

= m(1− dTDCfς),

with dTDC = dmd2p, the compensated delay. We can use this new time-delay-compensated

base-station controller block BTDC instead of a normal controller in the old block diagram

of Figure 2.3b. The derived time-delay compensation is very similar to the one derived

I Dp

Fι DBB Dp DMM

M Dp DMM

Fς Dp

Fς Dp

γ††

σ

e u

pι

−

−

+

T

Figure 6.1: Block Diagram with Additional Feedback Paths to Cancel Delays

We added two additional inner loops to Figure 2.3b. The dashed line marks the Time-Delay Compensation(TDC) block T. It ideally cancels the lower loop completely and adds the feedback of the ideal version ofthe mobile. Note that we can not cancel the base station’s delay DB with this TDC.


in [6]. Note, though, that we do not filter both feedback paths, which originates from our

block-diagram point of view and leads to smaller Nyquist curves and thus to better results.

DBB

M

Fς Dp DMM Dp

e u

−

−

T

(a) BS Controller with Every Subblock

DBB

T

e u

−

(b) Condensed BS Controller

Figure 6.2: The Base Station’s TDC Controller

Here, we isolated the additional paths to show the base station’s TDC controller in more detail. Note thatthis controller can be plugged into the the existing framework without any further modifications.

6.3 Extension

The weakness of this approach is that it cannot compensate for delays in the base-station

controller. Keeping that in mind, one can analogously derive a similar controller for the

mobile. We have thus found means to cancel all lower-loop delays except the ones in the

base station; or to cancel all lower-loop delays except the ones in the mobile.

The logical next step is to combine those two time-delay compensators. One has to

keep in mind, however, that the TDC in the BS is dependent on the transfer function of the

mobile and vice versa. Applying such kind of double TDC is not practicable since both the

BS and the MS exactly need to know whether the other does compensate, and if, for which

delay. Especially when considering mobiles being in soft handover, i.e. being connected to

more than one base station while transitioning between cells, this is not feasible due to

possibly varying delays and compensations.

Let us reconsider the normal TDC again. Reducing its block diagram (Figure 6.1)

6.3 Extension 57

yields Figure 6.3 which has the following lower-loop dynamics:

hTDC =dbb

1 + dbbmdmmd2pfι. (6.2)

We see that the TDC does not actually cancel delays but merely shifts them out of the

feedback path. This results in less-delayed lower-loop dynamics with the same input-output

delay. The BS delay, however, remains and threatens to destabilize the system. Naturally,

analogous results hold for TDC in the MS.

An ultimate goal of a TDC is a totally undelayed feedback, i.e. a system where

h!=

b

1 + bmdbdmmd2pfι.

Using the transfer function (2.6) of the dynamics of the uncompensated system

h =dbbdmmd2pfι

1 + dbbdmmd2pfς

I

Fι DBB Dp DMM Dp

M

γ††

σ

e u

pι

−

Figure 6.3: Reduced Block Diagram of the Time-Delay-Compensated System

Considering the cancellations from Figure 6.1 gives this block diagram. Note that this analogously worksfor TDC in the mobile. The only difference in the block diagram would be switched B:s and M:s.


we find the following ansatz:

dbb∗TDCdmmd2pfι

1 + dbb∗TDCdmmd2pfς

!=

b

1 + bmd∆mfι, (6.3)

where d∆ is the yet-unknown delay of the the time-delay-compensated dynamics. The

compensated base station’s transfer function dbb∗TDC has the following structure:

dbb∗TDC =

dbbt∗

1 + dbbt∗

and we search for a stable and proper t∗. Solving (6.3) gives

t∗ =d∆

dbdmd2p + dbdmd2pbm− dbdmd2pd∆fςbm− dbd∆b.

The question that arises is: are the introduced blocks (t∗ and dbb∗TDC) stable? If this was

the case, we would have found a controller that performs independent of the considered

system’s delay.

Note that if there are no delays in the base station, the traditional and the extended

TDC coincide. We now move on to an evaluation of the classical as well as the just derived

TDC controllers. A focus will lie on their robustness against errors in the prediction of the

time delays and the mobile gain as well as their internal stability.

6.4 Investigation

In this section, we will investigate the advantages and disadvantages of applying a Time-

Delay Compensation to our system. The tools we use will be the Nyquist analysis as

introduced in Chapter 4 as well as Matlab simulations to confirm e.g. rate-of-convergence

predictions (see Chapter 5). We will compare the traditional TDC to the just derived

extended TDC. Furthermore, we will investigate how both kinds of TDC perform when

too high/too low delays/gains are assumed, i.e. when one over- or undercompensates.

6.4 Investigation 59

6.4.1 Traditional TDC

Our stability analysis is based on Theorem 5 which is only applicable when the lower loop is

stable. Thus, we will first investigate the stability of the time-delay-compensated dynamics

block hTDC (6.2) in dependence of the delays and controller parameters. Remember that

in the investigation of the uncompensated lower loop, see Figure 4.3, we did not have to

distinguish where the delays occur. The reason for this is clear from the block diagram:

all delay blocks are found in linear succession and thus interchangeable in their position

due to their diagonal form and time invariance. The commutability, however, is destroyed

by the feedback loop introduced with BTDC . Hence, we have to investigate the impact of

delays in the base station and delays elsewhere separately.

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

κP

κI

δb = 1δb = 2

δb = 3δb = 4

δb = 5δb = 6

Figure 6.4: Stability Region of the BS-TDC Dynamics Matrix

Consider hTDC with a PI-controller and a delay of δb = 1. Now, increase δb step by step. The shadedregions then depict the regions of stable hTDC . The compensator delay δTDC is δm + 2δp. Note that thisplot captures any case of δm and δp. Those delays do not threaten stability, since they were compensatedfor by shifting them outside the feedback path.

Figure 6.4 shows the regions of stability for hTDC with a PI-controller in the mobile.

We begin with the case where there is one delay in each, the BS and the MS, and then

raise the delay δb in the base station. Not surprisingly, this plot is exactly the same as

Figure 4.3, when considering δb as δ. In fact, this single plot captures any combination of

delays outside the base station!

Keep in mind that the plot above only depicts the input-output dynamics of the ideal

case, i.e. the delay and gain of the system is known and consequently δTDC is chosen


correctly to be δm +2δp. Furthermore, the TDC-intern model of the mobile mTDC behaves

exactly like mobile in the network m. Keep in mind that errors in the compensation might

destabilize even stable systems.

As an example, consider hTDC with integral control. Here, we get the stability regions

of Figure 6.5a and 6.5b for over- and undercompensation for delay in dependence of the

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

κI,T

DC

κI

δε = 0

δε = −1

δε = −2

δε = −3

δε = −4

(a) Undercompensated TDC

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

κI,T

DC

κI

δε = 0

δε = 1

δε = 2δε = 3δε = 4δε = 5

(b) Overcompensated TDC

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

κI,T

DC

κI

δε = 0

δε = −1δε = −2

δε = −3

δε = −4

(c) Undercompensated Stable TDC

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

κI,T

DC

κI

δε = 0δε = 1

δε = 2δε = 3δε = 4δε = 5

(d) Overcompensated Stable TDC

Figure 6.5: Stability Region of the Under-/Overcompensated BS-TDC Dynamics Matrix

Consider hTDC with an I-controller and a delay of δm = δb = 2 as well as δp = 1. The compensator delayδTDC is δm + 2δp + δε. Thus, we consider undercompensation in the left plots and overcompensation inthe right ones. The shaded regions depict the regions of stable hTDC . In the upper plots, only hTDC wastaken into account. The lower plots show the same system when also considering the stability of bTDC .


internal and external mobile gain κI,TDC and κI . As one would expect, there is a stability

region roughly around the diagonal, i.e. where κI,TDC and κI are similar. When one,

however, also considers the internal stability, i.e. imposes that bTDC has to be stable,

one gets Figure 6.5c and 6.5d. Here, we see that the TDC introduces unstable pole-zero

cancellations for higher gains. Those have to be taken into account when designing a TDC

controller.

So far, we have only investigated the lower-loop stability which is not sufficient for

the overall stability. In order to investigate the stability of our system, we need to take

the interference into account which is done by once more utilizing Theorem 5. We also

calculate the bounds on the system’s rate of convergence pursuant to Chapter 5. Some

example plots can be found in Chapter 8.

The following observations were made during our investigation:

• Higher controller gains are almost always possible with TDC.

• Bounds on the rate of convergence suggest that TDC converges slower for systems

far away from the border of stability and faster for stability-critical systems. This

behavior could be confirmed by simulation.

• Undercompensation of the delay does not threaten stability, in the worst case, i.e. δε =

−δTDC , the Nyquist curves of h and hTDC are identical.

• Overcompensation of the delay can threaten stability in some cases, e.g. high delay

and/or high gain systems.

• Underestimating the mobile’s gain leads to a growing Nyquist curve and can thus lead

to instability. Note that the curve of the compensated system grows; in cases with

considerable delay this might consequently still be preferable to an uncompensated

system.

• Overestimating the mobile’s gain leads to a shrinking Nyquist curve. This would be


desirable, but the internal stability deteriorates rapidly with growing ∆κI and hence

has to be considered carefully.

6.4.2 Extended TDC

The extended TDC, introduced in Section 6.3, promises delay independent performance.

This comes to the cost of complexity. Its advantages and drawbacks are discussed in this

section.

In an ideal case we have from (6.3) that

h∗TDC =bm

1 + bmd∆fι,

which for the standard case (2.7) with integral control yields

h∗TDC =zκIz−1

1 + zκIz−1

1

zδ∆fι

=zκI

z(κI + 1)− 1

1

zδ∆fι

=

κIκI+1

z − 1κI+1

1

zδ∆−1fι. (6.4)

Apart from known perturbations by the filter dynamics, this obviously has only poles

inside the unit disk and the lower-loop gain is always smaller one for all κI > 0. For

the standard case, the extended TDC has thus restored the DPC property that feasibility

implies stability.

Analogous to the discussion in the last section, we have to consider that the real delay

and gain in the system might not be known exactly. This might indeed have a devas-

tating effect since the base-station controller now is a high-order system itself. Hence,

introduce δ∗ε , the error in the time-delay compensation. We only needed to consider one

uncertainty for the delays since the delays outside the base station (δm and δp) enter the

equations in the same fashion. Furthermore, the base-station delay can in all conscience

be assumed to be locally known.


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0κI,T

DC

κI

δ∗ε = 0δ∗ε = −1δ∗ε = −2

δ∗ε = −3

δ∗ε = −4

δ∗ε = −5

(a) Undercompensated TDC*

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

κI,T

DC

κI

δ∗ε = 0

δ∗ε = 1

δ∗ε = 2

δ∗ε = 3δ∗ε = 4

δ∗ε = 5

(b) Overcompensated TDC*

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

κI,T

DC

κI

δ∗ε = 0δ∗ε = −1δ∗ε = −2

δ∗ε = −3

δ∗ε = −4

δ∗ε = −5

(c) Undercompensated Stable TDC*

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0κI,T

DC

κI

δ∗ε = 0

δ∗ε = 1

δ∗ε = 2

δ∗ε = 3

δ∗ε = 4

δ∗ε = 5

(d) Overcompensated Stable TDC*

Figure 6.6: Stability Region of the under-/overcompensated extended TDC DynamicsMatrix

Consider h∗TDC with an I-controller and a delay of δm = δb = 2 as well as δp = 1. The compensator delayδ∗TDC is δb + δm + 2δp + δ∗ε . Thus, we consider undercompensation in the left plots and overcompensationin the right ones. The shaded regions depict the regions of stable h∗TDC . In the lower plots, we took alsothe internal stability of b∗TDC into account.

In Figure 6.6, we plotted the regions where h∗TDC is stable. This is done in dependence of

the error in the assumed delay and gain. In the upper plots, we only consider the stability

of h∗TDC , in the lower plots also the stability of t∗ is taken into account. If one considered

the base station block (δbb∗TDC) as a line of code in the base station, only the stability of

this block has to be considered. Note that those cases are captured in Figure 6.6a and 6.6b.


Despite the complexity of the new controller design, the lower block seems to be relatively

robust to uncertainties in the delay and the mobile’s gain.

Since the lower-loop stability is not sufficient for stability, we move on to the Nyquist

analysis of the considered cases. Moreover, bounds on the rate of convergence are calculated

and compared to time-domain simulations. Some examples with concrete numbers can be

found in Chapter 8. During our tests, the following observations were made:

• The extended TDC shows remarkable stability to a wide range of parameters when

the internal model of the mobile and delay is exact.

• Bounds on the rate of convergence suggest that the extended TDC performs slower

than the traditional TDC. However, time-domain plots show that those bounds might

be conservative for some parameters.

• Undercompensation of the delays does not threaten stability, in the worst case,

i.e. δ∗ε = −δ∗TDC , the Nyquist curves of h and h∗TDC are identical.

• Underestimating the mobile’s gain can lead to instability, just like for the traditional

TDC.

• Overestimating the mobile’s gain may lead to internal instability when found in

combination with overcompensating for the delay. Otherwise, the extended TDC is

relatively robust to overestimation.

If one compares both compensations, huge similarities in the respective behavior can be

found. If there are delays in the base station, the extended TDC outperforms the traditional

TDC significantly. If there is, however, no delay in the base station, the new TDC offers

no advantages.

6.5 Conclusion 65

6.5 Conclusion

In this chapter, we introduced the concept of Time-Delay Compensation, both in the

framework of the time domain and our block diagram. The drawback of not being able

to compensate for base-station-internal delay motivated us to come up with an extended

version of the traditional TDC.

In our investigations, we focused on what happens if one over- or undercompensates

for delays and the mobile’s integrator. It seems that one should always compensate for the

lowest expected delay, since undercompensation does not threaten stability but increases

the maximum feasible delay. Also, the internal stability of our base-station controller has

to be taken into account when choosing the gain of the internal mobile. Furthermore,

the extended TDC seems more robust to an erroneous delay assessment than one would

expect. Some simulations backing our assessment can be found in Chapter 8. Still, we

want to point out that the extended TDC is of theoretical nature and its applicability has

to be checked carefully and problem-specific.

Keep in mind that so far, we have only considered information feedback. The next

chapter will elaborate on methods of investigating binary feedback.


Chapter 7

Describing Functions

One fact that has been neglected thus far is that in the real-world 3G system there is no

information feedback. Instead, the base station just sends one deciding bit which causes an

increase or decrease of the power-output level of the mobile. This provides faster update

rates at the price of precision.

In our framework, the binary feedback corresponds to a sign block after the controller

in the base station. Given that the update rate is sufficiently high, our model should give

a good approximation. If the rate, however, does not suffice, means of handling the non-

linear feedback have to be found. One way of tackling binary feedback is the method of

describing functions.

We will first present the basic idea behind describing functions. Then, we will derive

the system-specific describing function and show how period lengths and amplitudes of

different modes can be predicted. Our results will be backed up by simulations.

In [7], Gunnarson derives the describing-function theory for the lower loop, i.e. the SISO

case. However, he leaves the interference-feedback case open. This chapter is dedicated to

answer the left-open question by capturing the MIMO case locally. In particular, we can

predict and verify that here larger amplitudes can occur than in the SISO case.

67

68 7 Describing Functions

G(z)

f(·)

r e

uy

−

Figure 7.1: Split System

Consider a system which can be split into a linear part G and a nonlinear function y = f(u). The referencevalue r is chosen to be zero for the derivations.

7.1 Idea

We consider systems that can be split into a linear partG and a nonlinear function y = f(u),

see Figure 7.1. The general idea behind the describing-function method is to assume that

those systems show oscillatory behavior, both in the input and the output of the nonlin-

earity. Assuming that the fundamental frequency is dominating and that G has sufficient

low-pass behavior, one then derives, based on first-order Fourier estimation, an approxima-

tion for the system’s nonlinearity. This approximation is called describing function Ψ(u)

and, as the name suggests, will be a function of the assumed oscillation. Thus, if there

exists a solution to the approximated system, the assumed oscillation approximately exists.

Let us derive describing functions in a general framework. We use the following nota-

tion:

ui . . . signal u, i-th component

ui . . . amplitude of signal ui

ui . . . first Fourier coefficient of signal ui

The first discrete-time Fourier coefficient is computed as follows:

ui

(

j2π

N

)

∆=

1

N

N−1∑

t=0

ui[t]e−j 2π

Nt, (7.1)

7.1 Idea 69

where N ∈ N is the number of discrete time steps in one period. Now, assume the

input u to our nonlinearity f to be a n-dimensional vector of sinusoids with the same

frequency ω = 2π/N but possibly differing phase φi, i.e.

ui[t] = ui sin

(2π

N(t+ φi)

)

, i ∈ {1, 2, . . . , n},

where n is the system’s dimension. The first discrete Fourier coefficient of ui[t] is conse-

quently, using (7.1),

ui

(

j2π

N

)

=ui2jej

2πNφi =

ui2ej(

2πNφi−

π2 ). (7.2)

Moreover, the first Fourier coefficients for the output yi[t] of f(ui) has to be computed

case-specific from

yi

(

j2π

N

)

=1

N

N−1∑

t=0

f(ui[t])e−j 2π

Nt.

Finally, the describing function Ψ(u) for the nonlinearity follows from

y = Ψ(u)u. (7.3)

This approximation then replaces the nonlinearity f(·) in the block diagram and thus

allows to write

GΨ(u) = −1, (7.4)

which is frequently rewritten into two separate equations for phase and amplitude, see

e.g. [7, 8]. Note that the solution to (7.4) is a function of ui, φi and N . If solutions exists,

the set of parameters approximately describes possibly existing oscillation in the system.

Oscillations might in general not be desirable, but they necessarily exist for systems

with a sign function, which leads us to the specific usage in our case.


7.2 Problem Specific Usage

Fι ∇I Dp

DBB sign Dp DMM

Fς Dp

γ†

ζ

σ

e u c

pι

ς

−

Figure 7.2: Block Diagram with Binary Control

The block diagram for our system as derived in Chapter 2 with an additional sign block to model thebinary control. Here, c = sign(u) is introduced.

In the real-world 3G system there is only binary feedback deciding whether to raise or

to lower the power output of the MS. This is readily included into our block diagram as

sign block resulting in Figure 7.2, which directly maps to Figure 7.1 when considering the

linearized system with positive feedback as well as r = 0. The linear part of our system1,

G, follows from Figure 7.2 as

G = DBM(Fι∇I− Fς),

where D = DBDMD2p. First, we need to check whether this transfer function has low-pass

behavior or not. For this reason, consider the Bode plot of G where we consider integral

control with gain κI = 0.2 and an overall delay of two. Its magnitude plot compared to

a single-pole low-pass filter (pole at 0.8) can be found in Figure 7.3 and shows reasonable

comparability.

In order to find Ψ(u) we first need to compute yi for the case of f = sign(ui). Since

1Note that we are still considering the linearized interference and thus the linearized system.

7.2 Problem Specific Usage 71

10−2 10−1 100−20

0

20

Frequency (rad/s)

Magnitude(d

B)

Figure 7.3: Check of Low-Pass Behavior of g

Here, we compare the Bode plot of a low-pass filter (pole at 0.8, blue) with the Bode plot of g with integralcontrol and gain κI = 0.2 (black). We show the magnitude plot from a typical diagonal element of G, theoff-diagonal elements are cut off at lower frequencies. The plot suggests reasonable low-pass behavior.

its computation is not trivial, we will present the derivation in some detail. During the

course of the derivation we take into account that the sign of a discretized sinus wave,

i.e. the output of our nonlinearity, has an unobservable phase shift. This phase shift will

be called ε and is depicted in Figure 7.4. The observable part of the phase is ⌊φ⌋, where

the lower Gaussian brackets denote the floor operator.

æ

æ

æ

ææ

æ

æ

ææ

æ

-2 0 2 4 6

-1

0

1

timesteps

amplitude

ε

φ

⌊φ⌋

u[t]

c = sign(u[t])

Figure 7.4: Motivation for Introducing ε

We see that only ⌊φ⌋ has an influence on the binary control signal c. The rest of the phase ε is notobservable. We assume εi to be strictly greater than zero in order to avoid hitting the sign function at 0,which would, dependent on the definition of sign(0), lead to asymmetries. Note, however, that those caseswould correspond to an error of 0 and are thus, due to the discrete power levels and their connection viathe interference function, more than unlikely.


The computation of the describing function works as follows:

yi

(

j2π

N

)

=1

N

N−1∑

t=0

sign(ui[t])e−j 2π

Nt

=1

N

N−1∑

t=0

sign

(

ui sin

(2π

N(t + φi)

))

e−j2πNt (7.5)

=1

Nej

2πN

⌊φi⌋N−1∑

t′=0

sign

(

sin

(2π

N(t′ + εi)

))

e−j2πNt′ (7.6)

=2

Nej

2πN

⌊φi⌋

N2−1∑

t′=0

e−j2πNt′ (7.7)

=2

Nej

2πN

⌊φi⌋1− e−jπ

1− e−j2πN

(7.8)

=2

Nej

2πN

⌊φi⌋2 e

j πN

2j

ej πN −e−j π

N

2j

(7.9)

=2

N sin(πN

)ej(πN−π

2+ 2π

N⌊φi⌋). (7.10)

Here, we used the following steps:

(7.5)–(7.6): Define t′∆= t + ⌊φi⌋ and εi = φi − ⌊φi⌋. We can see from Figure 7.4 that

only ⌊φi⌋ has an influence on the sign while εi can not be observed. Also utilize the shift

theorem, i.e. take advantage of the periodicity of e−j2πN to sum from zero again.

(7.6)–(7.7): Make use of the sign change in the sinus as well as the exponential term to

sum over only half the period. Note that we implicitly assumed N to be even for N/2− 1

to be an integer. Uneven period lengths in a stable oscillation can be viewed as alternating

even periods2.

(7.7)–(7.8): Use the geometric series∑n

k=0 ak = 1−an+1

1−a, where a = e−j

2πN .

(7.8)–(7.9): Expand with ejπN /2j in order to get exponential representation of the sinus.

2We want to point out that those stable uneven periods could be observed in the system. However, inthe SISO case and for matrices with only real eigenvalues, the even-period conjecture in [7] holds.

7.2 Problem Specific Usage 73

Now, Ψ(u) follows from (7.3), with (7.2) and (7.10), as

Ψ(u)

u12ej(

2πNφ1−

π2 )

...

un2ej(

2πNφn−

π2 )

!=

2

N sin(πN

)

ej(πN−π

2+ 2π

N⌊φ1⌋)

...

ej(πN−π

2+ 2π

N⌊φn⌋)

.

This allows us to define Ψ(u) as follows:

Ψ(u) =4

N sin(πN

)

︸︷︷︸

∆=ψ

diag

(

ej(πN− 2π

Nεi)

ui

)

︸︷︷︸

∆=D

. (7.11)

Furthermore, we know that

GΨ(u)u!= u,

and thus directly that

λi(GΨ(u))!= 1, for some i. (7.12)

Note that the describing function is indeed only a function of ui, εi and N , not of ⌊φ⌋.

Utilizing the mainly diagonal structure of the linear part and assuming the filters to be

equal we get

G = DBM(Fι∇I− Fς) = dbmf︸︷︷︸

∆=g

(∇I− I),

which, combined with (7.11) and (7.12), yields

λi ((∇I− I)D)!=

1

gψ, for some i. (7.13)

We have thus found a system of equations which can be solved for N and ui and con-

sequently yields the period lengths and amplitudes of possible oscillations. The non-

observable εi ∈ (0, 1) can be seen as tuning parameter. However, finding solutions

to (7.13) is still computational expensive.


Suppose we model the mobile as an integrator with gain κI . Then, the power output

of the mobile p is an integrated square wave (see Figure 7.4), i.e. a discrete triangle wave.

Taking into account that our assumption of equal frequency ω implicitly introduced the

assumption that all amplitudes ui are equal, we know that the real amplitude of the power-

output oscillation is ˆu = κI4N .

Furthermore, the describing function (7.11) involves a phase shift of πN−2 π

Nεi. Since εi ∈

(0, 1) this phase shift could be relatively big. Supposing that the phase shift due to the

sign block is only marginal, we take εi ≈12.3 Knowing that those changes both make an

approximate analysis even more approximate, we conclude the following rule of thumb:

Rule of Thumb 2. Suppose there exists an approximate solution to

1

λi(Γ†F )

!=

gΨ

1 + gΨ, (7.14)

for some i, where the approximate describing function Ψ is defined as

Ψ∆=

4

uN sin(πN

) .

Then, there exists a triangular steady-state oscillation in the power output of the mobiles

with period N and amplitude κI4N .

Note that we could split the lower-loop dynamics from the interference feedback, just like

in Chapter 4. Furthermore, we used Theorem 2 to substitute λi(∇I) with λi(Γ†F ).

We can now plot the reciprocal eigenvalues of the feasibility matrix and check for

every N if the right-hand side of (7.14) hits one of them for varying u. The exact value

of u is of no importance, since the emerging solution is sufficiently described by the period

length N . Obviously, multiple solutions are possible. In the next section, we will consider

some examples and compare our approximate predictions with time-domain plots of the

system with binary feedback.

3In [7] the unobservable phase shift was fixed to be exactly 0.5. This was necessary due to the one-dimensionality and the integer nature of the solution of a discretized problem.


7.3 Investigation

In this section, we will consider a three-user network that has a feasibility matrix with

conjugate-complex reciprocal eigenvalues. We will predict and verify different steady-state

oscillations for that network, showing that even though approximative, the describing

function analysis can give valuable insight into the system’s steady-state dynamics.

Consider the following gain matrix

G3∆=

1 0.055 0.001

0.001 1 0.055

0.055 0.001 1

.

For a target SIR and receiver noise of γ†i = σi = 10 this yields the following reciprocal

feasibility-matrix eigenvalues:

λ(Γ†F3)−1 = {0.56,−0.28 + j0.47,−0.28− j0.47}.

As investigated controller, we chose the standard case (2.8) with an overall delay δ of 4

and gain κI of 0.5. The plot of Rule of Thumb 2 then yields Figure 7.5. Here, we see that

for N = {11.5, 14, 18} the Nyquist curve of right-hand side of (7.14) approximately hits

one of the reciprocal eigenvalues of Γ†F3. Naturally, N = 11.5 can not be obtained, since

the period length has to be an even integer. However, alternations of N = 10 and N = 12

can lead for example to a period length of N ≈ 11.33. In fact, we could find stable modes

with alternating 10 and 12, 10 and two times 12, and only 12 (last mode not shown). This

once more shows the approximate nature of this analysis. It should also be mentioned that

the vast majority of all initial condition led to the 18 mode. This observation is consistent

for all gains and delays we checked. Most interestingly, the mode Gunnarsson found for

SISO systems (N = 2 + 4n, where n is δ − 1 see [7]) still exists for the MIMO case and

can be obtained by initializing all mobiles approximately synchronous or anti-synchronous.

The simulations, obtained by using different initial conditions, can be found in Figure 7.6.


æ

æ

æ

-2 -1 0 1 2

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Re

Im

N = 11.5

N = 14

N = 18

Figure 7.5: Test of Rule of Thumb 2The reciprocal eigenvalues of Γ†F3 are depicted in red, the Nyquist plot of the system in blue. Each linerepresents one N for varying u. Approximate hits can be obtained for N = {11.5, 14, 18}. Note that thederivations of the harmonic balance are only valid for even N . Only to show the hit between 10 and 12the case N = 11.5 was plotted.

60 80 100 120 140

22

24

26

t

pi[t]

(a) Period Length of 11

60 80 100 120 140

22

24

26

t

pi[t]

(b) Period Length of 11.33

60 80 100 120 140

22

24

26

t

pi[t]

(c) Period Length of 14

60 80 100 120 140

22

24

26

t

pi[t]

(d) Period Length of 18

Figure 7.6: Time-Domain Plots of System with Binary FeedbackHere, the predicted modes from Figure 7.5 were obtained by using different initial conditions for thethree mobiles. We highlight one mobile by drawing the output powers of the other mobiles only dotted.They, however, always show identical behavior, which backs our homogeneity assumption which allowedfor using Ψ. Especially note the odd periods in the upper plots. Furthermore, we want to mention thatmost initial conditions lead to N = 18, which has indeed a bigger amplitude than the mode N = 14, whichwas predicted in [7].

7.4 Conclusions 77

7.4 Conclusions

In order to to make a better model of the 3G system, we introduced binary feedback.

Consequently, the steady state of our system became a steady-state oscillation. Describing

functions were introduced in order to describe those oscillations. Here, we successfully

answered the left-open question of the MIMO case in [7].

Taking the interference into account led to further possible oscillations. Note that

shorter as well as longer possible periods could be found. We then tested our predictions

by simulation. Here, all predicted modes could be found.

Even though the describing-functions analysis is very approximative, relatively good

predictions of the emerging steady-state oscillations could be made. Most noteworthy,

we could show that larger amplitudes than those found in the SISO analysis may occur.

Moreover, simulations suggest that the largest mode is most likely to occur. This might

proof useful if one e.g. wants to make sure that a certain minimum Signal-to-Interference

Ratio is obtained.


Chapter 8

Simulations

During the course of this thesis, we frequently used both Nyquist and time-domain plots.

The former were generated with self-written Mathematica routines. The latter were

created with a Matlab/Simulink model of Figure 2.3b, also built for this thesis. The

model is capable of simulating disturbances like changing G matrices and command errors.

Both were not explicitly taken into account in our analysis. However, simulations show

that our local analysis gives good insight into the systems behavior.

In this chapter, we give some intuition into the considered system’s response to changes

in the connectivity, portrait by the gain matrix G, and the lower-loop dynamics h, in

particular adding a Time-Delay Compensation. Furthermore, we will compare stability

results for the system with and without information feedback. Lastly, we will add some

simulations of systems with varying G in order to consolidate our results.

8.1 Influence of the Gain Matrix

In this section, we will investigate the influence of the gain matrix to our system. The

gain matrix G portrays the connectivity of a network. The diagonal elements depict the

channel attenuation while the off-diagonal elements represent the amount of cross-channel

interference.

We will base our investigation on a network without channel attenuation and with

79

80 8 Simulations

arbitrarily chosen cross-channel gains of 0.025. The basic network will then be perturbed

by a parameter ε. This parameter will be varied in position and value to illuminate

the influence of differently-structured gain matrices on system feasibility and stability.

The former can be investigated with the help of Γ†Fi and (4.3) only, the latter only in

combination with the dynamics h, which will be dealt with in the subsequent section.

The investigated gain matrices are

GSym =

1.000 ε ε

ε 1.000 ε

ε ε 1.000

GBlk =

1.000 ε 0.025

ε 1.000 0.025

0.025 0.025 1.000

GAsy =

1.000 ε ε

0.025 1.000 ε

0.025 0.025 1.000

GRnd =

1.000 ε 0.025

0.025 1.000 ε

ε 0.025 1.000

GOne =

1.000 ε 0.025

0.025 1.000 0.025

0.025 0.025 1.000

GAtt =

1.000 0.025 0.025

0.025 1.000 0.025

0.025 0.025 1− ε

We begin by considering symmetrical cases, namely GSym, where all cross-channel gains are

raised, and GBlk, where two mobiles increasingly interfere each other. Second, we consider

asymmetric cases. There, we have GAsy, where the upper triangular matrix is perturbed,

and GRnd, where the interference is augmented in a round-robin fashion. The last two cases

capture increasing interference from one mobile to one other mobile (GOne) and growing

attenuation (GAtt).

Before we move on to plotting the eigenvalues of the corresponding feasibility matrices,

we first have to calculate their respective feasibility regions, which are functions of ε.

Table 8.1 shows the maximal feasible ε, evaluated by both the of Rule of Thumb 1 (εappx)

and actual calculus (εcalc), cf. Section 3.2. The error factor in the right column is computed

simply by dividing the maximal approximated through the maximal actual ε. One can see

that the rule of thumb is getting ever more conservative with growing inhomogeneity of the

8.1 Influence of the Gain Matrix 81

Table 8.1: Comparison between the Approximated and the Actual Feasibility Region

Influence (G) max εcalc max εappx error factor

Symmetric (GSym) 0.050 0.050 1Round robin (GRnd) 0.075 0.075 1Block form (GBlk) 0.088 0.075 1.2Attenuation (GAtt) 0.833 0.500 1.7Asymmetric (GAsy) 0.095 0.050 1.9One entry (GOne) 0.275 0.075 3.7

gain matrix G. This was to be expected considering the rule was based on Gerschgorin’s

disks.

Plotting the reciprocal eigenvalues of the feasibility matrices for all six considered cases

gives Figure 8.1. Here, we can see that the biggest always-positive simple eigenvalue is

approaching the unit circle for ε → εmax. This was already known from the Perron-

Frobenius theorem (see Section 3.2). Unknown, however, were the whereabouts of the

smaller eigenvalues. We will use the condensed plot Figure 8.1g to show stability for all

six considered classes of networks in the subsequent section. Note that those are just

arbitrary examples. For a more rigorous discussion on the position of the eigenvalues of

the feasibility matrix Γ†F see [9]. There, we used the specific structure of Γ†F as well as

Perron-Frobenius and Gerschgorin results to find bounds on λ(Γ†F )−1 for whole classes of

networks. Since the examples here have known G, we compute the λi(Γ†Fj)

−1 directly.

Note that the analysis in this section is done without taking the auto-interference α into

account. The reason for this is that with growing α the region of feasible ε shrinks rapidly.

That alone does not influence the analysis at all. The side effect of decreasing the maximal

feasible cross-channel gains, however, is that the biggest eigenvalue increasingly dominates

in the sense that it grows, at least relative to the other eigenvalues. Consequently, the

reciprocals of the other relatively small eigenvalues threaten stability ever less.

82 8 Simulations

-6 -4 -2 0 2-2

-1

0

1

2

Im

Re

(a) Inverse eigenvalues of Γ†FSym

-6 -4 -2 0 2-2

-1

0

1

2

Im

Re

(b) Inverse eigenvalues of Γ†FBlk

-6 -4 -2 0 2-2

-1

0

1

2

Im

Re

(c) Inverse eigenvalues of Γ†FAsy

-6 -4 -2 0 2-2

-1

0

1

2

Im

Re

(d) Inverse eigenvalues of Γ†FRnd

-6 -4 -2 0 2-2

-1

0

1

2

Im

Re

(e) Inverse eigenvalues of Γ†FOne

-6 -4 -2 0 2-2

-1

0

1

2

Im

Re

(f) Inverse eigenvalues of Γ†FAtt

-4 -3 -2 -1 0 1 2-1.0

-0.5

0.0

0.5

1.0

Im

Re

(g) Case for later investigation: Γ†FBlk,Asy with ε = 0.08

Figure 8.1: Feasible Reciprocal Eigenvalues of Γ†FiFigures (a)–(f) show the development of the inverse eigenvalues of the feasibility matrices of the consideredcases for ε ∈ [0, εcalc) (see Table 8.1). The biggest eigenvalues are always simple and real (known fromPerron-Frobenius) and approach the unit circle on the positive real axis. The last figure shows the casewe will simulate later on. Here, we took λ(Γ†FBlk)

−1 (red) and λ(Γ†FAsy)−1 (blue) for ε = 0.08.

8.2 Influence of the Dynamics Matrix 83

8.2 Influence of the Dynamics Matrix

In order to simplify our analysis we condensed the lower loop of our system into the

dynamics block H in Chapter 2. This block includes the controller in the base station and

the mobile (B and M) as well as filters (Fς and Fι) and delays (D). Our investigation in

this section will show the influence of growing controller gains, delays and filtering.

As point of departure, we use (2.7) with low-gain integral control (κI = 0.2) and two

samples delay (δ = 2). The transfer function of this basic case is

hbase(z) =0.2

z2 − z + 0.2.

We will assume that there is always an integrator in the system. This leads to the maximal

eigenvalue of Γ†Fi being of no concern, when only considering feasible systems. The recip-

rocal of the maximal eigenvalue of the feasibility matrix always approaches the point (1, 0)

on the positive real axis. The Nyquist plot of h(z), however, is always exactly 1 for ω → 0.

This can simply be shown by including any controller with integral action into (2.6)

h(z) =mbfι

zδ +mbfς

=( bz−1

+ a)fι

zδ + ( bz−1

+ a)fς

=(b+ (z − 1)a)fι

zδ(z − 1) + (b+ (z − 1)a)fς,

where b and a are arbitrary polynomials in z; of course chosen such that the integrator

term is not canceled. Using z = eiω and assuming f(1) = 1 we get for ω → 0 that

h(1) =(b+ 0 · a)fι(1)

zδ · 0 + (b+ 0 · a)fς(1)=b

b= 1.

We now move on to investigating the influence of different choices of parameters.

84 8 Simulations

8.2.1 Changing the Integrator Gain

Let us first visualize the influence of a growing integrator gain. Figure 8.2 shows the

Nyquist plot of h(z) for gains κI from 0.2 to 0.8; the overall delay δ is always two. All

considered gains lead to a stable lower loop, see Figure 4.3. The plots show that the

Nyquist curve expands in a non-linear fashion when κI increases. Hence, we can conclude

different maximal cross-channel gains ε in dependence of the network configurations G and

the controller gains κI . The maximal stable gains will be considered in more detail later

in this section.

Note that the Nyquist plots always traverse through the point (1,0) and thus infeasible

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(a) κI = 0.2, δ = 2

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(b) κI = 0.4, δ = 2

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(c) κI = 0.6, δ = 2

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(d) κI = 0.8, δ = 2

Figure 8.2: Investigation of Increasing Integrator Gain κIFigures (a)–(d) combine Figure 8.1g (red, blue) with the Nyquist plot of hi (black). The unit circle is addeddashed. We see that increasing gain leads to a growing Nyquist curve. Eventually, there are encirclementsof the reciprocal poles of the considered networks. For Figure 8.2c we predict a stable red and unstableblue system. A time-domain plot using the same color conventions can be found in Figure 8.3.


systems are always unstable. Remember that in the case of the simple DPC algorithm,

both the feasibility and the stability condition coincided. However, as soon as one considers

more delays, the conditions diverge and the feasibility condition becomes only necessary

and is not sufficient for stability anymore.

Note, furthermore, that our time-domain plots in Figure 8.3 confirm the results of our

Nyquist analysis. Here, we show for the same controller parameter and initial conditions

the trajectories of the power outputs of all three mobiles for two networks, GAsy and GBlk.

As predicted, one network leads to instability while the other converges to its steady state.

0 20 40 60

30

40

50

t

pi[t]

(a) GBlk

0 20 40 60

30

40

50

t

pi[t]

(b) GAsy

Figure 8.3: Time-Domain Plot of Figure 8.2c

Corresponding to the frequency-domain plot in Figure 8.2 we used GBlk (red) and GAsy (blue) withε = 0.08 such that reciprocal eigenvalues without and with encirclements emerge. While the blue network(no encirclements) is stable, the red (two encirclements) is not. The time-domain simulation thus showsexactly the predicted behavior.

86 8 Simulations

8.2.2 Changing the Overall Delay

A factor that has a major effect on the networks performance is the overall delay δ. Using

a constant gain of κI = 0.2 and increasing the delay δ from two to six gives Figure 8.4.

Again, one should check the stability of the lower loops first; in this case the lower loop

is always stable (see Figure 4.3). Pursuant to the plots below, increasing delay seems to

have a more linear influence on the growth of the Nyquist curve than the increase in the

integrator gain. A big difference is, though, that each delay adds another encirclement of

the origin.

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(a) κI = 0.2, δ = 2

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(b) κI = 0.2, δ = 4

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(c) κI = 0.2, δ = 5

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(d) κI = 0.2, δ = 6

Figure 8.4: Investigation of Increasing Overall Delay δ

Here, again originating from hbase, increasing overall delays δ are considered. The reciprocal eigenvaluesof Γ†FAsy and Γ†FBlk are shown in blue and red, respectively. The Nyquist curve (black) grows withincreasing δ. Every delay adds an encirclement of the origin. Eventually, the border of stability is crossed.A time-domain plot of Figure 8.4c is given on the next page.


Figure 8.5 shows two networks, GBlk and GAsy, with the same controller and an overall

delay of five. The trajectories of the first network slowly diverge from the steady state

while in the second case, slow convergence can be observed. This time-domain plot, again,

validates our predictions. Just as one would expect, more delay leads to longer oscillation

periods. Consequently, the rate of convergence is relatively slow. We will see later on that

this effect can be mitigated by compensating for the delay.

0 20 40 60

30

40

50

t

pi[t]

(a) GBlk

0 20 40 60

30

40

50

t

pi[t]

(b) GAsy


Corresponding to the frequency-domain plot in Figure 8.4 we use GBlk (red) and GAsy (blue) with ε = 0.08such that reciprocal eigenvalues without and with encirclements emerged. While the blue network (noencirclements) is still stable, the red network (two encirclements) is not. Note that convergence for high-delay networks is relatively slow. The time-domain simulation thus shows exactly the predicted behavior.

88 8 Simulations

8.2.3 Adding a Proportional Part

The use of a PI- rather than an I-controller gives

hPI(z) =z(κI + κP )− κP

zδ(z − 1) + z(κI + κP )− κP.

Its plot can be found in Figure 8.6. For the sake of this illustration we used a proportional

part up to κP = 0.6. The integrator gain always remained at κI = 0.2. All investigated

lower loops are stable, see Figure 4.3.

Characteristic for the P part is the expansion of the inner encirclements rather than

the outer ones. Also, a left shift of the encirclements’ center can be observed. This holds

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(a) κP = 0.2, κI = 0.2, δ = 2

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(b) κP = 0.3, κI = 0.2, δ = 2

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(c) κP = 0.5, κI = 0.2, δ = 2

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(d) κP = 0.6, κI = 0.2, δ = 2

Figure 8.6: Investigation of Additional Proportional Part κPHere, a proportional part is added to the controller of hbase, its Nyquist plot is given in black. Thereciprocal eigenvalues of Γ†FAsy and Γ†FBlk are shown in blue and red, respectively. The proportionalpart leads to an expansion and a left shift of the innermost encirclement. A time-domain plot of Figure 8.6cis given on the next page.


true for increasing δ.

Especially the left shift can be seen as a major disadvantage of the PI controller since

the stability-critical reciprocal eigenvalues typically approach the Nyquist plot from the

left. Another disadvantage of PI-control in our particular case is that it is not realizable

with binary feedback. Thus it will be disregarded for most parts in the later analysis. In

the information feedback layout it also seldom contributed positively – only for a small set

of parameters the proportional part is stabilizing.

In Figure 8.7 we can see the influence of the proportional part in the time-domain. As

expected, the stepsize grows with κP . This should lead to a faster convergence of systems

far away from pss. The danger of the bigger steps is that there will be overcompensation

for small errors and thus prolonged oscillations may occur. This might even destabilize the

system, which is the case for the red plot below.

0 20 40 60

30

40

50

t

pi[t]

(a) GBlk

0 20 40 60

30

40

50

t

pi[t]

(b) GAsy


Corresponding to the frequency-domain plot in Figure 8.6, we used GBlk (red) and GAsy (blue) withε = 0.08 such that reciprocal eigenvalues without and with encirclements emerge. While the blue network(no encirclements) is stable, the red (two encirclements) is not. The time-domain simulation thus showsexactly the predicted behavior.

90 8 Simulations

8.2.4 Adding Filters

In order to deliver insight into the effect of filtering, we plot the Nyquist curves in Figure 8.8.

We use the exponential-forgetting and the local-average filter, both were introduced in

Section 2.3.2. The parameters were chosen such that three to five samples were considered

by both the forgetting and the averaging filter. For better comparability, the unfiltered

system was also plotted. Since the filter’s influence on the Nyquist curve is rather small,

we show all plots of one filter in one figure.

We can see that filtering leads to an expansion of on the Nyquist curve. The more

samples are considered, the larger the growth. We see that the local-average filter has a

bigger effect than the exponential forgetting filter. Since both filters fulfill f(1) = 1, our

derivations in 8.2.1 hold and only non-maximal eigenvalues threaten stability. Moreover,

filtering the signal has a bigger influence than filtering the interference. This was clear

from the relative gains of the upper and lower loop.

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(a) Fi = fef , κI = 0.2, δ = 2

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(b) Fi = fla, κI = 0.2, δ = 2

Figure 8.8: Investigation of the Influence of Filtering

Comparison between the exponential-forgetting filter (Figure 8.8a) and the local-average filter (Fig-ure 8.8b). The smallest curve always corresponds to the unfiltered system; then three to five samplesare considered by the filters, leading to growing Nyquist plots. We see, however, that the influence offiltering is only slight and that the biggest eigenvalue stays outside the Nyquist curve.


8.2.5 An Interesting Observation

The following observation was made during the course of our investigations. There seems

to be a linear relationship between the stability regions of the lower loop and those of

the whole system, including the interference. As a small example consider Table 8.2. For

consistency with the rest of the chapter we took the examples of GAsy and GBlk and a

delay from two to six. The regions of stability for the lower loop are depicted in Figure 4.3.

The stability for the MIMO system was investigated with Theorem 5. We see that the

maximal stable κI can be deduced for all delays δ when it is known for one δ. Note that,

for notational convenience, we denoted δ = i with δi and rounded all gains to two digits in

the table below.

Table 8.2: Maximal κI for different Systems

δ2 δ3 (δ3 : δ2) δ4 (δ4 : δ2) δ5 (δ5 : δ2) δ6 (δ6 : δ2)

h 1 0.62 (0.62) 0.44 (0.44) 0.35 (0.35) 0.28 (0.28)hGAsy

0.64 0.39 (0.62) 0.28 (0.44) 0.22 (0.35) 0.18 (0.28)

hGBlk0.55 0.34 (0.62) 0.24 (0.44) 0.19 (0.35) 0.16 (0.28)

This observation is solely of heuristic nature and thus in need of solid proving. Unfor-

tunately, this task has to be handed to the next researcher due to the time constraints of

a Master’s thesis.

92 8 Simulations

8.3 Influence of Time-Delay Compensation

Here, we will show the advantages of implementing TDC in our system as well as its

drawbacks. The focus, however, will lie on the effect of errors in the internal model of the

TDC.

8.3.1 Under- and Overcompensating the Delay

One thing that we assume to be not precisely known in the base station is the overall delay.

It might even be variable. Since we do not have any means of measuring the delay in real

time, the TDC has to be designed with a compensation that actually benefits the systems

performance for all expected delays.

Consider the example from Chapter 6. There we assumed a total delay of six samples,

two per controller and one per propagation. This might be extremely high for the 3G

system but more delays make it easier to showcase the benefit of compensating for it.

Furthermore, wireless communication with higher update rates or cheaper hardware might

experience delays in this magnitude.

Figure 8.9 gives some insight into under- and overcompensation. Both the classical

and the extended TDC are drawn in each plot. In the first plots, where there is only

slight under/overcompensation, also the uncompensated system is drawn. We see that

with growing error δε the Nyquist plots of both the traditional and the extended TDC

grow. We also see that for undercompensation, the worst case equals the uncompensated

case. Furthermore, we see that the extended TDC is more robust to errors in the delay

than the traditional TDC.

8.3 Influence of Time-Delay Compensation 93

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(a) δε = −1, δ∗ε = −1

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(b) δε = +1, δ∗ε = +1

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(c) δε = −3, δ∗ε = −3

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(d) δε = +3, δ∗ε = +3

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(e) δε = −4, δ∗ε = −6

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(f) δε = +4, δ∗ε = +6

Figure 8.9: Nyquist plot of Under and Overcompensation by TDC

Figure (a)–(f) show the reciprocal eigenvalues of the considered systems’ interference in red and blue. Allplots also feature the unit circle, plotted dashed. The uncompensated system is only given in the firstrow (black). We compare the behavior of the traditional TDC (lila) and the extended TDC (green) withrespect to under/overcompensation. The errors are given below the plots. Here, the star denotes theerror in the extended TDC. Note especially Figure 8.9e where the compensated systems as well as theuncompensated coincide.

94 8 Simulations

8.3.2 Under- and Overestimating the Gain

The next source of errors in the compensation is under- or overestimation of the mobiles’

gains. This might happen when user use different update-laws which might not be exactly

known in the base station. We, again, consider the example system from Chapter 6. The

gain in the mobiles’ controller is assumed to be κI = 0.2. A higher gain would benefit the

rate of convergence but also lead to instability in an uncompensated high-delay network.

Intuition on the effect of under and overestimation can be found in Figure 8.11. Again,

both the classical and the extended TDC are plotted in comparison. We see that underes-

timating the mobile’s gain leads to a growth of the Nyquist curve for both compensators.

Most interestingly, the opposite effect can be observed when considering overestimation.

Since smaller curves are desirable, so seems to be overestimation. Note, however, that at

some point this error threatens the internal stability of the base-station controller. This

can be observed in Figure 8.11f as well as a time-domain simulation (Figure 8.10). For the

regions of internal stability, cf. Chapter 6.

0 20 40 60 80 100 120 140 160 180

25

30

35

t

pi[t]

Figure 8.10: Time-Domain Plot of Figure 8.11f

Note that no conclusions can be drawn from Figure 8.11f since the lila plot is based on an internally-unstable base-station controller. We see that the extended TDC converges slowly to the steady state whilethe internally-unstable traditional TDC does not. The time-domain plot thus shows exactly the predictedbehavior.

8.3 Influence of Time-Delay Compensation 95

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(a) ∆κI = −0.05

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(b) ∆κI = +0.05

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(c) ∆κI = −0.10

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(d) ∆κI = +0.20

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(e) ∆κI = −0.15

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(f) ∆κI = +0.30

Figure 8.11: Nyquist plot of Under and Overestimation by TDC

Figure (a)–(f) show the reciprocal eigenvalues of the considered systems’ interference in red and blue. Allplots also feature the unit circle, plotted dashed. The uncompensated system is only given in the firstrow (black). We compare the behavior of the traditional TDC (lila) and the extended TDC (green) withrespect to under/overestimating κI . The errors are given below the plots. Note especially Figure 8.11fwhere the traditional TDC gets internally unstable.

96 8 Simulations

8.3.3 Rule of Thumb

While investigating the effects of the Time-Delay Compensation we made the following

observation: it is preferable to undercompensate in contrast to overcompensate for delays.

Furthermore, the opposite holds for the estimation of the mobiles’ integrator gains. Thus,

we conclude the following rule of thumb:

Rule of Thumb 3. When implementing Time-Delay Compensation, always compensate

for the lowest assumed-possible delay and the highest assumed-possible gain that leads to

an internally stable controller for improved stability.

As an example, consider once more the example from the last section with a com-

pensation for only half the delay and overestimation the mobiles’ by 100%, i.e. δε = −3

and κI,TDC = 0.4. Even though the considered errors where vast, the Nyquist plot sug-

gests stability, which could be backed up by simulations. Note, though, that for delay-

compensated systems higher gains in the mobiles are possible. For example, doubling the

mobiles gains and assuming we make the same errors, we still get a stable system – at least

for an extended TDC (for both observations see Figure 8.12).

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(a) δε = −3, ∆κI = +0.2

-4 -3 -2 -1 0 1 2-2

-1

0

1

2

Im

Re

(b) Same with κI = 0.4

Figure 8.12: Additional TDC examples

In both plots we see the Nyquist curves of the traditional TDC (lila) and the extended TDC (green), theunit circle (black, dashed) and the reciprocal eigenvalues of two considered linearized interferences. Theleft plot showcases the stability against sizable errors in the internal model of the TDC:s; the right plotdoubles the mobiles gain while having the same errors. We see that the extended TDC allows for highergains even though errors of 50% were made for both model parameters.

8.4 Influence of Binary Feedback 97

8.4 Influence of Binary Feedback

So far, we have just considered information feedback in our simulations. Adding binary

feedback to our block diagram leads to Figure 7.2 which is readily included into our

Simulink model. Note, however, that the sign of 0 has to be defined manually to ei-

ther plus or minus 1 since we only have a binary signal c which leads to an increase or

decrease in the mobile’s power output. This is no inconvenience, though, since an error of

zero is virtually impossible to achieve with binary feedback.

The assumption of information feedback can be seen as a linearization of the sign

function. Consequently, the obtained stability results only hold locally for binary feedback.

In general, good mapping is given for high update rates and small step sizes/integrator

gains in the mobile.

In Figure 8.13, we show three possible cases. First, a rather good approximation where

both, the information and the binary feedback have similar trajectories up to the point

where the binary feedback converges to its steady-state oscillation while the information

feedback converges to the actual steady state. Second, a case where we predicted instability,

but the binary feedback indeed stabilizes our system. We could also construct the inverse,

a case where the binary feedback destabilizes the system. This, however, was only possible

by having a certain stepsize for the binary feedback. As a fourth plot we present an

observation we frequently made when simulating binary feedback. Here, the steady-state

oscillation of the output powers is below the expected pss. The reason for this gets clear

when considering the SIR instead of the powers. Keep in mind that a small increase in the

target SIR leads to a sizable increase in the steady-state powers due to their non-linear

relationship. Thus, there exist oscillations of the SIR around a γ < γ†. Those oscillations

are stable even though our controller reference is γ†. The reason for this is the rough

discretization and high minimal stepsize entailed by the binary feedback. Note that in

those cases, the actual SIR is more often than not below the target SIR. This has to be

taken into account when deciding on the target SIR. In the information case it may be the

minimal SIR leading to sufficient quality of service. Introducing binary feedback leads to

98 8 Simulations

a target SIR that has to be significantly above that.

The reason for the binary feedback often being stable, even though the linearization

constitutes the opposite, is the following. In the information feedback, we basically feed

back the error. If the error grows, so does the input to our integrator. Consequently, it

happens easily that we leave the region of stability. In the binary case, however, we only

increment or decrement the mobiles powers by a fixed number. To include this into the

linearization would mean to have a variable gain, κvarI,i [t] = κI/ei[t]. This gets obvious from

the following small example:

Consider a case where the controller error ei is large, e.g. 6. With information feedback

and gain κI = 0.5 we have a variation in the power output of 3. The binary feedback this

variation is always κI = 0.5 and thus less likely to leave the region of stability. A drawback,

however, is that for small errors the same stepsize holds and thus near the steady state it

is more dangerous to use binary feedback than information feedback.

0 20 40 60 80 100

30

40

50

t

pi[t]

(a) Unexpected Stability

0 20 40 60 80 100

40

60

t

pi[t]

(b) Unexpected Instability

0 10 20 30 40 5022

23

24

25

t

pi[t]

(c) Good Approximation

0 10 20 30 40 5028

30

32

34

t

pi[t]

(d) Unexpected Lower SIR

Figure 8.13: Information versus Binary Feedback

We plot the trajectories of the system with binary feedback in blue, the one with information feedback inred. The steady states are given in black, dashed. We show the following three possible cases: Instabilityof the binary system even though the Nyquist analysis predicted stability, stability even though instabilitywas predicted and good tracking. Additionally, in Figure 8.13d, we show a common feature with binaryfeedback, oscillation around a lower SIR.

8.5 Influence of Varying the Gain Matrix 99

8.5 Influence of Varying the Gain Matrix

In order to show in how far our results – which where all based on having a fixed G – hold

for a system with varying gain matrix we show the following plots where we consider two

systems. In Figure 8.14, we investigate an overall delay of 2, where δb = 1 and δm = 1 and

compare different base-station controllers. In Figure 8.15, we compare the performance of

the two considered TDC:s on a system with an overall delay of 6, where δb = δm = 2 and

δp = 1. Lastly, we compare binary feedback to information feedback in Figure 8.16. The

gain of the mobile is 0.6, if not mentioned otherwise.

In all plots we show the output powers (dotted) as well as the resulting SIR (solid). The

gain matrix is initialized having a mean value of 0.8 on the diagonal and 0.025 for all other

entries. The random values are drawn uniformly with a deviation of 0.2 on the diagonal

and 0.02 for all other entries. The target SIR and noise is fixed to be γ†i = σi = 10.

Generally, we can state the following: Our predictions, which where solely based on the

linearization around the steady state, map adequately to the non-linear case. The TDC

improves the system’s stability to the cost of rate of convergence when far away from the

steady state. However, once near the – here varying – steady state, better performance can

be observed. Also, higher gains are possible (stable) but do not drastically improve the

rate of convergence (not shown). The extended TDC outperforms the traditional TDC if

there are delays in the base station. When considering binary feedback, slower convergence

far away from the steady state and oscillation in the steady state can be observed. Note,

however, that due to a rapidly changing steady state, the behavior of information feedback

is comparable to the behavior of the binary feedback.

100 8 Simulations

0 10 20 30 40 50 60 70 80 90 100

10

20

30

t

γ[t],pi[t]

No TDC (17.568)

TDC (14.8358)

(a) No Compensation vs. TDC

0 10 20 30 40 50 60 70 80 90 100

10

20

30

t

γ[t],pi[t]

No TDC (17.568)

TDC* (14.4922)

(b) No Compensation vs. TDC*

Figure 8.14: Investigating TDC for a Changing Gain Matrix (low delay)

Here we compare the performance of uncompensated, traditional and extended TDC for δb = 1, δm =1, δp = 0 and κI = 0.6. We plot both the output powers (dotted) and the emerging SIR (solid). Addition-ally, the target SIR of 10 is given by a black line. For better comparability we compute the root-mean-squareerror of the SIR for 200 timesteps (in parenthesis). We see that the faster slope of the uncompensatedcase leads to overshot and thus to worse performance once in a region around the varying steady state.

8.5 Influence of Varying the Gain Matrix 101

0 10 20 30 40 50 60 70 80 90 100

10

20

30

t

γ[t],pi[t]

TDC (27.3817)

TDC* (16.0693)

(a) TDC vs. TDC* (no error)

0 10 20 30 40 50 60 70 80 90 100

10

20

30

t

γ[t],pi[t]

TDC (24.8746)

TDC* (16.9294)

(b) TDC vs. TDC* (high error)

Figure 8.15: Investigating TDC for a Changing Gain Matrix (high delay)

Here we compare the performance of traditional and extended TDC for δb = 2, δm = 2, δp = 1 and κI = 0.6in the upper and κI = 0.35 in the lower plot. The lower plot also includes the following prediction errors:δε = −3, ∆κI = 0.35. We plot both the output powers (dotted) and the emerging SIR (solid). Additionally,the target SIR of 10 is given by a black line. For better comparability we compute the root-mean-squareerror of the SIR for 200 timesteps (in parenthesis). In the upper plot we can observe what looks likemarginal stability for the traditional TDC due to the high delay. Consequently, when also introducingerrors in the TDC in the lower plot, we lowered the mobile’s gains to stabilize the TDC. Note that theextended TDC has no issues with the higher gain since it compensates for 2 delays more.

102 8 Simulations

0 10 20 30 40 50 60 70 80 90 100

10

20

30

t

γ[t],pi[t]

Information Feedack (17.568)

Binary Feedback (18.9967)

(a) Information Feedback vs. Binary Feedback (low delay

0 10 20 30 40 50 60 70 80 90 100

10

20

30

t

γ[t],pi[t]

Information Feedback (25.1996)

Binary Feedback (22.6442)

(b) Information Feedback vs. Binary Feedback (high delay)

Figure 8.16: Investigating Binary Feedback for a Changing Gain Matrix

Here we compare the performance of control with information and binary feedback for δb = 1, δm = 1,δp = 0 and κI = 0.6 in the upper and δb = 2, δm = 2, δp = 1 and κI = 0.2 in the lower plot. We plotboth the output powers (dotted) and the emerging SIR (solid). Additionally, the target SIR of 10 is givenby a black line. For better comparability we also compute the root-mean-square error of the SIR for 200timesteps (in parenthesis). For high delay we lowered the mobile’s gain in order to get a stable informationfeedback – binary feedback tolerates approximately five times higher gain.

8.6 Conclusion 103

8.6 Conclusion

In this chapter, we used simulations to back up our earlier assessment. In the course of

doing so, we gave some insight into the impact of various parameter changes. Since there

are a lot of variables in the considered system, we mostly changed the parameters one by

one.

We could see that our analysis, which was only done based on a linearization around

the steady state, gives good insight into the system’s behavior. This holds as long as we do

not consider the binary feedback. For the binary feedback, we saw new phenomena emerge

(e.g. the oscillation around a lower target SIR and improved overall stability).

Lastly, we took into account that our analysis was based on the assumption that the

gain matrix is static in the timescale of the control feedback. Some simulations with

randomly changing gain matrices showed that our analysis holds in that framework to a

certain extend.

104 8 Simulations

Chapter 9

Conclusions

During the course of this work, we modeled the uplink power control of the well-established

3G network. Our point of departure was Gunnarsson’s PhD thesis [6].

The model we derived is a MIMO system. However, we exclusively analyzed the homo-

geneous decentralized case where the actual power control loop is identical and decoupled

for every mobile and the only existing coupling is done by the interference between the

mobiles. The first part, the homogeneity assumption, could be generalized since channel

properties, e.g. delays, might very well vary. The second part, coupling only by interference,

is due to the distributed nature of our system and thus valid.

Note that we did focus on the analysis of the linearized system. It is linearized in

two ways: on the one hand, we linearized the interference around the steady state of the

mobile’s output powers at the desired Signal-to-Interference Ratio. On the other hand, we

considered information feedback, i.e. the mobile receives a control signal u ∈ R, while in the

real-world 3G system there is only binary feedback; that means only a decrease/increase bit

is sent. The validity of our first linearization could be seen through simulation, where the

actual interference was calculated. The second assumption proved to be less maintainable.

Therefore, we added a chapter on describing-function analysis. Note, however, that due to

similarities in communication networks the former analysis still has ample applications.

The analysis of the linearized system comprised stability as well as performance analysis.

The former was done with a modified Nyquist criterion. Here, we could split the dynamics

105

106 9 Conclusions

of the control loop from the dynamics of the linearized interference and thus facilitate the

analysis notably. The latter was done by finding bounds on the rate of convergence of

the linearized system. In order to do so, we utilized the results of the Nyquist analysis in

combination with scaling the system.

Issues of the binary feedback were investigated using describing functions. They could

give a good approximation of possible steady-state limit cycles of our system. The weakness

of this approach is that only the first harmonics are captured. For higher-order control

this might not suffice and a more involved approach, e.g. using more than the first Fourier

coefficient, could be needed.

Considerable efforts were made to find means to handle delays in our system. Again, de-

parting from Gunnarsson’s thesis, we investigated Time-Delay Compensation approaches.

While investigating the classical TDC we came up with an extended version. Both com-

pensators were then subject to internal-model-error robustness analysis. It turns out that

there a constraints on the applicability of both TDC:s. Nevertheless, if applicable, a com-

pensation can greatly improve stability to a small cost of performance.

9.1 Contributions 107

9.1 Contributions

While working on this Master’s thesis the following original results were obtained:

• A proof that the eigenvalues of the linearized interference function and the feasibility

matrix are the same.

• An easy-to-use case-specific Nyquist theorem (see Theorem 5) that allows to split

the interference from the lower loop. Note that ideas from [2] where used in its

derivation.

• A high-order time-delay compensator especially for the case that there exist delays

in the base station due to computation or less frequent power-update signaling was

found.

• Specific describing functions methods for the MIMO case could be found, using sim-

ilar separation as in the Nyquist analysis. This was a left-open question in [7].

• Code for the implementation of the discussed contributions was generated in Math-

ematica and a Matlab/Simulink model was build to validate our predictions.

9.2 Possible Extensions

Further work is possible extending this thesis in a number of areas. Most noteworthy are

the following:

• Relaxing the homogeneity assumption in the modeling, i.e. allowing each channel to

behave differently.

• Finding a more general approach to linearizing the interference, e.g. linearize it

around the manifold of feasible steady states.

• Verifying our heuristic observation from Section 8.2.5 or proving it wrong.

• Extending the describing functions method to include more harmonics.

108 9 Conclusions

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Delay-Stability of Power Control in Wireless Networks