stability analysis of multiple time-delay systems with ......stability analysis of multiple...

STABILITY ANALYSIS OF MULTIPLE TIME-DELAY SYSTEMSWITH APPLICATIONS TO SUPPLY CHAIN MANAGEMENT

A Dissertation Presented

by

Ismail Ilker Delice

to

The Department of Mechanical and Industrial Engineering

in partial fulfillment of the requirementsfor the degree of

Doctor of Philosophy

in the field of

Mechanical Engineering

Northeastern UniversityBoston, Massachusetts

May, 2011

Contents

List of Figures ix

List of Tables x

Nomenclature xi

Acronyms xiv

Abstract xvii

1 Motivation of the Research 1

2 Problem Statements and Preliminaries 5

2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Delay-Dependent Stability (DDS) . . . . . . . . . . . . . . . . 8

2.2.2 Delay-Independent Stability (DIS) . . . . . . . . . . . . . . . 13

2.3 Existing Limitations in Analyzing Stability . . . . . . . . . . . . . . . 15

2.3.1 Stability Analysis in Laplace Domain . . . . . . . . . . . . . . 16

3 Opportunities 20

3.1 CTCR Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 Identification of Critical Hypersurfaces and Crossing Frequency

Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

i

CONTENTS

3.1.2 Observation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Resultant Theory, Discriminant and Descartes Rule concepts . . . . . 23

3.2.1 Geometric Interpretation of Discriminant in a 3D Topology . . 25

3.2.2 Observation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Supply Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Observation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Delay-Dependent Stability Analysis of Multiple Time-Delay Sys-

tems 30

4.1 General Approach: Advanced Clustering with Frequency Sweeping

Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1.1 Theoretical Construct of ACFS Methodology . . . . . . . . . . 33

4.1.2 Algorithmic Construct of ACFS Methodology . . . . . . . . . 38

4.1.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.4 Changes in PSSC for perturbations in fixed delay values . . . 44

4.1.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Specific Problem: Extraction of 3D Stability Switching Hypersurfaces 47

4.2.1 Features of Stability Switching Curves . . . . . . . . . . . . . 50

4.2.2 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Delay-Independent Stability Analysis 57

5.1 Delay-Independent Stability Analysis for MTDS . . . . . . . . . . . . 58

5.1.1 Discriminant of Resultant RT` with Repeated Factors . . . . . 62

5.1.2 Delay-Independent Stability Test on the L-D delay domain . . 67

5.1.3 Delay-Independent Stability Test on the 2D delay domain . . 69

5.1.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.1.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Delay-Independent Controller Synthesis with Sufficient Conditions . . 73

5.2.1 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

ii

CONTENTS

6 Time-Delay Systems in Supply Chain Management 83

6.1 Literature Review of Supply Chains . . . . . . . . . . . . . . . . . . . 83

6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.2.1 Mathematical Modeling of Delays . . . . . . . . . . . . . . . . 87

6.2.2 Mathematical Modeling of the Supply Chain . . . . . . . . . . 88

7 Contribution to Supply Chain Management 91

7.1 Inventory Dynamics in Supply Chains with Three Delays . . . . . . . 91

7.1.1 Characteristic Equation of APIOBPCS with Three Delays . . 91

7.1.2 Stability Analysis of a Supply Chain with Three Delays . . . . 93

7.1.3 Extracting Stability Switching Curves . . . . . . . . . . . . . . 94

7.1.4 Ordering-Policy Design for Delay-Independent Stability . . . . 97

7.1.5 Repercussions to Supply Chain Management . . . . . . . . . . 102

7.2 Generalized Supply Chain Model . . . . . . . . . . . . . . . . . . . . 106

7.2.1 Development of the Model . . . . . . . . . . . . . . . . . . . . 106

7.2.2 ACFS Application to Inventory Regulation Problem . . . . . . 111

7.2.3 Supply Chain Management in the Presence of Multiple Time-

Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8 Conclusions and Future Work 123

8.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

A Derivation of Line Equation in Section 4.2 127

References 129

iii

List of Figures

2.1 Pure delay model and its effect. . . . . . . . . . . . . . . . . . . . . 6

2.2 Schematic representation of 1D and 2D stability maps on delay do-

main. Green points or curves show stability switchings. . . . . . . . 9

2.3 Correspondence between complex s-plane, delay domain and time

domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1 a) 3D figure of F (ν, µ1, µ2) = 0, b) 2D figure of F (ν, µ1, µ2) = 0 on

the ν − µ1 plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Discriminant of F (ν, µ1, µ2) = 0 with respect to µ2, Dµ2(F ) = 0. . . 26

4.1 Case 1: Stability map for τ3 = 1.5. Shaded regions are stable. . . . . 40

4.2 Case 1: Stability map for τ3 = 4.0. Shaded regions are stable. . . . . 41

4.3 Case 1: Amplitude of frequency versus index of frequency for τ3 = 1.5

and τ3 = 4.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4 Case 2: Stability map for τ3 = 0.169 and τ4 = 0.26. Shaded region is

stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Case 3: Stability map for τ3 = 0.0. Shaded region is stable. . . . . . 43

4.6 Case 3: Stability map for τ3 = 0.06. Shaded region is stable. . . . . . 44

4.7 Part of the kernel curve of (4.14) for τ3 = 8 (red color) and τ3 = 8.05

(magenta color); dτ3 = 0.05. Larger arrows indicate bigger changes

in PSSC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

iv

LIST OF FIGURES

4.8 PSSC of (4.15) for τ3 = 0.3, τ4 = 0.16 (red and blue color) and

τ3 = 0.29, τ4 = 0.17 (magenta and yellow color); dτ3 = −0.01 and

dτ4 = 0.01. Larger arrows indicate bigger changes in PSSC. . . . . . . 46

4.9 Flow chart of the proposed procedure in Section 4.2. . . . . . . . . . 51

4.10 Case 1: 3 dimensional depiction of ℘kernel in (τ1, τ2, τ3) for τ4 = 0.197,

τ5 = 0.076, τ6 = 0.013, τ7 = 0.1, τ8 = 0.147, τ9 = 0.228 and τ10 =

0.11. Gray-scale color coding represents ω ∈ Ω1 correspondence. . . 53

4.11 Case 2: 3 dimensional depiction of a part of ℘kernel in (τ1, τ2, τ3) for

τ4 = 0.0197, τ5 = 0.076, τ6 = 0.13, τ7 = 0.03, τ8 = 0.026, τ9 =

0.022 and τ10 = 0.1. Gray-scale color coding represents ω ∈ Ω1

correspondence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


τ4 = 0.0197, τ5 = 0.076, τ6 = 0.13, τ7 = 0.03, τ8 = 0.026, τ9 =


correspondence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55


τ4 = 0.0197, τ5 = 0.076, τ6 = 0.13, τ7 = 0.03, τ8 = 0.026, τ9 =


correspondence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.14 Case 2: 3 dimensional depiction of ℘ and the stability map in (τ1, τ2, τ3)

for τ4 = 0.0197, τ5 = 0.076, τ6 = 0.13, τ7 = 0.03, τ8 = 0.026, τ9 =

0.022 and τ10 = 0.1. Gray-scale color coding represents ω ∈⋃3k=1 Ωk

correspondence. System is asymptotically stable at the origin. . . . . 56

5.1 Case 1: Boundaries formed by α2k(k1, k2) coefficients. Controller

gains from the shaded region render the system delay-independent

stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

v

LIST OF FIGURES

5.2 Case 1: Comparison of the proposed method (color curves) and DDE-

BIFTOOL result (gray shaded regions) for τ1 = 100 and τ2 = 100 on

k1 − k2 domain. Gray color coding indicates the number of unstable

roots. White region indicates stability. . . . . . . . . . . . . . . . . . 78

5.3 Case 1: DIS regions are obtained for a1 = 7.1 (outer curve, damping

ratio ξ > 1), a1 = 4.5 (dashed red curve, damping ratio ξ = 0.9186),

and a1 = 3.4 (inner curve, damping ratio ξ = 0.694). Controller gains

from the closed regions render the system delay-independent stable

for a given a1 parameter. . . . . . . . . . . . . . . . . . . . . . . . . 79

5.4 Case 1: DDE-BIFTOOL result for τ1 = 0.1 and τ2 = 0.15 on k1 − k2

domain. Gray color coding indicates the real part σ of the rightmost

root. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.5 Case 2: Implicit functions of α2k(k1, k2) coefficients and delay-free

system stability condition (black color). Controller gains from the

shaded region render the system DIS. . . . . . . . . . . . . . . . . . 81

5.6 Case 3: Block diagram of closed-loop system, ξ > 0, ωn > 0. . . . . . 82

6.1 Combination of pure (dead-time) and first-order delay model and its

effect on step input. This type of model can represent decision making

delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.1 Block diagram representation of inventory dynamics displaying only

the parts leading to homogeneous delay differential equation (7.1). . 92

7.2 Simulation of block diagram in Figure 7.1 for λ = 1.0, h1 = 1, h2 = 4,

h3 = 3 weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.3 Intersection of unit circle and ω-dependent line equation as per (7.9)

and (7.13), ω is fixed. . . . . . . . . . . . . . . . . . . . . . . . . . . 97

vi

LIST OF FIGURES

7.4 Given τ3 delay value, the maximum αi is computed for different λ

values as part of the conditions guaranteeing the delay independent

stability of the supply chain. . . . . . . . . . . . . . . . . . . . . . . 99

7.5 Policy design for delay independent stability. . . . . . . . . . . . . . 102

7.6 Case 1: Stability map on h2 − h3 domain for fixed αi = 0.4 1/weeks,

λ = 2.5 weeks and dead-time h1 = 0 weeks. . . . . . . . . . . . . . . 103


λ = 2.5 and τ3 = 8 weeks. . . . . . . . . . . . . . . . . . . . . . . . . 104

7.8 Case 2: Inventory levels are adapting to a change of 10 units from an

initial 200 units to 210 units in Figure 7.1. h1 = 0.5, h2 = 5.5, h3 = 2

and λ = 2.5 weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105


λ = 2.5 and τ2 = 5 weeks. . . . . . . . . . . . . . . . . . . . . . . . . 105

7.10 Case 3: Inventory levels are adapting to a change of 10 units from an

initial 200 units to 210 units in Figure 7.1. h1 = 1, h2 = 4, h3 = 3

and λ = 2.5 weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.11 Schematic representation of the flow of products and information. h1,

h2, h3, h4 and h5 respectively denote human decision-making, produc-

tion, transportation, information of inventory level and information

of products in shipment delays. . . . . . . . . . . . . . . . . . . . . . 107

7.12 Block diagram representation of the supply chain model (7.33). C(s)

is either αi for proportional control or αi + αI/s for proportional-

integral (PI) control. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

vii

LIST OF FIGURES

7.13 Case 1: Stability map on h1−h2 domain for different β values, β = 0.5

(blue), β = 0.7 (black), β = 1.0 (red). Parameters αi = 0.4 1/weeks,

λ1 = 2.0 weeks, λ2 = 0.5 weeks, λ3 = 0.4 weeks and delays h3 = 1

week, h4 = 0.15 weeks, h5 = 0.4 weeks are fixed. For a given β value,

delay values chosen from the regions that include the origin reveal

stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.14 Case 1: Simulation of block diagram in Figure 7.12 for various β

values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.15 Case 1: Stability map on h1−h3 domain for different β values, β = 0.5

(blue), β = 0.7 (black), β = 1.0 (red). Parameters αi = 0.4 1/weeks,

λ1 = 2.0 weeks, λ2 = 0.5 weeks, λ3 = 0.4 weeks and delays h2 = 4

weeks, h4 = 0.15 weeks, h5 = 0.4 weeks are fixed. For a given β

value, delay values chosen from the regions that include the origin

reveal stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.16 Case 2: Stability map on h1 − h2 domain for different αI values,

αI = 0.0 (red), αI = 0.02 (black), αI = 0.04 (blue). Parameters

β = 1.0, αi = 0.4 1/weeks, λ1 = 2.0 weeks, λ2 = 0.5 weeks, λ3 = 0.4

weeks and delays h3 = 1 week, h4 = 0.15 weeks, h5 = 0.4 weeks are

fixed. For a given αI , delay values chosen from closed regions which

include the origin reveal stability. . . . . . . . . . . . . . . . . . . . . 119

7.17 Case 2: Simulation of block diagram in Figure 7.12 for various αI

values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

viii

LIST OF FIGURES

7.18 Case 2: Stability map on h1 − h3 domain for different αI values,

αI = 0.0 (red), αI = 0.02 (black), αI = 0.04 (blue). Parameters

β = 1.0, αi = 0.4 1/weeks, λ1 = 2.0 weeks, λ2 = 0.5 weeks, λ3 = 0.4

weeks and delays h2 = 4 weeks, h4 = 0.15 weeks, h5 = 0.4 weeks are

fixed. For a given αI , delay values chosen from closed regions which

include the origin reveal stability. . . . . . . . . . . . . . . . . . . . . 121

ix

List of Tables

2.1 Computing potential stability switching hypersurfaces: anticipated

computation times of existing techniques that perform point-wise

sweeping with nested loops. . . . . . . . . . . . . . . . . . . . . . . . 15

x

Nomenclature

C, C−, C+ Complex plane, open left half of complex plane

and open right half of complex plane

R, R+, R0+, RL+ The set of real numbers, the set of positive real

numbers, the set of nonnegative real numbers

and the set of L-vectors with components in R+

Z+, N The set of positive integer numbers and the set

of natural numbers including zero

j Imaginary unit, j =√−1

jR Imaginary axis of complex plane

s Laplace variable, s ∈ C

C+ C+ ∪ jR

ω Frequency, imaginary part of eigenvalue

¯ω, ω, ¯ω Exact lower bound, exact upper bound and con-

servative upper bound of frequency, respectively

f , g, h Original characteristic function, transformed char-

acteristic function, and characteristic function

for some delays are fixed in f or g

xi

NOMENCLATURE

Ω, Ω, Ω Crossing frequency set of f , g and h, respec-

tively

℘, ℘ Potential stability switching hypersurfaces and

projection of these hypersurfaces on to 3D/2D

delay domain

<(), =() Real part and imaginary part of a variable or a

function

I Identity matrix with appropriate dimensions

sup, inf Supremum of a set (smallest upper bound), In-

fimum of a set (greatest lower bound)

• Fixed value of a variable •

L The number of delays

N The system order

~T = T`L`=1 = (T1, . . . , TL) Pseudo-delay vector

~τ = τ`L`=1 = (τ1, . . . , τL) Delay vector

c` The commensurate degree of τ`

Rµi(p1, p2) The resultant of multivariate polynomials p1(ν, ~µ)

and p2(ν, ~µ) with eliminating µi, where i ∈ [1, r ],

and ~µ = µiri=1 = (µ1, . . . , µr)

End of proof

xii

Acronyms

ACFS Advanced Clustering with Frequency Sweeping

APIOBPCS Automatic Pipeline Inventory and Order Based

Production Control System

CFS Crossing Frequency Set

CTCR Cluster Treatment of Characteristic Roots

DDS Delay-Dependent Stability

DIS Delay-Independent Stability

LTI Linear Time-Invariant

(M)TDS (Multiple) Time-Delay System

PSS(C/H) Potential Stability Switching (Curves/Hypersurfaces)

SC Supply Chain

SCM Supply Chain Management

xiii

Abstract

Time-Delay Systems (TDS) arise in many applications from diverse areas such as

economy, biology, population dynamics, traffic flow and communication systems.

Asymptotic stability analysis of even linear time-invariant time-delay systems is

a notoriously complex task due to the NP-hard nature of the stability problem.

Additionally, consideration of multiple delays totally hampers the existing stability

analyses which are limited to less than three delays. There is still no comprehensive

treatment for the most general time-delay systems where the system order, the

number of delays or the rank conditions of the system matrices are not limited.

All the existing techniques are case-specific and derived only for lower order time-

delay systems. The main goal of this dissertation is to develop a stability analysis

procedure for the most general linear time-invariant multiple time-delay systems,

relaxing all the mentioned limitations.

A novel methodology, Advanced Clustering with Frequency Sweeping (ACFS),

is introduced for the delay-dependent stability (DDS) analysis of the most general

class of linear time invariant (LTI) time delay systems (TDS) with multiple delays.

Different from the literature, ACFS does not impose any restrictions in system

order, the number of delays and the ranks of the system matrices in the LTI-TDS

considered. ACFS owes these superiorities to an elegant way of cross-fertilizing the

resultant theory, frequency sweeping technique and the root clustering paradigms.

ACFS can achieve to directly extract the 2D cross-sections of the stability views

xiv

ABSTRACT

in the domain of any of the two delays. ACFS reveals the complexity measures of

the stability views as a function of system properties and a new formula that can

compute the precise lower and upper bounds of the only parameter, the frequency,

that ACFS sweeps. Furthermore, another stability technique is developed for the

treatment of sub-class of the general LTI-TDS.

Delay-independent stability (DIS) of a general class of LTI multiple time-delay

system (MTDS) is then investigated in the entire delay-parameter space. Stability of

MTDS may change only if their spectrum lies on the imaginary axis for some delays.

An analytical approach, which requires the inspection of the roots of finite number

of single-variable polynomials, is built in order to detect if the spectrum ever lies on

the imaginary axis for some delays, excluding infinite delays. The approach enables

to test the necessary and sufficient conditions of the delay-independent stability

of LTI-MTDS, technically known as weak DIS, as well as the robust stability of

single-delay systems against all variations in delay ratios. Moreover, general class

LTI-MTDS is investigated in order to obtain a control law which stabilizes the LTI-

MTDS independently of all the delays.

Delays exist in supply chains due to decision-making, production lead-time,

transportation times and lags in flow of information. Finally, developed techniques

are applied to inventory regulation problem. The presence of delays may cause poor

management in the supply chains which eventually leads to undesirable behavior (i.e.

oscillations) of inventory levels. These behaviors are examined via new developed

methodologies, which can characterize the inventory oscillations as a parameter of

multiple delays in supply chains. A generalized supply chain model is developed

departing from a commonly studied simpler one based on fluid-flow dynamics. In

the generalized model, in order to eliminate drift, deficit or surfeit of stocks in the

inventory levels, a proportional-integral (PI) decision-maker is implemented. In-

ventory oscillations are then characterized with respect to the parameters of the

xv

ABSTRACT

PI and some parameters inherent to the supply network. New stability techniques,

combined with the generalized supply network model, could provide both thorough

insight into better controlling the inventories in supply chains as well as manage-

rial interpretations. Hence, a novel ordering policy design with which the inventory

variations can be rendered insensitive to detrimental effects of delays is presented.

xvi

Acknowledgements

First and foremost, I would like to thank my advisor Prof. Rifat Sipahi, not only

for his contributions to this research, but also for the outstanding guidance from

the starting date of my studies until the graduation. He has been supportive, en-

couraging advisor to me all the past four years. He is also a very good listener and

I am very thankful to him for listening my new ideas and I am very grateful to him

for fostering my professional growth in this dissertation by teaching every informa-

tion that he has. I would also like to thank my committee members, Prof. Dinos

Mavroidis, Prof. Nader Jalili and Prof. Surendra M. Gupta for their invaluable

ideas which undoubtedly augmented this research.

Secondly, I express gratitude toward our engineering department chair, Prof.

Hameed Metghalchi, National Science Foundation ECCS 0901442 for the support

in part and Prof. Sipahi’s start-up fund available at Northeastern University. I

would like to thank Dr. Elias Jarlebring for providing us the characteristic function

analyzed in Case 4.1.3 of Chapter 4. I also would like to thank the members of our

control laboratory, Payam Mahmoodi Nia, Wei Qiao, Andranik Valedi and Melda

Sener who have formed a very friendly and pleasant environment during my research.

Finally, I would like to thank my wife, Senay Demirkan Delice who is also a PhD.

candidate in the same department. Besides her own studies, she has supported me

in every situation. This dissertation would never have been completed without her

presence. I dedicated this research to my wife.

xvii

Chapter 1

Motivation of the Research

Time-delay systems (TDS) have been studied for its some certain properties partic-

ularly after Picard (1908) and Volterra (1931) works, see Hale (2006) for the detailed

history. Stability of TDS is one of the fundamental problems that triggered a 50 year

research effort in control systems community with an increasing intensity in the last

decade (Bellman and Cooke, 1963; Oguztoreli, 1966; Stepan, 1989; Hale and Ver-

duyn Lunel, 1993; Chen et al., 1995b; Dugard and Verriest, 1998; Richard, 2003;

Michiels and Niculescu, 2007). It is a fundamental problem since delays inevitably

exist in each part of the systems. In order to perform more accurate stability anal-

ysis, methods to cope with delays are needed. Since they affect system stability, the

presence and effects of delays can not be ignored, whether their magnitudes be very

small or quite large. In this dissertation, it is intended to develop new procedures

for stability analysis of TDS, and this section is devoted to explain how and where

these delays exist in real-life systems.

Delays arise in population dynamics (Kuang, 1993) due to fact that any species

(human or animal) need time to digest their food for their activities or to become

mature. Thus, their mathematical models have to consider these time lags. Simi-

1

CHAPTER 1. MOTIVATION OF THE RESEARCH

larly, biological systems have delays. Incubation models (Sharpe and Lotka, 1923),

neuron models (Plant, 1981), chemostat models (Monod, 1950; MacDonald, 1982)

and human respiration models (Michiels and Niculescu, 2007) with delays were in-

vestigated. Moreover, other biological topics e.g., epidemiology, neurophysiology

and microbiology were studied widely (MacDonald, 2008). In these days, there are

two more interesting and very crucial topics, HIV dynamics (Yi et al., 2008) and

leukemia (Niculescu et al., 2006; Peet et al., 2009), where traditional analysis can

not be applied due to time delay, as indicated in Yi et al. (2008).

Traffic flow problem represents human-in-the-loop dynamics (Sipahi et al., 2009b).

Since the human is a part of the dynamics, human add delays to the system due to

sensing and performing the appropriate actions in driving. Average delay range is

between 0.6 seconds and 2 seconds and its value depends on the drivers’ cognitive

and physiological states (Sipahi et al., 2011). These decision based delays inherently

exist in traffic flow and were noticed since 1950s (R. E. Chandler, 1958). Moreover,

different drivers bring heterogeneity, and traffic flow can not be considered as single

delay problem. Finally, delays in the traffic flow can not be ignored due to the fact

that stable delay-free traffic flow models can be unstable in the presence of delays

as reported in Sipahi et al. (2007).

Supply chains (SC) are an interconnection of various dynamics contributed by

customers, suppliers, manufacturing units, assembly lines, parallel running processes

and sources (Forrester, 1961; Sterman, 2000; Simchi-Levi et al., 2000; Delice and

Sipahi, 2009b; Sipahi and Delice, 2010). Supplies in supply chains flow towards the

direction of increased demand (from inventories to customers), while the informa-

tion about the demand flows in the opposite direction (from customer forecasting

to company headquarters). Among many objectives in managing SC, one of the

most critical ones is to regulate the inventory levels while successfully responding

2


to customer demand. This may seems like a simple task, however, in presence of

delays, supply chain management is known to be a challenge (Siegele, 2002).

Delays in SC arise from various different physical reasonings and constraints,

such as decision-making, manufacturing lead times, transportation and information

flow. Due to the presence of delays, what is currently occurring in the supply chain

is the after-effects of what has happened earlier. Consequently, any decision based

only on current observations in the SC is likely to be unsuccessful as those obser-

vations represent the past. The consequences are very well known in management

science and business (Sterman, 2000). Prof. Kalecki’s business cycle model is the

first mathematical treatment of business cycles which are basically self-sustained

oscillations (Kalecki, 1935). He observed that the delay between decision and in-

stallation of investment goods causes the business cycles. Delays also lead to ex-

cessive/depleted inventories and synchronization problems across parallel-running

processes and these effects may cost companies billions of dollars (Marion et al.,

2008).

From above statements, one can think that delays have detrimental effects on

supply chains or on the other systems; but it is not always true. For example,

decision-making delays may have positive effects on SC management, because wait-

ing may reveal a clear picture to managers regarding sale trends and the market

behavior. This wait time also contains the required time for perception of human

behavior towards deciding a new order (Sterman, 2000). With these fundamental

observations, it is impossible to conclude intuitively how delays may affect inven-

tory behaviors, in a positive or negative way. Hence, counter-intuitive results may

happen.

Time delays also exist in chemistry, mechanical vibrations, combustion engines,

steel rolling mill control, semiconductor laser systems, distributed systems, telema-

3


nipulation systems, congestion avoidance in high-speed internet (Niculescu, 2001;

Gu et al., 2003; Chiasson and Loiseau, 2007; Erneux, 2009; Sipahi et al., 2011).

As explained in SC example, delays may have positive effect on the stability of

these areas. For example, in order to reduce vibrations from blasting for break-

ing rock, delay is added between each blasts (Duvall et al., 1963). Moreover, time

delays on positive feedback loop can stabilize oscillatory systems (Abdallah et al.,

1993). Furthermore, there are systems that single delay can not stabilize, however,

adding a second delay can stabilize the same system (MacDonald, 2006). In order

to completely understand these complex and counter-intuitive effects of time delays,

stability analysis techniques for treating multiple time delays have to be developed.

Ignoring some of the delays is not a choice since like the case in MacDonald (2006),

removing second delay from the system makes it unstable for every value of the first

delay.

This chapter ends with quotations from Kuang (1993) on the importance of time

delays: “... time delays occur so often, in almost every situation, that to ignore them

is to ignore reality”, and he continues “... any model of species dynamics without

delays is an approximation at best”.

4

Chapter 2

Problem Statements and

Preliminaries

2.1 Problem Formulation

In this dissertation, one of the most important and unresolved problems of TDS is

studied: the asymptotic stability of linear time invariant (LTI) multiple time delay

system (MTDS) with respect to delays τ`. The system is expressed in state space

form as,

d~x(t)

dt= A ~x(t) +

L∑`=1

B` ~x(t− τ`) , (2.1)

where A ∈ RN×N , B` ∈ RN×N are constant system matrices; ~x(t) ∈ RN×1 is the

state vector. τ` are the nonnegative pure delays and they are basically shift operator

in time as shown in Figure 2.1. Different than the literature cited in Section 2.2, no

restriction is imposed here on the system order N , the ranks of A and B` matrices

as well as the number of delays L considered.

Recall that when the general class of multi-input LTI systems,

~x(t) = A ~x(t) + B ~u(t) , (2.2)

5

CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES

Pure delay ( )

model

Outflowat time

Time Time

Inflowat time 0

0

Figure 2.1: Pure delay model and its effect.

where B ∈ RN×M are the control matrix, M is the number of inputs and the system,

is closed by a feedback control law ~u(t), which is affected by multiple delays

~u(t) =L∑`=1

K` ~x(t− τ`) ∈ RM , (2.3)

where K` ∈ RM×N , ` = 1, . . . , L, are the control laws, LTI-MTDS in (2.1) is recov-

ered, B` = B · K`.

Characteristic function of the system in (2.1) is given by:

f(s, ~τ) =K∑k=0

Pk(s) e−s

∑L`=1 υk` τ` , (2.4)

where Pk are polynomials in terms of s with real coefficients, K ∈ Z+ and υk` ∈ N.

MTDS in (2.1) is a retarded class LTI-TDS since the highest order derivative of the

state is not influenced by delays. This corresponds to the case where P0 does not

multiply any terms carrying delays, υ0`L`=1 = ~0, and P0 has the highest power of

s in (2.4). Definition of asymptotic stability is provided next.

Definition 1. For a given ~τ = ~τ , MTDS (2.1) is asymptotically stable if and only

if

f(s, ~τ) 6= 0 , ∀s ∈ C+ . (2.5)

Due to the presence of transcendental terms, the characteristic function (2.4)

possesses infinitely many roots for a given set of delays, τ1, . . . , τL. The LTI-MTDS

is asymptotically stable for a given ~τ = ~τ if and only if the measure α (τ1, . . . , τL) =

6


sup<(s) |f(s, ~τ) = 0 is negative for ~τ , α(~τ) < 0 (Bellman and Cooke, 1963). Note

that this measure states that all roots of the system must be on the left hand side

of the complex plane for asymptotical stability. In other words, none of the roots

of the system must place at the right hand side of the complex plane as depicted in

Definition 1. Furthermore, the continuity of α holds with respect to the imaginary

axis (Datko, 1978) and with this knowledge stability transitions of the dynamics

can be studied via α (τ1, . . . , τL) = 0. This requires to investigate the imaginary

roots s = jω of (2.4), where ω ∈ R0+ without loss of generality. All nonnegative ω

values, where s = jω is a root of (2.4) for some positive delays, define the crossing

frequency set (CFS),

Ω = ω ∈ R0+ | f(jω, ~τ) = 0 , for some ~τ ∈ RL0+ , (2.6)

and ω ∈ Ω maps to at least a point ~τ as well as to all the infinitely many solutions

of (2.4),

(τ1, τ2, . . . , τL) + (η1, η2, . . . , ηL) .2π

ω, η`L`=1 ∈ NL , (2.7)

where (τ1, τ2, . . . , τL) are the minimum nonnegative delays in (2.7) without loss of

generality. The solutions in (2.7), considering all ω ∈ Ω, lie on the L dimensional

potential stability switching hypersurfaces (PSSH). They are denoted by ℘,

℘ = ~τ ∈ RL0+ | f(jω, ~τ) = 0 , ∀ω ∈ Ω . (2.8)

Among all the PSSH, there exists a special subset which constitutes the kernel

hypersurfaces, defined by ℘kernel = ℘ | η`L`=1 = ~0. It is easy to see that given

ω ∈ Ω and a point τL`=1 ∈ ℘kernel, one can generate the remaining infinitely many

solutions by increasing the counter η` in (2.7). In other words, kernel hypersurfaces

are the generators of infinitely many hypersurfaces called the offspring and defined

as ℘offspring = ℘ \ ℘kernel.

7


Delay-independent stability (DIS) lemma is given next.

Lemma 1 (Gu et al. (2003)). The system in (2.1) is delay-independent stable if

and only if condition in (2.5) is satisfied for all ~τ ∈ RL0+.

Lemma 1 explains the DIS concept, however, verifying the conditions in the

lemma is impossible due to the transcendental nature of (2.4). Instead, different

logic is followed by checking whether Ω is empty set in the following chapters. The

notions of weak and strong delay-independent stability is explained below. (Chen

et al., 2008).

Definition 2. If the system in (2.1) is weakly delay-independent stable, then ω = 0

solutions can be neglected. This is because, under these conditions, ω = 0 may

satisfy (2.4) only when some delays approach infinity. Such a possibility is, however,

excluded (or included) in the analysis of weak (or strong) delay-independent stability.

2.2 Stability

2.2.1 Delay-Dependent Stability (DDS)

Study of time-delay systems (TDS) has been an attractive research field since the

18th century with the works of Euler, Bernoulli, Lagrange and Poisson on functional

differential equations (Gu and Niculescu, 2003). Notable developments in the field

start with Volterra (1931) and Pontryagin (1942). Volterra’s and Pontryagin’s stud-

ies had breakthrough effects on TDS field and many other milestone studies have

followed subsequently (Bellman and Cooke, 1963; Oguztoreli, 1966; Halanay, 1966;

Hale and Verduyn Lunel, 1977; Gorecki et al., 1989; Stepan, 1989; Marshall et al.,

1992; Niculescu, 2001; Gu et al., 2003; Michiels and Niculescu, 2007).

Presence of delays leads to an infinite dimensional spectrum in (2.1) making

the stability assessment of (2.1) in delay parameter space a non-trivial task. The

8


2

Uns

tabl

e

Stab

le

Uns

tabl

e

Uns

tabl

e Unstable

Stable

(b) 2D stability map (a) 1D stability map

Stab

le

1

0

(0,0)

Figure 2.2: Schematic representation of 1D and 2D stability maps on delay domain.Green points or curves show stability switchings.

stability problem with respect to a single delay L = 1 is concerned with finding

intervals along the delay axis, where in these intervals any choice of delay leads to

asymptotic stability of (2.1) (Chen et al., 1995b; Olgac and Sipahi, 2002; Michiels

and Niculescu, 2007). The display of the stability with respect to delay parameter

is called as ‘stability map’ or ‘stability chart’ (Stepan, 1989), where this map is a 1D

nonnegative delay axis along which stable and unstable delay intervals are marked,

see Figure 2.2a (Cooke and van den Driessche, 1986). It is crucial to surface all

these intervals with their precise lower and upper bounds for the necessary and

sufficient conditions of asymptotic stability. In the case with L = 2, stability maps

are the displays of 2D stability/instability regions on the plane of two delays, see

Figure 2.2b (Stepan, 1989; Hale and Huang, 1993; Sipahi and Olgac, 2005; Gu et

al., 2005; Sipahi, 2008).

It is important to note that the literature review below is related to the scope

of this dissertation and thus it is narrowed down to those existing techniques that

avoid introducing any conservatism in assessing the stability of (2.1) with respect

to delays, see Richard (2003); Gu and Niculescu (2003) for a review of conservative

techniques. The case with single delay L = 1 with different difficulty levels can

be solved by numerous methods (Sipahi and Olgac, 2006b). Some developments

9


are Rekasius transformation (Rekasius, 1980), the solutions of argument and mag-

nitude conditions of the corresponding characteristic functions (Cooke and van den

Driessche, 1986), elimination of transcendental terms (Walton and Marshall, 1987),

resultant technique (Chiasson, 1988; Wang et al., 2004), utilization of matrix poly-

nomials and matrix pencil techniques (Niculescu and Ionescu, 1997; Niculescu, 1998;

Fu et al., 2006; Chen et al., 2007), kronecker product techniques (Louisell, 2001), fre-

quency sweeping ideas (Hsu and Bhatt, 1966; Olgac and Holm-Hansen, 1994; Chen

and Latchman, 1995), and surfacing clustering identifiers of the characteristic roots

(Olgac and Sipahi, 2002). Despite the existence of a variety of methods for L = 1

cases, stability analysis on the plane of two delays follows different paths, as direct

extensions of L = 1 case are prohibitive. The reason is that reducing a two delay

problem to a single delay problem by assuming τ2/τ1 is a rational number leads to

cumbersome analysis. Even sweeping this ratio infinitely many times will not cover

the entire (τ1, τ2) ∈ R2+ plane. This issue has been discussed from implementation

and mathematical points-of-view in the work Sipahi (2008); Niculescu (2001).

The main objective in 2D stability analysis is to construct all the potential stabil-

ity switching curves (PSSC) which partition the delay space into stable and unstable

regions. Obviously, the accuracy and completeness of the analysis strongly depends

on finding all the existing PSSC without any approximations. To the best of author

knowledge, the first attempts in analyzing stability for L = 2 delays are found in

Nussbaum (1978); Cooke and van den Driessche (1986); Stepan (1989); Hale and

Huang (1993). The most recent methods along this line start to arrive from 2002

on, with the work of Niculescu (2002); Sipahi and Olgac (2003b). Needless to say,

the cited works are implemented on case specific problems, limiting their extensions

to general treatment of two delay problems. This gap was bridged in 2005 by two

different methods, Sipahi and Olgac (2005) and Gu et al. (2005). In Sipahi and

Olgac (2005), 2D stability maps are extracted by using the Rekasius transformation

10


and an adaptation of Routh-Hurwitz tableau for a corresponding finite dimensional

problem and in Gu et al. (2005), the authors depart from the geometry of the

complex vectors, which is a triangle on the complex plane, in order to develop a fre-

quency sweeping approach (Chen and Latchman, 1995) for constructing the PSSC.

Three new techniques are observed after these publications, where in Ergenc et al.

(2007) and Jarlebring (2009), the stability problem is initially formulated differently,

but leads to the computation of generalized eigenvalues of a matrix pencil and in

Fazelinia et al. (2007), the authors identify PSSC by using the ‘Building Block’

concept.

The methodology in Sipahi and Olgac (2005) is called as Cluster Treatment

of Characteristic Roots (CTCR), and in the cited study the authors revealed some

properties about PSSC, such as invariance features of stability switching (sensitivity

analysis) behaviors of PSSC and presence of finite number of kernel curves, which are

actually the generators of all PSSC. In other words, detection of kernel curves suffices

to finding all PSSC, and stability analysis follows using the invariance property of

stability switching behavior once all PSSC are identified. It is important to state that

kernel curves and invariance property of PSSC exist independently of the approach

taken to analyze the stability as these properties are inherent to LTI-TDS.

In the case of three delays, L = 3, there have been only six studies in the liter-

ature Sipahi and Olgac (2006a); Almodaresi and Bozorg (2008); Jarlebring (2009);

Sipahi and Delice (2009); Sipahi et al. (2009a); Gu and Naghnaeian (2011), and see

Sipahi and Delice (2009) for the case with arbitrarily large number of delays. These

advancements are case-specific and there still exists no method to treat the stability

of the most general system in (2.1). As recognized in Sipahi and Delice (2009);

Jarlebring (2009), the limitations in the existing methodologies can be summarized

as follows;

(i) they require exponentially increasing computation times as they perform mul-

11


tiple parameter sweeping in nested loops when extracting the potential stability

switching hypersurfaces (PSSH) of the L dimensional stability maps,

(ii) they are case-specific,

(iii) they cannot extract the 2D or 3D cross sections of an L dimensional stability

map and therefore they are limited to treat L ≤ 3 problems.

One exception to (c) is our recent work Sipahi and Delice (2009) which still falls

short to treat the stability of (2.1). Although, some existing methods in theory can

be claimed to resolve the stability problem of (2.1), these methods cannot extract the

stability maps of (2.1) for L > 3. This can be partially attributed to the NP-hard

nature of the problem (Toker and Ozbay, 1996).

When there are more than three delays in the stability problem, the only venue is

to extract 2D and 3D cross sections of L dimensional stability maps. This idea was

introduced for the first time in the milestone work of Cooke and van den Driessche

(1986), where the respective authors attempted to solve a two delay problem with

their knowledge of solving a one-delay problem. They fixed the second delay and

investigated the stability intervals along the first one. This philosophy is also the

backbone of recent work Sipahi and Delice (2009) that extends the stability treat-

ment of a sub-class of (2.1) to arbitrarily large number of delays. In this research,

the same lines is followed with the difference that a new method is proposed to

extract the 2D PSSC (3D PSSH), that is, the 2D (3D) cross sections of the L di-

mensional stability maps of the most general MTDS (2.1), without needing to obtain

the L dimensional PSSH. In accomplishing this non-trivial effort, no conservatism

is introduced and computing in multiple nested loops is avoided by adapting the

frequency sweeping technique (Chen and Latchman, 1995) to the novel approach

developed to construct the PSSC.

12


2.2.2 Delay-Independent Stability (DIS)

The stability analysis of (2.1) requires to investigate the eigenvalues of (2.1) that are

on the imaginary axis of the complex plane for some critical delay values τ ∗ (Datko,

1978). It is these eigenvalues which may cross the imaginary axis at ∓jω and may

cause stability reversals/switches as τ ∗ is perturbed (Stepan, 1989; Gu et al., 2005).

The frequency parameter ω indicates the pathways of the eigenvalues across the

imaginary axis. In this sense, the set of all nonnegative ω values, called the crossing

frequency set Ω carries key information about the stability and spectral properties

of (2.1). In the delay-dependent stability case, Ω 6= ∅, that is, system’s stability

may change with respect to the delay parameter. Since the finite upper-bound of

Ω is known to exist (Hale and Verduyn Lunel, 1993), one can sweep ω in a range

starting from zero up to a conservative upper-bound in order to solve all the ∓jω

eigenvalues of a TDS. Although this is a graphical-based approach, frequency sweep-

ing methodology (Chen and Latchman, 1995) is applicable to robustness analysis

(Chen et al., 2008) and to extracting the stability features of MTDS in 2D (L = 2)

(Gu et al., 2005) and 3D (L = 3) delay space (Sipahi and Delice, 2009).

When Ω = ∅, however, system’s stability/instability becomes delay-independent.

Many papers are published along these lines, where delay-independent stability

(DIS) sufficient (Chen and Latchman, 1995; Chen et al., 1995a), and necessary

and sufficient conditions are proposed (Chen et al., 2008). The starting point in

many studies is that TDS cannot possess imaginary eigenvalues with respect to the

entire delay-parameter space. When L 6= 1, graphical display in all these analyses,

however, is inevitable in order to easily verify, by sweeping ω, whether or not larger

ω values reveal any eigenvalue solutions. There are other techniques to test DIS

of TDS. DIS conditions are studied in Gu et al. (2003) for subclasses of (2.1). In

another study, one of the most complicated MTDS is studied for robustness via fre-

quency sweeping (Chen et al., 2008), but the characteristic function treated in the

13


cited work does not cover the general problem in (2.1). Furthermore, the studies in

Kamen (1980); Thowsen (1982); Hertz et al. (1984); Chiasson et al. (1985); Gu et

al. (2001); Wang et al. (2004); Wei et al. (2008); Souza et al. (2009) are applicable

for only single-delay cases (L = 1), and Hu and Wang (1998); Wang and Hu (1999);

Wu and Ren (2004) are feasible only for two-delay cases (L = 2).

When L = 1; Kamen (1980) studies the DIS problem by means of two-variable

zero criterion, which is limited to single-delay problems. Since some trigonometric

identities are utilized in Kamen (1980) and Thowsen (1982), these methods remain

restricted to scalar TDS (N = 1), as recognized in Thowsen (1982), see also Chiasson

et al. (1985). Moreover, the resultant theory is applied to the DIS problem in Hertz

et al. (1984); Chiasson et al. (1985), followed by Gu et al. (2001) and Wang et

al. (2004) which use a similar logic, yet different set of two polynomial equations

for the resultant computation. Procedures in Hertz et al. (1984); Chiasson et al.

(1985); Gu et al. (2001); Wang et al. (2004) are applicable to only TDS with single

time-delay, with no restriction on system order. Furthermore, Wei et al. (2008)

transforms frequency sweeping conditions in Hale et al. (1985) to easily testable

algebraic conditions by utilizing the resultant theory. These conditions are, however,

valid for single-delay cases. Finally, Souza et al. (2009) concludes DIS property of

TDS, but with a single-delay, based on the roots of a polynomial constructed by

utilizing bilinear transformation.

When L = 2; Hu and Wang (1998) considers a specific second-order damped

vibration problem, which has only two time delays. The techniques in Wang and

Hu (1999); Wu and Ren (2004) are also limited to a specific dynamic system with

two delays. In all the cited papers, extensions to L > 2 cases is restrictive due to

two main reasons;

(i) The DIS test for L > 1 is an NP-hard problem since each delay needs to be

treated as an independent parameter (Toker and Ozbay, 1996).

14


(ii) The number of available equations to be solved for DIS analysis is less than

the number of unknowns in the respective analysis.

Finally, Linear Matrix Inequality (LMI) based conservative approaches (Gu et al.,

2003; Baser, 2003) and systems with time-varying delays (Zhang et al., 2006) are

kept outside the scope of this dissertation.

2.3 Existing Limitations in Analyzing Stability

The key for finding ℘ is the detection of Ω. For this detection, in Ergenc et al.

(2007); Jarlebring (2009), the substitution e−jωτ` := κ` ∈ C is utilized with |κ`| = 1;

and, in Sipahi and Olgac (2005) and Fazelinia et al. (2007), it is proposed that

e−jωτ` := (1 − jωT`)/(1 + jωT`), with κ` = T` ∈ R and κ` = ωτ` ∈ [0, 2π), re-

spectively. These choices, however, require to sweep L − 1 number of parameters

κ1, . . . , κL−1 in nested loops to solve for s = jω. The disadvantage of such a choice,

as recognized in Jarlebring (2009); Sipahi and Delice (2009); Sipahi (2007), is the

exponentially growing computation times, which are known to be in the order of

years, see Table 2.1 for their computation times, even for sweeping three nested

loops (Sipahi, 2007).

Considering the computational burden, one needs different ways to approach

Table 2.1: Computing potential stability switching hypersurfaces: anticipated com-putation times of existing techniques that perform point-wise sweeping with nestedloops.

L− 1Number of grid

Time needed to sweeppoints to sweep for a fixed N

1 103 30 seconds

2 (103)2 ≈ 8 hours

3 (103)3 ≈ 347 days

4 (103)4 ≈ 951 years

15


the stability problem. The visualization of ℘ is impossible, and 3D visualization is

restrictive. Hence, the best option is to extract the 2D cross-sectional views of the

L-dimensional ℘. In other words, L − 2 number of delays are kept fixed, and the

projections of ℘, denoted here by ℘, are extracted in the plane of the remaining

two delays. There are two ways of taking cross-sections;

(i) compute the projections ℘ directly in any pre-determined 2D delay parameter

space without the need to extract ℘, or

(ii) extract the L-dimensional ℘ first, and then numerically project ℘ onto a

considered two-delay space.

Clearly, option (i) is direct and less involved. This is what the develop methods

in this research follow for L > 3, which is in essence similar to what the authors in

Cooke and van den Driessche (1986) do when L = 2. Option (ii) is impossible to im-

plement due to exponentially growing computations times. Moreover, as recognized

in Jarlebring (2009); Sipahi and Delice (2009), none of the existing methodologies

can be adapted to option (i) when L > 3. The main reason for this is that the ex-

isting methodologies cannot take the cross-sections of the stability views, since they

cannot set some of the delays as constants prior to the stability analysis. This can

be easily seen in the algorithmic steps of the methods in Sipahi and Olgac (2005);

Ergenc et al. (2007); Jarlebring (2009); Fazelinia et al. (2007), where all the delays

are to be computed as an end result of the specific approach, that is, (ω,~κ)→ ~τ .

2.3.1 Stability Analysis in Laplace Domain

Stability analysis starts with identifying the stability of the origin of the L-D delay

space, ~τ = ~0. Let the number of unstable roots for ~τ = 0 be denoted by NU ≥ 0.

Due to continuity of α(~τ), NU in delay space may change only across PSSH. Since

delay values on PSSH render s = ∓jω roots of (2.4), the sensitivity of these roots

16


with respect to ~τ will show whether ∓jω roots tend to favor stability or instability.

This sensitivity expression when computed along any one of the delay axes τ` exhibits

some invariance properties for a given ω ∈ Ω. For instance, sensitivity of the s =

∓jω roots becomes independent of the delays that create these roots as a solution

of (2.4) (Michiels and Niculescu, 2007). By means of this invariance property, the

particular segments of PSSH can then be labeled as stability favoring or instability

favoring. This practical procedure enables a rapid way of identifying NU within

each closed L-D space encapsulated by PSSH. Obviously, when NU = 0, the TDS is

asymptotically stable, otherwise it is unstable. It is noted that the identification of

NU in the delay space is straightforward, once all the PSSH are precisely identified.

Detailed discussions on the calculation of NU can be found in Sipahi and Olgac

(2005).

Outline of Analyzing the Stability in the Presence of Two Delays

Before proceeding to multi-delay treatment in the following chapters, let us sketch

the outline of analyzing the stability in the presence of two delays τ1 vs τ2 (L = 2).

In this way, solving the complicated multi-delay problem in the next sections will

be appreciable.

À It is proven in (Datko, 1978) that the roots, s, of the characteristic equation

exhibit continuity property with respect to delay values ~τ = (τ1, τ2), i.e. s(~τ).

This means that if delay value is increased/decreased slightly from ~τ to ~τ ∓ ~ε

(|~ε| 1), roots s(~τ + ~ε) will be in ~ε-neighborhood of s(~τ) (Niculescu, 2001;

Gu et al., 2003).

Á Due to the continuity property, stability may only change when the roots

cross the imaginary axis since the imaginary axis is the boundary separating

the stable vs. unstable regions on the complex plane, Figure 2.3a. Conse-

quently, stability may only change when <(s) = 0. For detecting the stability

17


Im

Re

(a) Complex s -plane

(b) Delay domain

* *1 2,

s jfor

Region 2

Region 1

* *1 2( , )

1

Stable

Region

( )

Point C Point B

Point A 2

* *1 2,

s jfor

(c) Time domain

Time

Syst

em

resp

onse

Point A

Point B

Point C

Figure 2.3: Correspondence between complex s-plane, delay domain and time do-main.

18


transitions, one should analyze the characteristic function on the imaginary

axis, <(s) = 0, by setting s equals to jω, ω ∈ R.

Â Next, one detects all ω and delay(s) that satisfy the characteristic equation,

i.e., that render the configuration in Figure 2.3a. On the plane of τ1 and

τ2, these delay solutions form some special curves called potential stability

switching curves, ℘(~τ), see Figure 2.3b. These curves are the only geometric

locations in delay domain representing all possible stability transitions of the

dynamics. For instance, ℘(~τ) in Figure 2.3b separates stable region 1 from

the unstable region 2.

The main aim is to identify the potential stability switching curves ℘(~τ) com-

pletely and precisely in order to reveal the complete stability features. In each region

encapsulated by these curves (similar to Figure 2.3b), any choice of delays will ei-

ther cause stable or unstable system behavior. Connection between time domain

behavior of system levels and the stability maps is immediate by comparison of

Figure 2.3b and Figure 2.3c.

19

Chapter 3

Opportunities

In this section, several important advancements from the literature are highlighted

with the aim to prepare the reader for the main contributions of this dissertation.

One of these advancements is Cluster Treatment of Characteristic Roots (CTCR)

methodology, which has been introduced to address the stability of linear time-

invariant time-delay systems. Strength of the CTCR method is to convert infinite

dimensional control problem to algebraic control problem. Hence, the properties

of algebraic polynomials (i.e. resultant and discriminant concepts) will prove to be

useful in dissertation development and has to be reviewed. Finally, supply chain to

which the new results can be applicable is highlighted. With these reviews, their

limitations and the differences from the developed method in the following chapters

will be clear.

3.1 CTCR Methodology

First step of CTCR methodology constructs Ω and ℘kernel starting from (2.4). Con-

struction is done as in the following.

20

CHAPTER 3. OPPORTUNITIES

3.1.1 Identification of Critical Hypersurfaces and Crossing

Frequency Set

First, the exponential terms in (2.4) are replaced by Rekasius transformation (Reka-

sius, 1980),

e−τ` s :=1− T` s1 + T` s

, s = jω , T` ∈ R , ` = 1, . . . , L . (3.1)

Transformation (3.1) is exact for imaginary roots s = jω with the following back

transformation rule found by developing the argument conditions on both sides of

(3.1),

~τ =

(2 tan−1(ωT1)

ω, . . . ,

2 tan−1(ωTL)

ω

)+ (η1, . . . , ηL) .

2π

ω, (3.2)

where 0 ≤ tan−1(.) < π, ω ∈ Ω and the counters η` are defined in (2.7). Moreover,

transformation (3.1) is different from the first-order Pade approximation of e−τ` s,

which is e−τ` s ≈ (1 − 0.5 τ` s)/(1 + 0.5 τ` s), (Silva et al., 2005, pg. 83). Since the

Rekasius transformation (3.1) is exact for s = jω, it proves to be convenient for

solving s = jω roots of (2.4). Upon substitution of (3.1) into (2.4) and with the

following manipulation,

g(s, ~T ) =

(f(s, ~τ)

∣∣∣∣e−τ` s:= 1−T` s1+T` s

, `=1,...,L

)L∏`=1

(1 + T` s)c` , (3.3)

one obtains

g(s, ~T ) =M∑m=0

Qm(~T ) sm , (3.4)

where Qm(~T ) are multinomials in terms of T1, . . . , TL; c` = rank(B`) ≤ N and

M = N +∑L

`=1 c` ≤ N(L+ 1). Let us define a similar set as in (2.6), but now over

equation (3.4),

Ω = ω ∈ R0+ | g(jω, ~T ) = 0 , for some ~T ∈ RL . (3.5)

21


Lemma 2 (Sipahi and Olgac (2005)). The identity Ω ≡ Ω holds.

Lemma 2 indicates that instead of finding Ω from the infinite dimensional equa-

tion (2.4), alternatively one can obtain Ω by finding Ω from the algebraic equation

(3.4). In the pursuit of finding Ω, CTCR builds a Routh’s array using the coeffi-

cients Qm(~T ). The entries of this array are parameters of L different pseudo delays

T`, and by exploiting the standard rules of the array, one can express the s = jω

roots of equation (3.4) with the following procedure:

1. Denote the only entry on the s1 row of the array with R11(~T ); the only two

entries on the s2 row of the array with R21(~T ) and R22(~T ), where R21 is on

the first column of the array.

2. Find all ~T ∈ RL such that R11 = 0.

3. If R22R21 > 0 holds for the solutions found from the previous step1, then

ω ∈ Ω is found by ω =√R22/R21; otherwise ω does not exist.

4. Denote all ~T ∈ RL that leads to ω ∈ R+ at step 3 with T. Vector T can also

be expressed as the solutions of (3.4); T = ~T ∈ RL | g(jω, ~T ) = 0 , ω ∈ Ω.

5. Once Ω and T are determined at steps 2 and 3, back transform to delay space

using (3.2). These delays construct the aforementioned L dimensional PSSH.

3.1.2 Observation 1

¬ Algebraic equation is obtained in CTCR method via Rekasius transformation.

However, this transformation is applied to all delays. This choice require to

sweep L − 1 number of parameters in nested loops to solve for s = jω from

transformed characteristic function.

1Notice that the completion of Step 2 requires numerical sweeping of L− 1 number of nestedloops.

22


Due to visualization problems explained in subsection 2.3, extraction of 2D

stability maps is logical. For this extraction, some of the delays can be fixed

prior to stability analysis. For the fixed delays, the transformation does not

require the Rekasius substitution.

® When above choice combined with frequency sweeping, exponential terms are

just known complex numbers and it facilitates the stability analysis. Also,

frequency based sweeping technique has never applied to CTCR method.

¯ Delay-independent stability analysis is also convenient in algebraic domain.

Instead of checking whether there exist stability switching delay τ values, one

can easily check corresponding T values in algebraic domain.

3.2 Resultant Theory, Discriminant and Descartes

Rule concepts

Consider the two multi-variate polynomials in terms of ν, ~µ with real coefficients,

p1(ν, ~µ) =m∑i=0

ai(ν, µ1, . . . , µr−1)µir = 0 , am 6= 0 , (3.6)

p2(ν, ~µ) =n∑i=0

bi(ν, µ1, . . . , µr−1)µir = 0 , bn 6= 0 , (3.7)

where p1 and p2 have positive degrees in terms of µr, and m, n > 0. The resultant

of p1 and p2 with respect to µr is defined by

Rµr(p1, p2) =

∣∣∣∣∣∣∣∣∣am am−1 . . . a0 0 0 00 am am−1 . . a1 a0 0 0. . . . . . . . .. . . . . . . a1 a0bn bn−1 . . . b0 0 0 00 bn bn−1 . . b1 b0 0 0. . . . . . . . .. . . . . . . b1 b0

∣∣∣∣∣∣∣∣∣ , (3.8)

23


which is the determinant of the well-known Sylvester matrix (van der Waerden, 1949;

Bocher, 1964; Barnett, 1973; Gelfand et al., 1994; Cohen, 2003; Prasolov, 2004).

Theorem 1 (Collins (1971)). If (ν, µ1, . . . , µr) is a common zero of (3.6)-(3.7), then

Rµr(p1, p2) = 0 for some (ν, µ1, . . . , µr−1). Conversely, if Rµr(p1, p2) = 0 for some

(ν, µ1, . . . , µr−1), then at least one of the following four conditions holds:

(I) There exists (ν, µ1, . . . , µr) which is a common root of both (3.6) and (3.7),

(II) leading coefficients of both p1 and p2 vanish, am(ν, µ1, . . . , µr−1) = bn(ν, µ1, . . . , µr−1) =

0,

(III) all the coefficients in p1 vanish, am(ν, µ1, . . . , µr−1) = . . .= a0(ν, µ1, . . . , µr−1) =

0,

(IV) all the coefficients in p2 vanish, bn(ν, µ1, . . . , µr−1) = . . .= b0(ν, µ1, . . . , µr−1) =

0.

Definition 3. (a) Let F = F (µ`) = F (ν, µ1, µ2, . . . , µr), ` ≤ r, then the discrimi-

nant of the polynomial F with respect to µ` is defined as

Dµ`(F ) , Rµ`(F, ∂F/∂µ`) . (3.9)

(b) Let F = F (ν, µ`) = F (ν, µ1, µ2, . . . , µr), ` ≤ r, then the discriminant of the

polynomial F with respect to ν and µ` is defined as

Dν, µ`(F ) , Rµ`(∂F/∂ν, ∂F/∂µ`) . (3.10)

Polynomial F is treated as a univariate polynomial in (3.9) and a bivariate polyno-

mial in (3.10) (Gelfand et al., 1994; Sturmfels, 2002; Wall, 2004).

Geometric interpretation of discriminant in Definition 3a is presented on an

example next.

24


3.2.1 Geometric Interpretation of Discriminant in a 3D Topol-

ogy

Assume that there exists a polynomial F (ν, µ1, µ2) = 0 that implicitly depends on

three variable and the aim is to find the maximum/minimum of ν ∈ R+ satisfying

F for some (µ1, µ2) ∈ R2. Since F is implicit, it is not possible to solve ν from F ,

however, this polynomial can visualized in (ν, µ1, µ2) domain as shown in Figure 3.1a.

In order to assist the reader, Figure 3.1b is provided to show the view of F from

ν−µ1 plane. If ν exhibits an extremum in µ1−µ2 domain, then it is necessary that

∂ν/∂µ1 = 0 and ∂ν/∂µ2 = 0. Let us focus on ∂ν/∂µ2 = 0 condition. This condition

can be formulated using F , paying attention to singularities. The regular points of

F = 0 satisfy

∂ν

∂µ2

= −∂F/∂µ2

∂F/∂νwith

∂F

∂ν6= 0 .

That is, ∂ν/∂µ2 = 0 can be alternatively studied with ∂F/∂µ2 = 0. For the singular

points of F = 0; ∂F/∂µ2 = 0 and ∂F/∂ν = 0. Such points can also be eligible to be

Figure 3.1: a) 3D figure of F (ν, µ1, µ2) = 0, b) 2D figure of F (ν, µ1, µ2) = 0 on theν − µ1 plane.

25


one of the extrema points, as it is known that functions can exhibit their extrema

either at regular or singular points (Larson, 2007). However, regardless of being

regular or singular, for the extrema points to exist, it is necessary that F = 0 and

∂F/∂µ2 = 0.

At this point, one can eliminate µ2 from the last two equations using the re-

sultant, which is called the discriminant of F by eliminating µ2, Dµ2(F ) = 0, see

equation (3.9). Geometrically speaking, Dµ2(F ) = 0 is a curve in ν − µ1 domain,

and all the critical points of the surface F = 0 are among those (ν, µ1) points on

these curves. These critical points are the projections of tangent points and singu-

lar points of F = 0. Here tangent points are all those points at which lines drawn

parallel to µ2 axis become tangent to the surface F = 0. In other words, among all

(ν, µ1) points satisfying Dµ2(F ) = 0 are those that are candidates for ν to exhibit an

extremum, compare Figure 3.1b and Figure 3.2. With a similar logic, one can next

eliminate µ1 by computing Dµ1(Dµ2(F )), which becomes a polynomial in terms of

only ν. This time, tangent points are all those points at which lines drawn parallel

to µ1 axis become tangent to the curve Dµ2(F ) = 0, see Figure 3.2. The zeros of

Dµ1(Dµ2(F )) = 0 are ν1, . . . , ν4, which are candidate ν values where ν makes an

extremum. The existence of the extremum can be checked by confirming that νi

maps to (µ1, µ2) ∈ R2 numerical values via back substitutions into the polynomial

pairs forming the resultants.

Figure 3.2: Discriminant of F (ν, µ1, µ2) = 0 with respect to µ2, Dµ2(F ) = 0.

26


It is noted that singularity points of F = 0 and Dµ2(F ) = 0 can be identified,

although this is not necessary in the process of detecting the extrema points. The

singular points of F = 0 satisfy both ∂F/∂µ2 = 0 and ∂F/∂ν = 0, while the singular

points of Dµ2(F ) = 0 satisfy both ∂Dµ2(F )/∂µ1 = 0 and ∂Dµ2(F )/∂ν = 0.

The example presented above is to explain visually the concepts of discriminant

using a 3D topology. In Chapter 5, iterated discriminants, which is discriminant of

discriminants, see Henrici (1866); Lazard and McCallum (2009); Buse and Mourrain

(2009).

Finally, Descartes’s rule of signs is presented.

Theorem 2 (Sturmfels (2002)). The number of positive real roots of a polynomial

is at most the number of sign changes in its coefficient sequence, which is the se-

quence of the coefficients sorted with respect to ascending/descending powers of the

polynomial variable.

It is noted that zero (missing) coefficients are ignored when counting the number

of sign changes in a sequence. For instance, a sequence +, 0, −, 0, +, 0, + has two

sign changes (Sturmfels, 2002).

3.2.2 Observation 2

¬ In delay-dependent stability analysis, for each frequency, transformed equation

with some fixed delays is complex function, which has real and imaginary

parts. Common points of these real and imaginary parts can be calculated via

resultant concept.

In delay-dependent stability analysis, computing crossing frequency sweeping

range is crucial and it can be calculated via iterated discriminant concept.

This concept for the first time applied to TDS.

27


® In delay-independent stability analysis, computing crossing frequency sweep-

ing set is also important, because one can conclude that MTDS is stable in-

dependent of multiple delays if this set is empty.

¯ Control parameters can also be incorporated into delay-independent stability

analysis. Using Descartes’s rule of signs, the set of conditions, which make

the signs of the coefficients of a polynomial, whose roots are lower and up-

per bounds of crossing frequency sweeping set, identical to each other, can

be computed without solving the roots of the polynomial. These conditions

guarantee that the polynomial has no positive real roots, which means that

lower and upper bounds of crossing frequency sweeping set do not exist.

3.3 Supply Chains

Stability of a linear time-invariant (LTI) system is determined by investigating the

s roots of its characteristic equation. The characteristic equation arises from the

Laplace transform of the delay differential equation with zero initial conditions (since

initial conditions do not play role on stability (Nise, 2004; Ogata, 2002)). The

characteristic equation of SC models in the literature is in the form of

s + αWIP + (αi − αWIP ) e−h s = 0 , (3.11)

where delay h in time domain becomes e−h s in Laplace domain and αi, αWIP are

positive constant control parameters. Equation (3.11) is a single delay scalar char-

acteristic equation and it is identical to what Kalecki (1935) and Koopmans (1940)

attempt to solve for analyzing supply chain dynamics. By studying (3.11), the work

in Riddalls and Bennett (2003, 2002b); Warburton (2004); Warburton et al. (2004)

extracted criteria in αi vs. β = αWIP/αi domain where inventory dynamics is stable

in the presence of given delay h (delay is fixed).

28


In order to assess the stability of (3.11) with respect to any delay h, one should

know where all s roots of (3.11) lie on the complex plane. Equation (3.11) has in-

finitely many roots on the complex plane due to the transcendental term e−h s. This

makes the analytical stability assessment intractable as also stated in Riddalls and

Bennett (2002a). Since solving all the infinitely many roots of (3.11) is impossible,

one should come up with a practical procedure.

3.3.1 Observation 3

¬ Stability analysis in SC has been performed on scalar single delay time-delay

system so far. It is known that multiple delays from different sources exist in

SC and assuming all delays identical (homogeneous) in a dynamical system or

combining all delays as one single parameter may lead to misinterpretations

and poor understanding of reality since each delay may contribute to stability

/ instability in different sense and coupling of these delays may change the

dynamic behavior.

Inventory regulation problem in SC can be analyzed in linear-time invariant

framework. Observations in 3.1.2 and 3.2.2 are convenient for inventory regu-

lation problem.

® Inventory regulation problem in SC is a crucial and challenging problem. As-

sisting the manager is needed since each delay may play different stability

roles. Therefore, mathematical analysis is fundamental to understand what

happens internally in SC.

29

Chapter 4

Delay-Dependent Stability

Analysis of Multiple Time-Delay

Systems

4.1 General Approach: Advanced Clustering with

Frequency Sweeping Methodology

The new method called Advanced Clustering with Frequency Sweeping (ACFS) is

introduced in this section. It is important to note that ACFS is not only an elegant

numerical algorithm that can extract the 2D stability maps, but it is also a platform

to advance the stability theory of TDS. ACFS reports the following new results:

(i) the maximum number of kernel points can be computed as a function of the

ranks of system matrices and this number is also a measure of computational

complexity,

(ii) necessary and sufficient conditions which yield the exact lower and upper

bounds of the crossing frequency set (CFS) can be formulated via a sequential

30

CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS

formula, and

(iii) delay-dependent stability analysis on any two-delay domain for the general

class LTI-MTDS.

ACFS uniquely stands out from the existing approaches as it is able to accommo-

date fixed delays in its theoretical construct. This ultimately allows us extract the

cross sectional views of the PSSH without running into the problems of the existing

approaches, see Section 2.3. In other words, ACFS is able to achieve (L > 3) what

the work in Cooke and van den Driessche (1986) achieved (L = 2) by analyzing the

stability of a TDS along τ1 via fixing τ2.

ACFS sweeps ω numerically, reducing the number of unknowns by one. It is

noted that this is the only parameter that needs to be swept in the procedural

steps of ACFS. Although ACFS is entirely different from the existing methods, its

frequency sweeping part is inspired by Chen and Latchman (1995); Gu et al. (2005);

Sipahi and Olgac (2006a); Sipahi and Delice (2009). The objective of ACFS is

then stated as follows: compute the projections of PSSH on any 2D delay plane

when the remaining delays are numerically fixed. ACFS starts similar to CTCR,

however, for the fixed delays it does not require the Rekasius substitution (Rekasius,

1980; Sipahi and Delice, 2011). This innocent looking choice when combined with

frequency sweeping and the resultant theory offers unmatched strength in revealing

the PSSC.

Assumptions

1. Delays τ3 = τ3, . . . , τL = τL are given.

2. It is assumed that s = jω with ω = 0 is not a root of (2.4) when τ1 = τ2 = 0.

This assumption can be removed by studying the degenerate cases (Fazelinia

et al., 2007). Other degeneracies, such as those studied in Sipahi and Olgac

31


(2003a); Jarlebring and Michiels (2010) are kept outside the scope of this

section.

3. Frequency ω ∈ R+ is a given sweep parameter.

4. ACFS extracts the projections of PSSH on τ1−τ2 domain by assuming, without

loss of generality, that rank(B2) = c2 ≤ rank(B1) = c1 ≤ N .

In light of the introduction and the assumptions above, the characteristic func-

tion to be studied in ACFS framework becomes,

h(jω, T1, T2, e−jωτ3 , . . . , e−jωτL) =(

f(jω, ~τ)

∣∣∣∣e−jωτ` := 1−jωT`1+jωT`

, `=1,2.

)2∏`=1

(1 + jωT`)c` . (4.1)

For any given sweep parameter ω, all the exponential terms e−jωτ3 , ... , e−jωτL are

known complex numbers, hence they are dropped from the arguments. It is started

by decomposing (4.1) as

h(jω, T1, T2) = h<(ω, T1, T2) + j h=(ω, T1, T2) , (4.2)

where h< = <(h) and h= = =(h), and the crossing frequency set of (4.2) is denoted

by Ω, which is obviously a subset of Ω.

For ω to be a zero of (4.2), h< and h= should be concurrently zero for some

(T1, T2). Let us investigate this next,

h< =

c2∑i=0

ai(ω, T1)T i2 = 0 , (4.3)

and

h= =

c2∑i=0

bi(ω, T1)T i2 = 0 . (4.4)

32


Note that all ai’s and bi’s are real polynomials in T1 for a given ω. h< and h=, which

have positive degrees in terms of T2, are assumed to have no common factors. Such

common factors, if they exist, can be separately studied.

Remember from (3.8) that RT2 with respect to ω and T1 is the resultant of h<

and h= by eliminating T2. Following corollary is obtained from the multi-variate

polynomial resultant theorem, Theorem 1, which reveals the common roots of h<

and h=.

Corollary 1. If (ω, T1, T2) is a common zero of (4.3)-(4.4), then RT2(h<, h=) = 0.

Conversely, if RT2(h<, h=) = 0, then at least one of the four conditions holds:

(i) there exists (ω, T1, T2) that is a common zero of (4.3)-(4.4). (ii) ac2(ω, T1) =

bc2(ω, T1) = 0, (iii) a0(ω, T1) = · · · = ac2(ω, T1) = 0, (iv) b0(ω, T1) = · · · =

bc2(ω, T1) = 0,

Detection of the common roots of (4.3)-(4.4) corresponds to Condition (I) in

Corollary 1, and the remaining Conditions (II)-(IV) can be identified for a given

(ω, T1, T2) triplet.

4.1.1 Theoretical Construct of ACFS Methodology

Theorem 3. For the general control system (2.1), and for a given ω ∈ Ω such that

h< and h= are not identically zero, the number of points generating the kernel points

on τ1− τ2 plane is bounded by 2 c1c22; twice the product of the larger commensurate

degree and square of the smaller commensurate degree associated with the delays

defining the 2D delay plane.

Proof. For a given ω ∈ Ω, construct the resultant using h< and h= with eliminating

T2. As per Corollary 1, the common roots of h< and h= satisfy the resultant, which

is RT2(h<, h=) = (ac2)c2 (bc2)

c2∏

i,k(δi − ξk), where δi and ξk are the zeros of h< and

h=, respectively (Gelfand et al., 1994, pg. 398). Since the maximum degree of T1 in

33


both ac2 and bc2 is c1, the degree of RT2(h<, h=) is 2 c1c2 indicating that there can

be at most 2 c1c2 number of T1 solutions (von zur Gathen and Gerhard, 2003, pg.

147)). For each T1 solution, h< and h= admit at most c2 number of T2 solutions,

thus the maximum number of points generating the kernel points is 2 c1c22 with the

fact that each (T1, T2) ∈ R2 solution point generates one kernel point on τ1 − τ2

plane (Sipahi and Olgac, 2005).

If either h< or h= vanishes (becomes identically zero) for a given ω, then the

resultant theory cannot be utilized. For this degenerate situation, the number of

kernel points can be computed from the non-vanishing function for ω = ω. For the

case of c1 < c2, one can obtain the maximum number of kernel points as 2 c2 c21.

Lemma 3. For all ω ∈ Ω, let all the real T1 zeros of RT2(h<, h=) be represented by

V = T1 ∈ R |RT2(h<, h=) = 0,∀ω ∈ Ω, and let all the real T1 zeros of h(jω, T1, T2)

be defined by V = T1 ∈ R | h = 0, for some T2 ∈ R ,∀ω ∈ Ω. The set V is a

subset of V, but not vice versa.

Proof. Proof follows from the fact that RT2(h<, h=) = 0 is a necessary condition for

h< and h= to have common roots.

Based on Lemma 3, it is chosen to study the zeros of RT2(h<, h=) instead of

studying the zeros of h(jω, T1, T2). Before presenting the main theorem, it is needed

to establish the differentiability of ω with respect to T1 in RT2(h<, h=), which is

a polynomial in T1 and which implicitly depends on both ω and T1. Notice that

∂RT2(h<, h=)/∂T1 and ∂RT2(h<, h=)/∂ω are continuous in ω−T1 domain (Rogawski,

2008). Consequently, besides few singularity points that the curve RT2(h<, h=) = 0

may possess, the regular points of this curve guarantee the differentiability of ω

with respect to T1, that is, ∂ω/∂T1 always exists. This claim follows from the

implicit function theorem (Courant, 1988, pg. 114), and immediately guarantees

34


the continuity of ω with respect to T1 (Rogawski, 2008). A separate discussion is

provided below on the singular points of the curve RT2(h<, h=) = 0.

Using Definition 3a, Corollary 1 and Lemma 3, the theorem proving the precise

nonzero lower bound¯ω and upper bound ω of Ω in the case of

¯ω 6= 0 is presented.

Theorem 4 (Exact Lower and Upper Bounds of Ω). Minimum and maximum

positive real roots of the discriminant of the resultant of h< and h= with respect to

ω, that correspond to (T1, T2) ∈ R2 solutions in (4.2), are the exact positive lower

and upper bounds of Ω.

Proof. For the delay-dependent case, finite lower bound¯ω and upper bound ω of Ω

are known to exist (Stepan, 1989). To find the global maximum ω and the global

minimum¯ω, it is started studying the extrema of ω via ∂ω/∂τ1 = 0, which is

identical to studying

∂ω

∂τ1

=∂ω

∂T1

∂T1

∂τ1

= 0 , (4.5)

where ∂T1/∂τ1 = 0.5(1 + ω2T 2

1

)as per (3.2). Since ∂T1/∂τ1 6= 0, one can study

∂ω/∂τ1 = 0 alternatively on ∂ω/∂T1 = 0. At this point, the differentiability of ω

with respect to T1 is essential as established above, and holds for the regular points

of RT2(h<, h=) = 0. Under this condition, one can write

∂RT2(h<, h=)

∂T1

+∂ω

∂T1

∂RT2(h<, h=)

∂ω= 0 . (4.6)

From (4.6), for ∂ω/∂T1 = 0 to hold; ∂RT2/∂T1 = 0, since ∂RT2/∂ω 6= 0 for regular

points. Two equations are obtained, RT2 = 0 and ∂RT2/∂T1 = 0. It is necessary that

these two equations are simultaneously satisfied such that (4.3)-(4.4) have common

solutions, and ω exhibits an extremum. This requires to study the zeros of the

resultant of these two equations. Focusing on ω and eliminating T1, the resultant

of RT2 and ∂RT2/∂T1, called the discriminant of RT2 by Definition 3a, becomes a

function of only ω, Z(ω) = RT1(RT2 , ∂RT2/∂T1) = 0. Real ω roots of Z(ω) = 0

35


are candidates to be the extrema points. Of those real ω roots, the minimum and

maximum positive ones that correspond to (T1, T2) ∈ R2 solutions in (4.2) are the

exact positive lower and upper bounds of Ω, respectively.

Explanatory Example:

A scalar example with L = 3 is presented in order to demonstrate the application

of Theorem 4. The characteristic function to be studied is taken as

f(s, ~τ) = s+ 3 + e−s τ1 + 4 e−s τ2 + 2.6 e−s τ3 . (4.7)

Next, τ3 is arbitrarily chosen as 1.0, and obtain (4.2), where

h< =[(2T1 − 1)ω2 + 2.6ω

(sin(ω)− cos(ω)ω T1

)]T2

2.6 cos(ω) + 2.6 sin(ω)ω T1 − ω2 T1 + 8 , (4.8)

and

h= =[2.6ω

(cos(ω) + sin(ω)ω T1

)− ω3 T1

]T2

+ 2.6 cos(ω)ω T1 − 2.6 sin(ω) + ω + 6ω T1 . (4.9)

Using the resultant command in MAPLE software package, eliminate T2 from h<

and h=,

RT2(h<, h=) =[−ω5 + 5.2 sin(ω)ω4 −

(10.4 cos(ω)− 5.24

)ω3]T 2

1

+[4ω3 − 10.4 sin(ω)ω2

]T1 − ω3 + 5.2 sin(ω)ω2 −

(20.8 cos(ω) + 6.76

)ω .

(4.10)

36


Discriminant of the resultant of h< and h= in Theorem 4 is the resultant of RT2 and

∂RT2/∂T1 with eliminating T1,

RT1(RT2 , ∂RT2/∂T1) = ω7−4ω6 + 62.4 sin(ω)ω5 +

[324.48 cos2(ω)− 166.4 cos(ω)

−293.6] ω4 + sin(ω)[−562.432 cos2(ω) + 1730.56 cos(ω) + 241.28

]ω3

+[4499.456 cos3(ω)− 3106.3552 cos2(ω)− 3587.584 cos(ω) + 1032.864

]ω2

+ sin(ω)[11811.072 cos2(ω)− 4741.7344 cos(ω)− 1028.9178

]ω

− 10123.776 cos3(ω) +6710.2464 cos2(ω) + 1787.5354 cos(ω)− 1309.2119.

(4.11)

The minimum and maximum positive real roots of Z(ω) = RT1(RT2 , ∂RT2/∂T1)

are computed as 0.7550 and 3.6590. Corresponding real T1 and T2 values are

calculated from and satisfy RT2 , ∂RT2/∂T1, h<, and h=. They are (T1, T2) =

(−12.7614,−12.7614) for ω = 0.7550, and (T1, T2) = (0.3173, 0.3173) for ω = 3.6590.

Since (T1, T2) ∈ R2 solutions exist, it is concluded that [¯ω, ω] is [0.7550, 3.6590]. Note

that four-digit precision is used for numerical values in order to conserve space.

Notice that Theorem 4 is in T1− T2 domain, hence the detection of¯ω and ω is

valid in τ1−τ2 domain for a given set of τ3, . . . , τL. Furthermore, under Assumptions

1-2, it is possible that¯ω → 0 can be a solution of h = 0. If such a case exists, it

can be detected via Fazelinia et al. (2007); Sipahi and Olgac (2007), and the lower

bound¯ω can be set to zero.

Remark 1. If RT2(h<, h=), ∂RT2(h<, h=)/∂T1, and ∂RT2(h<, h=)/∂ω are all zero in

(4.6), “singular points” occur. Singularities can be treated using multivariable resul-

tant theory, particularly by analyzing the multiplicity of solutions, and the common

factors of RT2 and ∂RT2/∂T1, see pg. 142 of Abhyankar (1990) for this treatment.

These modifications do not change the essence of discriminant computation, but only

require to factor out the greatest common divisor (gcd) of RT2 and ∂RT2/∂T1 from

37


the analysis, before constructing the discriminant. Hence, the structure of Theo-

rem 4 does not change. Notice that Theorem 4 does not exclude singular points

since they are known to be candidates for extrema (Larson, 2007).

4.1.2 Algorithmic Construct of ACFS Methodology

The PSSC ℘ needed for the delay-dependent stability analysis of the system can be

detected following the algorithmic steps of ACFS given below. Firstly, Theorem 4

is used to compute¯ω and ω of Ω. For each ω ∈ [

¯ω, ω] with an appropriately chosen

step size, the following steps are performed:

¬ Solve the polynomial equation RT2(h<, h=) = 0 for T1 ∈ R values.

For each T1 ∈ R found from above, if T2 ∈ R values exist satisfying h< = 0

and h= = 0, then proceed to the next step, otherwise increase ω by the step

size, and restart from the step above.

® Via (3.2), calculate the delay values (τ1, τ2) corresponding to (T1, T2) ∈ R2

pairs, and restart from Step 1 increasing ω by the step size.

For a given ω, if all ai’s are identically zero, a modification is needed in the

above algorithm. This can be done by simply analyzing the (T1, T2) ∈ R2 solutions

in h= = 0 when ω = ω. Similar approach can be implemented when all bi’s are

identically zero. In these degenerate cases, the resultant in Corollary 1 cannot be

utilized, however, (T1, T2) ∈ R2 solutions can be detected from the non-vanishing

function (either h< or h=) for the given ω = ω. Moreover, ac2(T1) and bc2(T1) may

become zero for a given ω, indicating that T2 → ∓∞. Corresponding T1 solutions in

this case can be identified from the common solutions of ac2(T1) and bc2(T1). Once

these scenarios are considered, the computed (ω, T1, T2) triplets can be used in Step

3 of ACFS. All the (τ1, τ2) pairs found in this step, including those found with the

modifications explained above, construct the complete PSSC ℘.

38


Remark 2. For a given ω and T1 ∈ R satisfying RT2(h<, h=) = 0, the existence of

common T2 solutions in Step 2 depends on the roots of the greatest common divisor

of h< = 0 and h= = 0 (von zur Gathen and Gerhard, 2003, pg. 162), (Uspensky,

1948). If a real root of gcd(h<, h=) exists, this root is the admissible T2 ∈ R solution.

When gcd(h<, h=) = 1, there exists no common T2 solutions, either real or complex.

Feasible T2 solutions can also be obtained by solving T2 from h< and h= for a given

(ω, T1) pair.

4.1.3 Case Studies

First, a simpler yet nontrivial stability problem with N = 2 and L = 3 is studied.

Then, a complicated case study with N = 4, L = 4, c1 = 4 and c2 = 2 is studied

in order to demonstrate the capabilities of ACFS. Third case study is borrowed

from Jarlebring (2009), N = 35 and L = 3. In these examples, the resultants

are constructed using homomorphism resultant algorithm (Collins, 1971). Stability

maps are extracted by showing kernel curves with red color and offspring curves

with blue color when viewed in color. Stability regions are shaded by means of

Sipahi and Olgac (2005).

Case 1:

Let the state matrices in (2.1) be

A =

0 1

−20.91 −9.2

, B1 =

−0.968 0.01

3.1 −2.6

,

B2 =

0 0

0.127 5.86

, B3 =

0 0.26

0.28 −2.7

,

where N = 2, L = 3, and the ranks of B1 and B2 are c1 = 2 and c2 = 1, respectively.

Next, the ACFS is implemented for arbitrarily chosen two τ3 delay values, τ3 = 1.5

and τ3 = 4.0. Following Theorem 4, corresponding frequency ranges are found as

39


[¯ω, ω]1.5 = [1.4774, 6.5821] and [

¯ω, ω]4.0 = [2.0241, 8.5652], which are computed in

approximately 1 second. Upon sweeping ω in these ranges, the PSSC are extracted,

see Figure 4.1 and Figure 4.2.

It is worthy to note that identifying the PSSC in each one of these figures requires

28 seconds of computation time on a standard laptop with 2.1 GHz CPU speed and

3 GB RAM, and to the best of our knowledge, none of the existing techniques

can extract the precise cross sections that ACFS can capture in Figure 4.1 and

Figure 4.2.

The following step is the stability analysis, which commences with identifying

the stability of the origin of the 2D delay space, τ1 = τ2 = 0. Using the technique in

Olgac and Sipahi (2002), the origin is found to be asymptotically stable independent

of the values of τ3. This indicates that all the regions that can be connected to the

origin with a continuous path without intersecting any PSSC are asymptotically

stable. The stability features in the remaining regions are identified by computing

the number of unstable roots NU of the system in these regions (Sipahi and Olgac,

2005). From these analyses, all the stable regions are identified and shaded, see

Figure 4.1 and Figure 4.2.

Figure 4.1: Case 1: Stability map for τ3 = 1.5. Shaded regions are stable.

40


Figure 4.2: Case 1: Stability map for τ3 = 4.0. Shaded regions are stable.

Remark 3. There can also be multiple frequency ranges instead of a single range

for MTDS, Ω =⋃nf`=1 Ω`. To the best of the author’ knowledge, situation nf > 3 is

observed for the first time in this example; nf = 2 when τ3 = 1.5 and nf = 5 when

τ3 = 4.0, see Figure 4.3. Moreover, it is observed that nf increases as τ3 increases

in this case study. Readers may consult Michiels and Niculescu (2007) for studies

on multiple number of admissible frequency ranges.

Case 2:

State matrices in (2.1) are taken as

Figure 4.3: Case 1: Amplitude of frequency versus index of frequency for τ3 = 1.5and τ3 = 4.0.

41


A =

0 1 0 0

0 0 1 0

0 0 0 1

−29.17 −56 −36.7 −10.1

, B1 =

−1.55 1 0 0

−1 −0.3 0 0

0 0 0.5 0

−0.7 0 −0.34 −2.6

,

B2 =

0 0 0 0

1 1.5 4 0

0 0 0 0

−0.33 0 0 −1.1

, B3 =

0 0 0 0

0 0 0 0

0 0 0 0

−0.08 −0.7 0 −1

and B4(4, 3) = −3 with its remaining entries being zero; c1 = 4 and c2 = 2. Next,

τ3 and τ4 are arbitrarily chosen as 0.169 and 0.26, respectively. From Theorem 4,

it is computed that [¯ω, ω] = [1.4004, 5.5849]. Upon sweeping ω in this range, the

PSSC of the system is extracted in Figure 4.4. It is noted that identifying the PSSC

in Figure 4.4 on average requires 40 seconds of computation time on average.

Figure 4.4: Case 2: Stability map for τ3 = 0.169 and τ4 = 0.26. Shaded region isstable.

42


Case 3:

Three-delay partial differential equation in equation (17) of Jarlebring (2009) is

discretized for N = 35 in cited work. Two scenarios are investigated, one with

τ3 = 0.0 and the other with τ3 = 0.06. For the case of τ3 = 0.0 when the problem

becomes a two-delay problem, it is computed that [¯ω, ω] = [3.7477, 24.6930] rad/sec,

and when τ3 = 0.06 it is revealed [¯ω, ω] = [3.5685, 11.7607] rad/sec. Sweeping ω

in the respective ranges found, stability maps on 2D delay domain are extracted,

Figures 4.5-4.6. It is confirmed that the two figures are consistent with those found

in Jarlebring (2009). It is noted that identifying the precise range for CFS takes

about 2 seconds and ACFS reveals PSSC in Figures 4.5 and Figure 4.6 on average

50 seconds and 65 seconds, respectively.

Figure 4.5: Case 3: Stability map for τ3 = 0.0. Shaded region is stable.

43


Figure 4.6: Case 3: Stability map for τ3 = 0.06. Shaded region is stable.

4.1.4 Changes in PSSC for perturbations in fixed delay val-

ues

In the sequel, it is shown that how small perturbations in τ3, . . . , τL affect PSSC.

For this analysis, total differential of characteristic function, df , is computed

df =∂f(jω, ~τ)

∂ωdω +

L∑`=1

∂f(jω, ~τ)

∂τ`dτ` , (4.12)

where dω and dτ`, ` = 1, . . . , L, are differentials with respect to ω and τ`, respectively

(Hildebrand, 1976). Without loss of generality, dω is taken as zero, because it is

interested that how (τ1, τ2) points are perturbed for the same frequency value and

df = 0 since f(jω, ~τ) = 0, it is a constant. Moreover, differential df = 0 reveals

dependencies of dτ1 and dτ2 on dτ3, . . . , dτL. Notice also that this analysis is valid

only for small values of dτ3, . . . , dτL, e.g., ∓0.01. Furthermore, f(jω, ~τ), ∂f(jω,~τ)∂ω

,

∂f(jω,~τ)∂τ`

, ` = 1, . . . , L, are continuous at each point of (ω, ~τ) domain (Datko, 1978).

Given dτ3, . . . , dτL, df = 0 in (4.12) is complex function with two unknowns, dτ1

and dτ2. Perturbations in the direction of τ1 and τ2 can be calculated from linear

44


system of equations,

<(df) = 0 and =(df) = 0 . (4.13)

The perturbation analysis is demonstrated on two examples. In the first example,

characteristic function

f(s, ~τ) = 2.5 s (2.5 s+ 1) + e−s τ1 − e−s τ2 + e−s τ3 , (4.14)

is considered. Figure 4.7(a) shows a part of the PSCC for τ3 = 8 (red color) and

τ3 = 8.05 (magenta color) for comparison purpose. Perturbation vectors are also

drawn in green color and their magnitudes are scaled for visual clarity. Larger

arrows indicate bigger changes in PSSC in Figure 4.7. Moreover, Figure 4.7(b)

shows zoomed plot to a perturbation vector and squares in this figure denote exact

location of starting and ending points of the perturbation vector. In the second

(a) Perturbation vectors are drawn in green colorand their magnitudes are enlarged for visual clar-ity.

(b) A perturbation vector is drawnwith original size.

Figure 4.7: Part of the kernel curve of (4.14) for τ3 = 8 (red color) and τ3 = 8.05(magenta color); dτ3 = 0.05. Larger arrows indicate bigger changes in PSSC.

45


(a) Perturbation vectors are drawn in green colorand their magnitudes are enlarged for visual clar-ity.

(b) A perturbation vector is drawnwith original size.

Figure 4.8: PSSC of (4.15) for τ3 = 0.3, τ4 = 0.16 (red and blue color) and τ3 = 0.29,τ4 = 0.17 (magenta and yellow color); dτ3 = −0.01 and dτ4 = 0.01. Larger arrowsindicate bigger changes in PSSC.

example, characteristic function

f(s, ~τ) = s2+1.5 s+60+8 e−s τ1−8 e−s τ2+3 e−s (τ1+τ2)+s e−2 s τ3+(s+3) e−s τ4 , (4.15)

is considered. Notice that characteristic function with four delays has a commensu-

rate delay and a cross-talk term. Figure 4.8(a) shows PSCC for τ3 = 0.3, τ4 = 0.16

(red and blue color) and τ3 = 0.29, τ4 = 0.17 (magenta and yellow color) and

Figure 4.8(b) shows zoomed version of Figure 4.8(a) to a perturbation vector.

4.1.5 Limitations

The measure in Theorem 3 can be at most 2N3 when B1 and B2 are full rank. With

the availability of numerically efficient real root isolation algorithms, T1 ∈ R roots

of the univariate polynomial RT2(h<, h=) can be easily computed, with degrees up

to 10000 (Parrilo and Sturmfels, 2003; Akritas and Strzebonski, 2005).

46


4.2 Specific Problem: Extraction of 3D Stability

Switching Hypersurfaces

In the sequel, the stability transitions of LTI-MTDS are studied with respect to

delays τ` of the characteristic function,

f(s, ~τ) = P0(s) +L∑`=1

P`(s)e−τ` s = 0 . (4.16)

A novel procedure which can reveal 3D stability switching hypersurfaces is studied

first. Next, some properties of these hypersurfaces is presented. It is worthy to note

that methodology in this section does not impose any limitations on the system order

N and the number of delays L; they can be arbitrarily large. Moreover, delays τ`

are independent from each other, thus the multiple dimensional nature of (4.16)

in delay parameter space is maintained, but this system is still a subclass of (2.4)

studied above. The special case arises under certain rank conditions of matrices B`,

` = 1, . . . , L.

In order to remove the aforementioned complications in subsection (2.3), a fre-

quency sweeping idea inspired by Gu et al. (2005); Chen and Latchman (1995) is

adapted here. Due to visualization constraints, 3D cross sections of ℘ will be de-

tected. These projections are denoted by ℘ and a versatile and efficient procedure

is developed to extract ℘ of specific characteristic function in (4.16), sub-class of

(2.4). The procedure can achieve this projection in any three-delay space without

obtaining the L-D ℘. In order to identify ℘, one needs to solve all (τ1, τ2, τ3) ∈ R3+,

without loss of generality, and ω ∈ R+ from the complex function,

h(jω, ~τ) = P0(jω) +3∑`=1

P`(jω) e−jτ` ω +L∑`=4

P`(jω) e−jτ` ω , (4.17)

given the delays τ`, ` = 4, . . . , L. Obviously, finding all the infinitely many (τ1, τ2, τ3) ∈

47


R3+ solutions from (4.17) is not trivial (4 unknowns τ1, τ2, τ3, ω, but one equation).

Since (4.17) represents a retarded-type dynamics, a conservative upper bound ¯ω,

such that sup Ω < ¯ω, exists (Hale and Verduyn Lunel, 1993). With this property,

it will be sufficient to sweep ω within the finite interval, ω ∈ (0, ¯ω], in order to solve

(τ1, τ2, τ3) that construct ℘ completely. Notice that one can compute lower and up-

per bounds of CFS from a theorem similar to Theorem 4 and sweep frequency in this

range. However, due to the treatment of sub-class problem, the stability analysis is

less involved compared to the treatment of general class LTI-MTDS since resultant

calculation is not needed for each frequency.

For a given τ4, . . . , τL, the procedure starts with the following sequential steps.

Define the real and imaginary parts of P`(s) in (4.17) for ` = 1, 2, 3 as:

P`<(ω) = <(P`(jω)), P`=(ω) = =(P`(jω)). (4.18)

Assuming that ω > 0 is given, the remaining terms in (4.17) are known,

φ(jω) := χ(ω) + jγ(ω) = P0(jω) +L∑`=4

P`(jω)e−jτ`ω , (4.19)

where (χ(ω), γ(ω)) ∈ R2. Next, let e−jτ`ω = x` + j y`, ` = 1, 2, 3 and define the unit

circles in R2 as,

C` = (x`, y`) ∈ R2 | x2` + y2

` − 1 = 0. (4.20)

Following the equations (4.18)-(4.20), the real and imaginary parts of (4.17) are

expressed as

3∑`=1

M`

x`

y`

+

χ(ω)

γ(ω)

=

0

0

, (4.21)

48


where M` =

P`< −P`=

P`= P`<

and det(M`) = P 2`< + P 2

`= = |P`(jω)|2 6= 0 since

P`(s) 6= 0 and ω 6= 0 by definition.

One can now solve the (x1, y1) pair from (4.21),

x1

y1

= −M−11

3∑`=2

M`

x`

y`

+

χ(ω)

γ(ω)

. (4.22)

For a solution to exist, it is necessary that (x1, y1) ∈ C1. This constraint yields a

line L in R2 (see Appendix A for the derivation),

L = (x2, y2) ∈ R2 | x2 Γ1(ω, x3, y3) + y2 Γ2(ω, x3, y3) + Γ0(ω, x3, y3) = 0 . (4.23)

The terms Γ`(ω, x3, y3), ` = 0, 1, 2 are frequency dependent coefficients,

Γ0 = |φ(jω)|2 − |P1(jω)|2 + |P2(jω)|2 + |P3(jω)|2

+ 2 x3(P3< χ(ω) + P3= γ(ω)) + 2 y3(−P3= χ(ω) + P3< γ(ω)) , (4.24)

Γ1 = 2(P2< χ(ω)+P2= γ(ω)

)+2 x3(P2= P3=+P2< P3<)+2 y3(P2= P3<−P2< P3=) ,

(4.25)

Γ2 = 2 (P2< γ(ω)−P2= χ(ω))+2 x3(P2< P3=−P2= P3<)+2 y3(P2= P3=+P3< P2<) . (4.26)

Among the frequency dependent coefficients Γ`(ω, x3, y3), ` = 0, 1, 2, one can verify

that the highest power of ω appears only in Γ0(ω, x3, y3).

The recursive part of the procedure then follows with the following three steps

for ω ∈ (0, ¯ω]:

49


Step 1. For a τ3 solution to lie on the ℘ hypersurfaces, it is necessary that (x3, y3) ∈

C3. These (x3, y3) pairs are used to obtain (x2, y2) ∈ C2 as follows. The (x2, y2)

solutions lie at the intersection points p1 and p2 between C2 and L, see Figure 4.9.

x2 components of these points are functions of ω, x3 and y3, and they are precisely

found as

x2 =−Γ0Γ1 ∓ Γ2

√∆(ω, x3, y3)

Γ21 + Γ2

2

, (4.27)

where ∆(ω, x3, y3) = Γ21 + Γ2

2 − Γ20.

Step 2. x2 solutions are used to obtain y2 solutions as per (x2, y2) ∈ C2. By back

substitution, one obtains the (x1, y1) pairs using (x2, y2) and (x3, y3) in (4.22).

Step 3. The delays (τ1, τ2, τ3) ∈ R3+ that construct the ℘ hypersurfaces are obtained

from

τ` = − 1

ω

/x` + j y` , ` = 1, 2, 3 , (4.28)

where the arguments above also carry ∓2πη`, η` ∈ N, shiftings as per trigonometric

properties.

Remark 4. The procedure presented above sweeps both ω and (x3, y3) ∈ C3 within

finite intervals in only two nested loops. Each (x3, y3) point and ω lead to numerical

values of Γ0, Γ1 and Γ2 which are used to compute x2 in (4.27), see the flow chart

in Figure 4.9. This assures that no approximation is imposed when detecting the

delays. There exist x2 solutions if and only if ∆(ω, x3, y3) ≥ 0 in (4.27). This

ultimately guarantees the existence of the delays in (4.28).

4.2.1 Features of Stability Switching Curves

In (4.19), it is clear that all the terms for ` = 0, 4, . . . , L are lumped into φ(jω), which

is only a function of the sweep parameter ω. Therefore, one expects no significant

challenges in computation times when extracting ℘ for a given L and N . It is noted

that the way the delay terms appear in the characteristic function plays a role in this

50


(0, ]w wÎ

3 3,x y

1 1,x y

Eq.(4.22)

Eq.(4.28)

2 2,x y

Frequency sweeping

___1

2

3

ttt

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷çè ø

Î Ã

Shaded region

3 3( , , ) 0x ywD ³

Frequency dependent

curve, 3 3( , , ) 0x ywD ³

2p

1p

2

2

2

3

Figure 4.9: Flow chart of the proposed procedure in Section 4.2.

simplification. Since there are no terms of the form e−s∑L`=1 υ` τ` , υ` ∈ N in (4.16)

(delay cross-talk and commensurate terms), identification of ℘ is computationally

less involved even for L > 3. Interested readers are referred to Jarlebring (2009),

Gundes et al. (2007), Fazelinia et al. (2007), and Sipahi and Olgac (2005) for delay

cross-talk treatments but with less than four delays. For MTDS with arbitrarily

large number of delays, the treatment of both cross-talk and commensurate delay

terms were presented in the previous section, see ACFS methodology in Section 4.1.

The maximum number of separate ℘ hypersurfaces needed to assess the sta-

bility of (4.16) is crucial. It indicates how complicated and intricate the stability

map is and how the infinite spectrum of (4.16) collapses onto a finite number of

hypersurfaces. In the following, this number which is an inherent feature of ℘ is

studied.

Definition 4 (Kunz (2005)). Given the general form of a quadratic polynomial

F (u, v) = a1 u2 + a2 u v + a3 v2 + a4 u + a5 v + a6 with real coefficients a` ∈ R, the

51


polynomial F (u, v) = 0 can only be in the form of one of four characteristic curves:

an ellipse, a hyperbola, a parabola or a pair of lines .

Lemma 4. For a given ω ∈ Ω, there exist at most two segments on C3 where a

point on these segments satisfies ∆(ω, x3, y3) ≥ 0.

Proof. Given ω ∈ Ω, it is easy to confirm that ∆(ω, x3, y3) = F (x3, y3), hence

∆(ω, x3, y3) = 0 is one of the four characteristic curves in Definition 4. Any two

curves with degrees m and n, respectively, have at most m · n intersection points

(Abhyankar, 1990; Kunz, 2005). In this case, m = 2 for ∆(ω, x3, y3) = 0 and n = 2

for C3. Hence, ∆(ω, x3, y3) = 0 can partition C3 into at most four segments. Points

on at most two of these four segments satisfy the inequality ∆(ω, x3, y3) ≥ 0, as per

the continuity with respect to x3 and y3.

Lemma 5. For a given ω ∈ Ω, the maximum number of points generating the kernel

hypersurface is four.

Proof. For a given ω ∈ Ω and a (x3, y3) pair, at most two (x2, y2) solution pairs

lie at the intersection of C2 and L. As per Lemma 4, there are at most two sep-

arate segments on C3 on which admissible (x3, y3) pairs reside. Consequently, the

maximum number of points generating the kernel hypersurface is four.

Remark 5. Since the sign of the coefficient of the highest power of ω in ∆(ω, x3, y3)

is negative, there exists ω∗ ∈ R+ such that ∆(ω, x3, y3) < 0 for ω > ω∗. The con-

dition ∆(ω, x3, y3) ≥ 0 can be satisfied in multiple and distinct frequency intervals,

Ω =⋃nfk=1 Ωk. Any ω, ω ∈ Ωk, gives rise to s = jω and (τ1, τ2, τ3) ∈ R3

+ solutions

to (4.16). This is a well-known argument (Michiels and Niculescu, 2007) and it

naturally arises in the context of our development.

52


4.2.2 Case Studies

In this section, two case studies are presented. In the first case, only ℘ is depicted

as kernel hypersurfaces disregarding the ∓2πη` shiftings in (4.28). The second case

takes into account these shiftings only on the τ1 − τ2 plane for visualization clarity

and presents the stability analysis in 3D. In both cases, the same system with N = 7

and L = 10 is considered, however, delays τ4, . . . , τ10 are taken differently, in order

to present how ℘ exhibits various 3D geometry. At the end of this section, the

computational complexities are discussed.

Case 1:

The polynomials P`(s) in (4.16) are chosen as P0 = s7 + 19s5 + 29s4 + 51s3 + 43s2 +

22s+ 5, P1 = 4s5 + 3s4 + 7s3 + 4.6s2 + 14s, P2 = s5 + 53s4 + 32s3 + 45s2 + 19s+ 4,

P3 = 3s2 + s+ 6, P4 = 85s3 + 10s, P5 = 28s4 + 2.3, P6 = 40s3 + 9.4, P7 = 23s5 + 50s,

P8 = 10s3 + 297s2, P9 = 4s6 + 26, P10 = 10s+ 2. Next, delay values are arbitrarily

chosen as τ4 = 0.197, τ5 = 0.076, τ6 = 0.013, τ7 = 0.1, τ8 = 0.147, τ9 = 0.228,

τ10 = 0.11 and the procedure above is implemented to detect ℘, see ℘kernel in

Figure 4.10. The procedure also reveals that nf = 1 and frequencies in the set Ω1

Figure 4.10: Case 1: 3 dimensional depiction of ℘kernel in (τ1, τ2, τ3) for τ4 = 0.197,τ5 = 0.076, τ6 = 0.013, τ7 = 0.1, τ8 = 0.147, τ9 = 0.228 and τ10 = 0.11. Gray-scale

color coding represents ω ∈ Ω1 correspondence.

53


range from 1.966 rad/sec to 2.641 rad/sec. The gray-scale color coding in the figure

indicates frequency distribution.

Case 2:

The polynomials P`(s) in (4.16) are kept the same as in the previous case study, but

the delay values are now taken as τ4 = 0.0197, τ5 = 0.076, τ6 = 0.13, τ7 = 0.03, τ8 =

0.026, τ9 = 0.022, τ10 = 0.1. Implementing the procedure reveals that there exist

three distinct ranges in the CFS, nf = 3. These ranges are as follows, [1.942, 2.136],

[2.811, 4.351] and [4.675, 5.581]. Portions of ℘ surfaces that correspond to each one

of the ranges are depicted separately, first, see Figure 4.11-4.13. This is done in order

to clearly label the frequency distribution of each individual surface with gray-scale

color coding. Next, these surfaces are combined together in 3D considering only η1

and η2 counters in (2.7) for visualization clarity, Figure 4.14.

Notice that it may not be easy to check whether the origin of Figure 4.14 is

asymptotically stable, since τ4, . . . , τ10 are non-zero. Both Nyquist stability criterion

(Ogata, 2002) and rightmost root solvers (Engelborghs, 2000) are exploited to assess

this. It is found that the origin of Figure 4.14 leads to asymptotic stability. Based on

Figure 4.11: Case 2: 3 dimensional depiction of a part of ℘kernel in (τ1, τ2, τ3) forτ4 = 0.0197, τ5 = 0.076, τ6 = 0.13, τ7 = 0.03, τ8 = 0.026, τ9 = 0.022 and τ10 = 0.1.

Gray-scale color coding represents ω ∈ Ω1 correspondence.

54






this information and the sensitivity of the characteristic roots across the ℘ surfaces

(Sipahi and Olgac, 2005), one can identify which portions of the 3D space correspond

to asymptotic stability. When inspecting this 3D stability map, one should recall

the τ -decomposition property which states that stability/instability behavior may

change only when one pierces through the surfaces. For the example at hand, any

3D delay point that can be connected to the origin of Figure 4.14 with a continuous

curve which does not pierce any of the ℘ surfaces leads to asymptotically stable

dynamics. The delay points in the remaining zones cause instability.

55


Remark 6. The proposed procedure runs on the same computer with the following

computation times. It successfully extracts the ℘kernel hypersurface in Figure 4.10

and the three kernel hypersurfaces in Figures 4.11-4.13 in less than 30 seconds and

2 minutes, respectively. Furthermore, the 3D example with N = 2 and L = 4 in

Sipahi (2007) requires about 25 seconds. One can clearly see that earlier claims

are coherent with the computation times observed in the examples. The only reason

for large computation times can be due to the size of increments in sweeping ω and

(x3, y3) ∈ C3, as well as the magnitude of upper bound frequency to which ω is swept.

The amount of computation time needed for solving two examples with two different

(N,L) pairs remains within the same order of magnitude under similar conditions.

Figure 4.14: Case 2: 3 dimensional depiction of ℘ and the stability map in (τ1, τ2, τ3)for τ4 = 0.0197, τ5 = 0.076, τ6 = 0.13, τ7 = 0.03, τ8 = 0.026, τ9 = 0.022 and

τ10 = 0.1. Gray-scale color coding represents ω ∈⋃3k=1 Ωk correspondence. System

is asymptotically stable at the origin.

56

Chapter 5

Delay-Independent Stability

Analysis

The main objectives of this chapter are to compute the lower bound¯ω and upper

bound ω of Ω, and to test the delay-independent stability in (2.1) based on the

necessary and sufficient conditions. Moreover, one can utilize DIS test for controller

design. Technically speaking, the DIS analysis here refers to the weak case, but not

the strong case, that is, delays at infinity are not considered here, see Definition 2.

This assumption, however, does not loose the essence of practical control problems,

where delays remain finite. Weak DIS, hereafter called DIS, means that, given A

and B`, system in (2.1) is asymptotically stable in the entire L-dimensional delay

parameter space, excluding the infinite delays (Michiels and Niculescu, 2007). DIS

holds when the following two conditions are satisfied,

(i) all the eigenvalues of A +∑L

`=1 B` have negative real parts, and

(ii) Ω in the entire delay parameter space is an empty set, Ω = ∅.

It is started by considering the DIS test of (2.1) as a problem of existence/absence

of the lower bound¯ω and upper bound ω of Ω. This problem formulation leads

to iterated discriminants development, which eventually yields a finite number of

57

CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS

single-variable polynomials. The roots of these polynomials are directly related to

the necessary and sufficient conditions of DIS of (2.1), or equivalently, to the absence

of the bounds ω and¯ω of Ω. Note that the approach does not impose any limitations

on the number of delays L, the system order N , or on the ranks and the entries of

system matrices in (2.1).

Remark 7. For the system in (2.1) to be delay-independent stable, it is necessary

that the system is asymptotically stable for zero delays (Michiels and Niculescu,

2007). Since the infinite delays are ignored for practical purposes, the analysis is

restricted to ω > 0 in the remaining of the text without loss of generality (Chen et

al., 2008; Fazelinia et al., 2007).

In some physical systems, there might be no information available about the

values of the inherently existing delays which can be very small or large. In these

circumstances, with the help of the DIS test in Section 5.1 (Delice and Sipahi, 2011),

one can guarantee stability robustness with respect to delays’ uncertainty (Bliman,

2002). In other words, in the worst-case scenario, the stability is guaranteed if the

system passes the DIS test. Moreover, the DIS test elicits how a controller, which

makes the system stable independently of delays, can be designed, see Section 5.2

(Delice and Sipahi, 2010b). By means of the DIS controller, one can ensure that the

system does not loose its stability property due to the detrimental effect of delays

on the stability.

5.1 Delay-Independent Stability Analysis for MTDS

Recall that studying Ω requires to study the roots of (3.3), which can be found from

g(jω, ~T ) = g<(ω, ~T ) + j g=(ω, ~T ) = 0 , (5.1)

58


where g< and g= are the real and imaginary parts of (3.3), respectively. When (5.1)

holds g< and g= are concurrently zero. It can be shown that these equations are in

the following form

g< =

cL∑i=0

ai(ω, T1, . . . , TL−1)T iL = 0 , (5.2)

andg= =

cL∑i=0

bi(ω, T1, . . . , TL−1)T iL = 0 . (5.3)

Notice the difference between ai versus ai, and bi versus bi by comparing (3.6)-(3.7)

with (5.2)-(5.3). The commensurate degree cL is known to be the largest power of

TL in both (5.2) and (5.3), and it is known that g< and g= do not simultaneously

vanish for all ω ≥ 0 when ~T = ~0, since (2.1) with ~τ = ~0 is asymptotically stable

(Sipahi and Olgac, 2005). Also it is assumed that, without loss of generality, g< and

g= do not have common factors. Such factors can be separately treated. Moreover,

in (5.2)-(5.3), acL and bcL terms can either vanish (identical to zero) or become zero

for some (ω, T1, . . . , TL−1) values. With this understanding, the highest power of TL

as cL in the summations is maintained.

Next, the resultant theory is utilized to eliminate TL from the two multivariate

polynomials g< and g= (Gelfand et al., 1994; Prasolov, 2004). A 2cL-order Sylvester

matrix is constructed via (3.8), and its determinant RTL(g<, g=) is a function of ω

and T1, . . . , TL−1.

Remark 8. The singularity of Sylvester’s matrix, RTL(g<, g=) = 0, is a necessary

condition for g< and g= to have common roots. Hence, studying the solutions of

RTL(g<, g=) = 0 is adequate for studying the solutions of g(jω, ~T ) = 0. This way is

followed in order to benefit the advantages of the resultant theory.

Based on the implicit function theorem (Courant, 1988), for the regular points

of the resultant and discriminant expressions calculated below, ω is differentiable

with respect to T1, . . . , TL, because the partial derivatives of these expressions are

59


multi-variable polynomials, and hence continuous with respect to ω (Johnston and

McAllister, 2009). The remaining few singular points, if any, can also be candidates

of extrema points (Larson, 2007), as explained in Section 3.2.1. With this knowledge,

the theorem that reveals the exact positive lower and upper bounds of Ω is now

provided.

Theorem 5. Minimum and maximum positive real zeros of the iterated discrimi-

nants

D(ω) := DT1

(DT2

(. . . DTL−1

(RTL(g<, g=))))

, (5.4)

that correspond to ~T ∈ RL are the exact positive lower bound¯ω and the exact upper

bound ω of the crossing frequency set Ω, respectively.

Proof. As per Remark 8, all ω that give rise to s = jω solution in (5.1) also satisfy

RTL(g<, g=) = 0 for some ω, T1, . . . , TL−1, where a mapping to TL exists through

(5.2)-(5.3). It is therefore adequate to seek¯ω and ω by studying RTL(g<, g=) = 0.

For the minima/maxima of ω to exist, it is necessary that ∂ω/∂TL−1 = 0. From

Courant (1988), for the regular points of RTL = 0,

∂RTL

∂TL−1

+∂ω

∂TL−1

∂RTL

∂ω= 0 . (5.5)

Since ∂ω/∂TL−1 = 0, for (5.5) to hold, a new equation, ∂RTL/∂TL−1 = 0, should

also hold as ∂RTL/∂ω 6= 0 for regular points1. One can now search for the common

solutions between RTL = 0 and ∂RTL/∂TL−1 = 0. Among these solutions lie¯ω and

ω for some T1, . . . , TL−1. For this search, one can eliminate TL−1 by constructing

RTL−1(RTL , ∂RTL/∂TL−1) via (3.8). With this,

¯ω and ω solutions are embedded into

the solutions of RTL−1= 0 in (T1, . . . , TL−2) domain, with mappings to TL−1 and TL

1Notice that RTL= 0 and ∂RTL

/∂TL−1 = 0 are also necessary conditions for singular pointsto exist. Hence, proceeding with the common solutions of these equations does not exclude thesingular points from the theorem, permitting us to capture also the singular points as candidateextrema points.

60


domains via RTL = 0, ∂RTL/∂TL−1 = 0 and g(jω, ~T ) = 0. If¯ω and ω exist, then

it is also necessary that ∂ω/∂TL−2 = 0, which can be analyzed with the same logic

used above in (5.5). The repetition of the same procedure until only the parameter

ω remains and all T` are eliminated leads to the following univariate polynomial in

ω

D(ω) := RT1

(RT2

(. . . RTL−1

(RTL , ∂RTL/∂TL−1))))

,

which is (5.4) as per Definition 3a. The minima/maxima,¯ω and ω, if they exist,

are among the roots of D(ω). For each root of D(ω), there exists ~T ∈ CL found via

sequential back-substitutions into single-variable polynomials RT2 = 0, ∂RT2/∂T1 =

0; RT3 = 0, ∂RT3/∂T2 = 0; ...; g = 0. The minimum and maximum positive real

zeros of the polynomial D(ω) that correspond to ~T ∈ RL are the exact positive lower

bound¯ω and the exact positive upper bound ω of Ω, respectively.

Note that equations (5.2)-(5.3) are interrelated and can be expressed as g2< +

g2= = 0. This new equation can be used to start the elimination procedure in

Theorem 5, instead of starting with RTL = 0. Nevertheless, this choice leads to

much higher powers of ω in (5.4) and is therefore not preferable from computational

efficiency point-of-view. Furthermore, the case of¯ω = 0 can be detected following

the extensions of Fazelinia et al. (2007).

It is stated that Theorem 5 treats both the regular and singular points of the

resultants, except when the singular points arise from repeated factors of the argu-

ments of the resultants. That is, so long the arguments of the discriminants do not

have repeated factors, Theorem 5 is applicable since the parametric discriminant

operation does not exclude the singularity points (Abhyankar, 1990). Furthermore,

the objective here is the detection of¯ω and ω regardless of identifying whether or

not the points are singular. For this objective, one only needs to check if the roots

of D(ω) have a mapping in ~T ∈ RL. When the arguments of the discriminants have

repeated factors, Theorem 5 needs to be modified.

61


5.1.1 Discriminant of Resultant RT`with Repeated Factors

When the arguments of the discriminants have repeated factors, the iterated dis-

criminants treatment in Theorem 5 needs a modification as explained next.

Lemma 6 (Wall (2004)). Let F = F (ν, µ`) = F (ν, µ1, µ2, . . . , µr), ` ≤ r, then the

discriminant Dν, µ`(F ) in (3.10) is identically zero if and only if F has a repeated

factor.

Lemma 6 states that partial derivatives ∂F/∂ν and ∂F/∂µ` will make the dis-

criminant Dν, µ`(F ) defined in Definition 3b vanish if and only if F has repeated

factors. Let us investigate how this information affects Dµ`(F ) = Rµ`(F, ∂F/∂µ`) in

Definition 3a, which is used iteratively in Theorem 5. In general, RT` = Qd`,1Q`,2Q`,3,

where d > 1, the polynomials Q`,1 and Q`,2 carry the variable T`−1, and the polyno-

mial Q`,3 has no T`−1 variable. It then follows that both ∂RT`/∂T`−1 and ∂RT`/∂ω

have a common factor of Qd−1`,1 . Therefore, all the roots of the repeated factor Q`,1

make the partial derivatives vanish. These roots are also some of the singular points

of RT` (Courant, 1988).

It is now easy to see that the discriminant DT`−1(RT`) = RT`−1

(RT` , ∂RT`/∂T`−1)

in Theorem 5 also becomes identically zero (always vanishes) due to the repeated

factorQd−1`,1 (Abhyankar, 1990, pg. 142). When this discriminant becomes identically

zero, the subsequent discriminant in Theorem 5 cannot be calculated. This issue

can be resolved with the following modification. The repeated factor Qd−1`,1 needs to

be eliminated and a modified resultant

R∗T` = RT`/Qd−1`,1 = Q`,1Q`,2Q`,3 , (5.6)

is to be found first. One should proceed with R∗T` in order to execute the remaining

steps of Theorem 5. Notice that this manipulation does not loose the insight of the

problem, but it carefully separates the multiplicity of the roots arising particularly

62


from repeated factors, incorporating them with multiplicity one into the discriminant

calculations in Theorem 5. Since R∗T` is square-free, that is, it does not have repeated

factors, the discriminant in Theorem 5 can be easily calculated. Once the analysis

provided in the proof of this theorem is complete, one can re-visit RT` = Qd`,1Q`,2Q`,3

to separately identify the multiplicity of the roots. This procedure is demonstrated

over two explanatory examples next.

Explanatory Example 1:

Consider the characteristic function of a MTDS given by

f(s, τ) = s2 + 13 s+ 20− 0.8 e−τ1 s + 29 e−2τ2 s . (5.7)

Using (5.7), the equation corresponding to (5.1) becomes

g(jω, ~T ) =((13T1 + 1)ω4 − 48.2ω2

)T 2

2 +(2T1 ω

4 + (16.4T1 − 26)ω2)T2

− (13T1 + 1)ω2 + 48.2 + j[(T1 ω

5 − (49.8T1 + 13)ω3)T 2

2

−((26T1 + 2)ω3 + 19.6ω

)T2 +

(−T1 ω

3 + (49.8T1 + 13)ω)]

= 0 . (5.8)

First, T2 is eliminated by calculating the resultant, RT2(g<, g=), which is

RT2 = −4ω4( (ω6+127.4ω4−408.36ω2

)T 2

1 + 41.6ω2 T1+ ω4+ 130.6ω2− 472.36)2

.

(5.9)

Notice that RT2 has a repeated factor in terms of T1 variable, thus the discriminant

of RT2 by eliminating T1, D(ω) = DT1(RT2), is identically zero not permitting us

to solve for ω. Therefore, a modification is needed as discussed above. By using

a symbolic manipulator, the resultant as explained in (5.6) is modified. It is now

63


possible to compute D(ω) = DT1(R∗T2

), which becomes

D(ω) = −256ω12 − 98662.4ω10 − 12343951.36ω8 − 457787783.168ω6

+ 5314120472.9856ω4 − 18202719158.8454ω2 + 20165057723.2527 , (5.10)

where ω = 0 roots are neglected, see Remark 7. It is found that the univariate

polynomial D(ω) has three positive real zeros, 1.7304, 1.7688 and 1.9179. By back

substitution of the minimum and the maximum of these roots into RT2 , ∂RT2/∂T1

and the characteristic equation (5.8), the corresponding T1 and T2 are found to

be real numbers. Therefore, it is concluded that¯ω = 1.7304 and ω = 1.9179.

The multiplicity of the roots are revealed by inspecting the roots of RT2 , which

really show multiple roots in T1 for both¯ω and ω. The results are as follows2:

(ω, T1, T2) = (1.7304, 1.1613, −0.9331), (1.7304, 1.1613, 0.3579), (1.9179, −0.2819,

−0.8746), (1.9179, −0.2819, 0.3108).

In this example, there exists an easier procedure, which does not need the resultant

modification. Since the order of elimination in (5.4) is immaterial, it is possible

to eliminate T1 before eliminating T2 by calculating D(ω) = DT2(RT1). This way

the repeated factor does not cause a problem in eliminating T2 in the discriminant

calculation, since multiple roots from the repeated factor do not arise in ω − T2

domain. Without the need for identifying the repeated factors, [¯ω, ω] is computed

directly as [1.7304, 1.9179]. One can now use this ω range and the ACFS method in

Chapter 4 (see also Sipahi and Delice (2011); Delice and Sipahi (2010a)) in order to

extract the stability maps on τ1− τ2 domain by sweeping the frequency from 1.7304

to 1.9179.

Although the factor Q`,3 of RT` is not repeated, it may also be eliminated. This

elimination can be done only if Q`,3 is a univariate polynomial in terms of ω. This

is because the derivatives of resultants with respect to ω are never calculated in

2Four-digit precision is used for numerical values in order to conserve space.

64


Theorem 5, and therefore the resultant RT` can always be modified without affecting

the results in the subsequent discriminants. In such cases, the roots of Q`,3 should

be separately studied. Moreover, the roots of Q`,3 may satisfy either one of the

arguments of RT` , i.e., either p1 or p2 polynomials (corresponding to Condition

(III)-(IV) of Theorem 1), or satisfy both p1 and p2 (corresponding to Condition (I)

of Theorem 1).

Explanatory Example 2:

Consider the characteristic function of a MTDS given by

f(s, τ) = s2 + 1.5− 0.35 e−τ1 s + 0.35 e−τ2 s . (5.11)

Equation (5.1) becomes

g(jω, ~T ) =(T1 ω

2 (ω2 − 1.5))T2 − (ω2 − 1.5)

+ j[(−ω (ω2 − 0.8)

)T2 − T1 ω (ω2 − 2.2)

]= 0 . (5.12)

First, T2 is eliminated by calculating the resultant RT2(g<, g=), which is

RT2 =((−ω4 +2.2ω2)T 2

1−ω2 +0.8)ω (ω2 − 1.5) , (5.13)

and next find

Q2,3 = ω (ω2 − 1.5) . (5.14)

Notice that this Q2,3 factor is arising from the non-repeated factor of RT2 , hence

the subsequent discriminant DT1 = RT1(RT2 , ∂RT2/∂T1) is not identically zero, and

the modified resultant R∗T2is not needed. It is preferred, however, to proceed by

constructing R∗T2with eliminating Q2,3 in order to ease the numerical computations.

65


Proceeding further, D(ω) = DT1(R∗T2

) is calculated as

D(ω) = −4ω6 + 20.8ω4 − 33.44ω2 + 15.488 , (5.15)

where ω = 0 roots are neglected, see Remark 7. The minimum and maximum

positive real root satisfying D(ω) = 0 are 0.8944 and 1.4832, respectively. Moreover,

positive real zeros of (5.14) should be separately studied. There is only one such

root,√

1.5 = 1.2247, which makes all the coefficients of g< in (5.12) zero. It can

be confirmed that for ω =√

1.5 , g= also becomes zero whenever T1 = T2. That

is, multiple roots of (5.12) occur in T1 − T2 domain where T1 = T2, and ω =√

1.5.

Finally, it is concluded that the lower and upper bounds of Ω are 0.8944 and 1.4832,

since it is found that ~T ∈ R2 satisfying RT2 = 0, ∂RT2/∂T1 = 0, and (5.12) for the

numerical values of these bounds.

Remark 9. Since, for the system in (2.1) to be DIS, it is necessary that the delay-

free system is asymptotically stable (Michiels and Niculescu, 2007), jω = 0 can be

a characteristic root only when some τ` → ∞, recall Remark 7. Moreover, because

τ` → ∞ is not a part of the weak DIS analysis, ω = 0 roots of D(ω) = 0 can be

ignored. Furthermore, ω → 0 and τ` → ∞ has a mapping in T` domain only when

T` → ∓∞ (Fazelinia et al., 2007). But in the converse, T` → ∓∞ can correspond

to a finite value of ω or to ω = 0. The case with finite ω is detectable from the

roots of D(ω) = 0, and the case with ω = 0 can be ignored since it corresponds to

τ` → ∞, which is not considered in the weak DIS analysis. Notice that Condition

(II) of Theorem 1 requires that a leading coefficient to vanish, that is, the parameter

multiplying this coefficient becomes unbounded, T` → ∓∞. In light of the above

discussions, such cases are taken care of by the iterated discriminants if ω is finite,

and ω → 0 solutions can be disregarded in the context of weak DIS. Finally, it is

noted that all the conditions of Theorem 1 are covered in our analysis since the

calculations are performed by studying the zeros of the resultants.

66


5.1.2 Delay-Independent Stability Test on the L-D delay

domain

The following theorem is the main result of this section.

Theorem 6. The MTDS in (2.1) is delay-independent stable in the entire L-D delay

domain if and only if the following two conditions are satisfied simultaneously:

(i) The matrix A +∑L

`=1 B` is Hurwitz stable.

(ii) The polynomial D(ω) in Theorem 5, with modified resultants when necessary,

has no positive real zeros corresponding to ~T ∈ RL.

Proof. Excluding τ` → ∞, Theorem 5 and the procedure of modified resultants

together form the first condition of the theorem guaranteeing that Ω is empty set.

Since Ω generates the stability switches/reversals, the condition Ω = ∅ does not

yield such switches, or vice versa, in the entire delay parameter space. As a result,

if there exist no stability switches, then the entire L-D delay domain exhibits the

delay free system’s stability behavior, which is stable by construction.

It is also noted that there exist studies on the analysis of DIS, see also Sec-

tion 2.2.2. What differentiates developed result is the combination of the following

two items: (a) DIS conditions are based on necessary and sufficient conditions, (b)

these conditions apply for the most general problem in (2.1) with L > 3.

Lemma 7. The polynomial D(ω) is an even and real polynomial.

Proof. RTL is a polynomial in terms of the coefficients of g< and g=. Ignoring its ω

factor, g= is an even polynomial as g< is. By inspection of the product formula of

the resultant (Gelfand et al., 1994),

RTL(g<, g=) = (acL)cLcL∏i=1

g=(δi) ,

67


where δi are zeros of g<, one sees that the resultant yields either an even or an odd

polynomial with respect to ω. If RTL is an odd polynomial, it can be converted to

an even polynomial by eliminating the ω factors. In this way, an even polynomial is

again obtained with respect to ω, excluding ω = 0 roots as per Remark 9. Moreover,

the power of ω remains even throughout the discriminant steps, since the multipli-

cation of two even polynomials is also an even polynomial. Hence, D(ω) is an even

polynomial with respect to ω. This polynomial is also a real polynomial, since the

coefficients of the resultants are all real.

Remark 10. The number of positive real zeros of even polynomials can be found

via a procedure proposed in Siljak (1969), and thus without numerically solving, one

can detect whether or not D(ω) = 0 has positive roots. The count of the number

of ω > 0 solutions can be used in place of the second condition of Theorem 6. If

this count is zero and the delay-free system is asymptotically stable, then the system

in (2.1) is guaranteed to be delay-independent stable. If this count is not zero, one

should use ω > 0 solutions to check whether or not corresponding ~T solutions are

real. If no real ~T solutions exist, one can still claim DIS of the system.

Lemma 8 (Louisell (1995)). Delay-independent stability of the single-delay TDS

~x(t) = A ~x(t) +L∑`=1

B` ~x(t− (`+ ε) τ) , (5.16)

is robust (well-posed) against all perturbations ε in the delay coefficients if and only

if the MTDS in (2.1) is delay-independent stable.

Lemma 8 states that testing the robustness of DIS property of (5.16) against

all delay ratios determined by ε ∈ R, with (` + ε) > 0, is equivalent to testing the

DIS property of MTDS in (2.1). Excluding ε → ∞ cases, the contributions of this

research also addresses the robust stability of (5.16) against all delay perturbations.

68


5.1.3 Delay-Independent Stability Test on the 2D delay do-

main

This subsection is the continuation of the delay-dependent stability analysis in Sec-

tion 4.1. With the procedures in this subsection and Section 4.1, delay-independent

and delay-dependent stability analysis can be performed on any 2D delay domain.

The DIS test developed above not only addresses the DIS problem on L-D delay

domain, but it is also applicable for specifically considered delay domain which

may contain fewer delays than L, (Delice and Sipahi, 2009a). This is especially

needed since visualization of stability maps is possible only in 2D and 3D domains

(Sipahi and Delice, 2009). For instance, one can check DIS conditions on any 2D

delay domain for a system with L > 2. Without loss of generality, the corollary to

Theorem 6 is proposed on τ1− τ2 delay domain.

Corollary 2. With Assumptions 1-4 in Sections 4.1, MTDS in (2.1) is delay-

independent stable on τ1− τ2 delay domain if and only if

(i) System in (2.1) is asymptotically stable when τ1 and τ2 are zero.

(ii) For both regular and singular points of the two resultants in Theorem 4, Z(ω) =

0 has no positive real roots that give rise to (T1, T2) ∈ R2 solutions in (4.2).

Proof. Condition (ii) guarantees that Ω = ∅. Since Ω generates ℘, the set Ω being

empty indicates that no ℘ exists, and vice versa. As a result, if there exists no ℘, the

entire 2D delay domain exhibits the stability behavior of the system at τ1 = τ2 = 0,

which is stable by the construction in condition (i).

Since ω = 0 solutions are ignored here, the DIS condition in Corollary 2 is

technically called weak (see Definition 2), that is, it excludes infinity delays. If

ω → 0 does not occur also for infinity delays, and if Ω = ∅, then the DIS condition

becomes strong (Chen et al., 2008).

69


5.1.4 Case Studies

Three case studies are presented to test the DIS property of TDS. In the first case,

the robustness of the DIS property against perturbations in delay ratios is investi-

gated. In achieving this, both single-delay dynamics and multiple-delay dynamics

are tested for DIS property. In the second case, three-delay dynamics with a com-

mensurate delay and a delay cross-talk term is considered. Finally, in the third case,

DIS analysis on 2D delay domain is demonstrated.

Case 1:

(A) DIS test: Consider the single-delay TDS (L = 1) governed by

~x(t) =

0 1 0

0 0 1

−20 −13 −4.1

~x(t) +

0 0 0.05

0.26 0 0

0 0.74 0

~x(t− τ) . (5.17)

The eigenvalues of the delay-free system are −0.9665∓ 2.8066 j and −2.1669, thus

the delay-free system is asymptotically stable. The characteristic function of the

system in (5.24) is

f(s, τ) = s3+41/10 s2+13 s+20−533/500 e−τ s+169/1000 e−2τ s−481/50000 e−3τ s .

(5.18)

The procedure to test the DIS property of the characteristic function (5.26) is

as follows. First, the Rekasius substitution in (3.1) is deployed in order to convert

(5.26) to (3.3) for τ1 = τ . Then, the parameter T1 in (3.3) is eliminated using

the resultant theory as shown in Theorem 5. The elimination leads to a univariate

polynomial which is given by

D(ω) =9∑

k=0

α2k ω2k ,

70


excluding ω = 0 solutions. In the above equation, all α2k’s are real constants and

omitted for brevity. After computing the zeros of D(ω) and it is found that none

of these zeros are real. As per Theorem 6, it is concluded that the TDS in (5.24) is

delay-independent stable.

(B) Robustness of DIS property against perturbations in delay ratios: Next, the

robustness of the DIS property of (5.26) against perturbations ε in delay ratios in

(5.16) of Lemma 8 is tested. To do this, as instructed in Louisell (1995), the terms

e−τ s, e−2τ s, and e−3τ s in (5.26) are replaced by e−τ1 s, e−τ2 s, and e−τ3 s, respectively,

where ~τ = (τ1, τ2, τ3) ∈ R30+. It is obtained a characteristic function of a MTDS

with three delays,

f(s, ~τ) = s3+41/10 s2+13 s+20−533/500 e−τ1 s+169/1000 e−τ2 s−481/50000 e−τ3 s .

(5.19)

Our approach follows from Theorem 5 and leads to

D(ω) =57∑k=0

β2k ω2k ,

excluding ω = 0 roots. In the above equation, all β2k ∈ R and they are omitted for

conciseness. It is verified that D(ω) has no positive real zeros. Since the delay-free

system is asymptotically stable, it is concluded as per Theorem 6 that the MTDS

in (5.19) is delay-independent stable. As per Lemma 8, it is concluded that the DIS

property of (5.26) is robust (well-posed) against all perturbations in delay ratios

(`+ ε) ∈ [0,∞).

(C) Computational efficiency: Remark that the computation times for testing

the DIS property of (5.24) and the MTDS represented by (5.19) are approximately

0.015 seconds and 0.35 seconds, respectively. What makes our approach extremely

fast is that it does not require any hand calculations, parameter sweeping and graph-

ical displays.

71


Case 2:

(A) DIS test: The DIS property for system (5.19) can also be checked via peer

methodologies, but only by using frequency sweeping or hand calculation tools.

An example, which is slightly different than (5.19), yet extremely difficult to treat

with the existing DIS test tools cited in Section 2.2.2 is presented. e−τ1 s and e−τ3 s

terms in (5.19) are multiplied by e−τ2 s and e−τ3 s, respectively, and the following

characteristic function is obtained

f(s, ~τ) = s3+41/10 s2+13 s+20−533/500 e−(τ1+τ2) s+169/1000 e−τ2 s−481/50000 e−2τ3 s .

(5.20)

Following the same procedure as in Case 1, it is found that D(ω) has no positive

real zeros. Since the delay-free system is asymptotically stable, it is concluded from

Theorem 6 that the MTDS represented by (5.20) is delay-independent stable.

(B) Computational efficiency: The computation time to test DIS of the MTDS

represented by (5.20) is approximately 0.6 seconds.

Case 3:

The system in Case 4.1.3 is considered for DIS analysis on 2D delay domain, and τ3

is chosen as 0.13. Following Corollary 2, the system is found to be delay-independent

stable on τ1− τ2 delay domain. The approach on average requires 4.5 seconds to

conclude on this DIS property. The detection of the DIS property is non-trivial. For

instance, using ACFS in Chapter 4, it is confirmed that the same system does not

have DIS property when τ3 = 1.5, see Figure 4.1 in Case 4.1.3.

5.1.5 Limitations

In order to make the DIS approach computationally more tractable, developments in

computer algebra on the computation of resultant and discriminant are extremely

72


important since these calculations, particularly iterated resultants and discrimi-

nants, need high computational power. Hence, improvements in this field favor

the feasibility and applicability of DIS approach. Interested readers are referred

to Buse and Mourrain (2009); Lazard and McCallum (2009) for details on iterated

discriminants.

5.2 Delay-Independent Controller Synthesis with

Sufficient Conditions

The objective in this section is to find controllers that render the stability of LTI-

MTDS insensitive to any delays in the closed-loop, that is, LTI-MTDS becomes

delay-independent stable (Delice and Sipahi, 2010b). This problem is investigated

on the general class of multi-input LTI systems,

~x(t) = A ~x(t) + B ~u(t) , (5.21)

where A ∈ RN×N and B ∈ RN×M are the constant system and control matrices,

respectively; system (5.21) is assumed to be controllable, ~x(t) ∈ RN is the state

vector, M is the number of inputs and the controller ~u(t) is affected by multiple

nonnegative delays τ`

~u(t) =L∑`=1

K` ~x(t− τ`) ∈ RM , (5.22)

whereK` ∈ RM×N , ` = 1, . . . , L, are the control laws, andK is defined as [K1, . . . ,KL] ∈

RM×M ·N .

The control synthesis of the general multi-input LTI-MTDS given by (5.21)-

(5.22), i.e., the selection of matrix K that stabilizes (5.21) for some delays τ` is a

challenging task, and is addressed in both frequency-domain (Michiels et al., 2002)

73


and in time-domain (Moon et al., 2001; Fridman and Shaked, 2002; Fridman et

al., 2003). In this dissertation, one step further is gone and the design methods

that reveal K matrix, which render the LTI-MTDS stable independent of all the

delays, are investigated. Similar problems are investigated in the context of H∞

control design based on Lyapunov-Krasovskii framework, see Baser (2003) and the

references therein. In this research, this non-trivial design problem is approached

from the frequency-domain stability analysis, which eventually leads to practical

and time-efficient algebraic design tools that do not require to solve Linear Matrix

Inequalities (LMI). The essence of our approach is as follows. It is known that

the imaginary eigenvalues of (5.21) may cause stability reversals/switches at some

delays ~τ (Datko, 1978). For a given K, if such eigenvalues do not exist for any τ`,

and if the delay-free system is asymptotically stable (when all τ` = 0), then the

controlled system (5.21)-(5.22) is DIS.

Remark 11. If ω = 0 is a root of (2.4), then system (5.21) is not DIS, and this

possibility can be checked and treated by Fazelinia et al. (2007) in the case of τ` →∞.

In the remaining of the text, such degeneracies are neglected, since τ` → ∞ is not

a practical case in control applications. It is also noted that ω = 0 can be a root of

(2.4) when ~τ = ~0. We prevent this possibility as well, by requiring that the delay-

free system is asymptotically stable, that is, A + B∑L

`=1K` being Hurwitz stable

should be satisfied as a necessary condition for DIS. This condition automatically

guarantees that a feasible K exists.

After relaxing the controller law K, the univariate polynomial in Theorem 6

reads

D(ω) =Kω∑k=0

α2k(K)ω2k , (5.23)

where α2k(K) coefficients are in terms of the controller gains in K, and Kω ∈ Z+.

Theorem 7. MTDS in (5.21)-(5.22) is stable independent of delays in the L-D

74


delay domain if all α2k(K) in (5.23) have the same sign, and A + B∑L

`=1K` is

Hurwitz stable.

Proof. According to Descartes’s rule of signs in Theorem 2, if all the coefficients of

the even polynomial (5.23) have the same sign, then there exists no positive real ω

roots of (5.23). Having no positive real roots of (5.23) indicates that all ω solutions

are complex conjugates since D(ω) is an even polynomial, see Theorem 7. When

there exists no positive real roots, Ω is ∅ from Theorem 5. Since CFS generates

the stability transitions, CFS being empty set indicates that there are no stability

transitions for all delays ~τ ∈ RL+, and the entire L-D delay domain exhibits the

delay-free system’s stability behavior, which is stable by construction.

Note that Theorem 7 requires us to inspect the coefficients of the polynomial

D(ω) without solving the roots of D(ω). This choice leads to sufficient conditions,

however, it offers a practical control synthesis approach constructed by algebraic

tools.

5.2.1 Case Studies

Case 1:

Consider the MTDS in (5.21) given by

A =

0 1

−6 −a1

, B =

1 0

0 1

, (5.24)

where a1 = 7.1, and the controller is given by

~u(t) =2∑`=1

K` ~x(t− τ`) , (5.25)

75


where K1 =

0 0

k1 0

, K2 =

0 0

0 k2

. The characteristic function of the closed

loop system is

f(s, ~τ,K) = s2 + 7.1 s+ 6− k1 e−τ1 s − k2 s e

−τ2 s , (5.26)

and it is easy to see that the delay-free system (when τ1 = τ2 = 0) is stable for

k1 < 6 and k2 < 7.1.

The approach commences with the manipulation in (3.3) for the two delays.

Then, (3.4) is found, and RT2 is constructed via (3.8) by eliminating T2. Next,

the discriminant of RT2 is calculated with respect to T1, DT1(RT2). This operation

is the iterated discriminant procedure introduced in Theorem 5, and it leads to a

single-variable polynomial (ignoring ω = 0 as noted earlier) given by

D(ω) =6∑

k=0

α2k(k1, k2)ω2k , (5.27)

where α2k(k1, k2) are listed with 4-digit precision,

α0(k1, k2) = (k1−6)2 (k1+6)4 > 0 ,

α2(k1, k2) = −0.01 (k1 + 6)2 (200 k31 + 1441 k2

1 + 53292 k1

+300 k21 k

22 − 1200 k2

2 k1 + 10800 k22 − 414828

),

α4(k1, k2) = −k41 + 129.64 k3

1 − 1547.3281 k21 + 13036.8972 k1 − 8296.56 k2

2 + 108 k42

− 777.84 k1 k22 + 3 k2

1 k42 − 76.82 k2

1 k22 + 12 k1 k

42 + 4 k3

1 k22 + 163223.4348 ,

76


α6(k1, k2) = 4 k31 − 76.82 k2

1 − 2172.8162 k1 − 4641.9843 k22 + 129.64 k1 k

22

+ 115.23 k42 − 2 k2

1 k22 − 2 k1 k

42 − k6

2 + 64963.9123 ,

α8(k1, k2) = −k21−141.64 k1−230.46 k2

2 +4 k1 k22 +3 k4

2 +4533.9843 ,

α10(k1, k2) = −2 k1−3 k22+115.23 ,

α12(k1, k2) = 1 > 0 .

Implicit functions α2k(k1, k2) are drawn on k1 − k2 domain next, see Figure 5.1.

Since α12 = 1 > 0, the shaded region in Figure 5.1 is found by imposing the

positivity of all α2k as well as by maintaining the stability of the delay-free system.

As per Theorem 7, it is concluded that (k1, k2) pairs chosen from the shaded region

guarantee that system in (5.24) with delayed state-feedback law in (5.25) is delay-

Figure 5.1: Case 1: Boundaries formed by α2k(k1, k2) coefficients. Controller gainsfrom the shaded region render the system delay-independent stable.

77


independent stable.

In order to validate our result in Figure 5.1, the numerical toolbox DDE-BIFTOOL

(Engelborghs, 2000) is implemented on the same system (5.24)-(5.25). Although

DDE-BIFTOOL is not designed for DIS test, we proceed to a case study where τ1

and τ2 are chosen as 100. The rightmost root distribution of (5.24)-(5.25) is found

with respect to k1−k2, and is depicted in Figure 5.2 using color coding that indicates

the number of unstable roots. The white region corresponds to the case when this

number is zero, that is, when the closed loop system is stable. Although Figure 5.2

is not conclusive to fully validate Figure 5.1, it provides a certain level of confidence.

The effects of damping ratio is then analyzed in the open loop system to the

shaded DIS region in Figure 5.1. The boundaries of the DIS regions are extracted

for different a1 values and are depicted in Figure 5.3. When a1 = 7.1, a1 = 4.5, and

a1 = 3.4, the corresponding damping ratios are ξ > 1 (solid black curve), ξ = 0.9186

(dashed red curve), and ξ = 0.694 (dotted blue curve), respectively. Controller gains

chosen from the closed regions in Figure 5.3 make the system delay-independent

Figure 5.2: Case 1: Comparison of the proposed method (color curves) and DDE-BIFTOOL result (gray shaded regions) for τ1 = 100 and τ2 = 100 on k1−k2 domain.Gray color coding indicates the number of unstable roots. White region indicatesstability.

78


Figure 5.3: Case 1: DIS regions are obtained for a1 = 7.1 (outer curve, dampingratio ξ > 1), a1 = 4.5 (dashed red curve, damping ratio ξ = 0.9186), and a1 = 3.4(inner curve, damping ratio ξ = 0.694). Controller gains from the closed regionsrender the system delay-independent stable for a given a1 parameter.

stable for the given a1 parameter or equivalently the damping ratio. Inspection

of Figure 5.3 shows that DIS regions in the space of controller gains are bounded.

These results are consistent with the earlier work (Michiels and Niculescu, 2007) on

bounded sets of stabilizing gains.

Finally, in Figure 5.4, the real part σ of the right most root with color code

is presented. In this figure, the boundary of the DIS region is displayed. With

a second-order system assumption, it is easy to see that settling time 4/σ of the

closed-loop system improves for some controller gain pairs chosen from the enclosed

DIS region. This is an interesting result as it shows that a closed-loop system can

be made DIS while still improving its settling time performance.

79


Figure 5.4: Case 1: DDE-BIFTOOL result for τ1 = 0.1 and τ2 = 0.15 on k1 − k2

domain. Gray color coding indicates the real part σ of the rightmost root.

Case 2:

Consider the same MTDS in Case 5.2.1, but this time, take the controller law as

K =

0 0 0.26 0

k1 1.7 k2 0

. (5.28)

Following the procedure in Case 5.2.1, the boundaries α2k(k1, k2) = 0 and the delay-

free system’s stability conditions (black color) are drawn in Figure 5.5. As per

Theorem 7, it is stated that (k1, k2) pairs chosen from the shaded region guarantee

that system in (5.24) is delay-independent stable.

Case 3:

Our methodology is also applicable to single delay DIS problems. Consider the block

diagram in Figure 5.6. The characteristic function of the closed-loop system is

f(s, τ,K) = s2 + 2 ξ ωn s+ ω2n + (kp + kd s)ω

2n e−τ s , (5.29)

80


Figure 5.5: Case 2: Implicit functions of α2k(k1, k2) coefficients and delay-free systemstability condition (black color). Controller gains from the shaded region render thesystem DIS.

where ξ > 0, ωn > 0; kp and kd are the proportional and derivative gains of the PD

controller, respectively. It is easy to see that the delay-free system is asymptotically

stable for kp > −1 and kd > −2 ξ/ωn.

Let ωn = 1 and follow the procedure as in Case 5.2.1 to obtain D(ω) (ignoring

ω = 0 as noted earlier)

D(ω) = ω4 + (−2− k2d + 4 ξ2)ω2 + 1− k2

p . (5.30)

As per Theorem 7, it is concluded that the closed-loop system in Figure 5.6 is delay

independent stable if

|kp| < 1 , |kd| < 2√ξ2 − 0.5 and ξ > 0.7071 .

We further analyze the effect of natural frequency on DIS condition. Let ωn = 5,

then the DIS condition is found as

|kp| < 1 , |kd| < 0.4√ξ2 − 0.5 and ξ > 0.7071 .

81


‐ + PD

Output Reference 2

2 22

sn

n n

e

s

twx w w

Figure 5.6: Case 3: Block diagram of closed-loop system, ξ > 0, ωn > 0.

Notice that the condition on the proportional gain of the PD controller does not

change with the natural frequency, and the range of the derivative gain of the con-

troller changes inversely proportional to the magnitude of the natural frequency.

Finally, note that sufficient amount of damping ratio, ξ > 0.7071, is needed for DIS,

independently of the natural frequency.

Remark 12. Given the complications in assessing DIS of linear time-invariant

multiple time-delay systems, our procedure based on Theorem 7 is efficient. It solves

the control synthesis problem under 0.3 seconds on average for all the three cases.

82

Chapter 6

Time-Delay Systems in Supply

Chain Management

6.1 Literature Review of Supply Chains

Inventory dynamics exhibit quite complex behavior in supply chains (SC) since in-

ventory level variations are the end results of combined decision-making, manufac-

turing, product shipment and information sharing activities which are dynamically

adapted against unpredictable and sometimes artificial consumer demand. While

excessive inventories (overshoot) cause increased stocking costs, undershoot of in-

ventory levels may increase freight costs and the risk of depletion of inventories, all

of which indicate inefficiency. Consequently, cost effective supply chain management

naturally requires thorough understanding of decision making, manufacturing, prod-

uct shipment dynamics and information sharing that directly affect the underlying

mechanisms of inventory behavior.

One of the most critical parameters in supply chain management (SCM) is the

delay (Sterman, 2000; Riddalls and Bennett, 2002b; Dejonckheere et al., 2004; Chat-

field et al., 2004; Kouvelis et al., 2006; Ouyang, 2007; Sipahi et al., 2009c; Marion

83

CHAPTER 6. TIME-DELAY SYSTEMS IN SUPPLY CHAIN MANAGEMENT

and Sipahi, 2009). Delay is inevitable in SC due to physical constraints related

to lead times (in manufacturing), transportation and delivery times (shipments),

decision-making durations (human behavior) and information availability (commu-

nication delays, data collection delays). In the presence of delays, what is known to

the SC manager is not what is happening in the chain, but it represents the infor-

mation regarding the SC’s behavior in the time history. When delays interfere with

the available information (Croson and Donohue, 2003) and the decisions, a supply

chain exhibits poor performance, improper synchronization, bullwhip effects (De-

jonckheere et al., 2004; Chatfield et al., 2004; Zhang and Burke, 2010), fluctuating

inventory levels (Helbing et al., 2004) and poor quality of service. Moreover, there

are multiple sources of delays in the SC and these delays are quantitatively different

(An and Ramachandran, 2005). Therefore, available information pertaining to SC

carries multiple delay signatures. What is detrimental to SCM is that delays mislead

decision-makers. This consequently prevents achieving successful SCM.

Although it is known that delays bring detrimental effects, in some cases it is

preferable that managers wait (adding delay) in order to observe the trends in the

SC and in the market before making critical decisions (Sterman, 2000). Clearly, it

is not straightforward to comment on the effects of delays to SCM. Riddalls and

Bennett (Riddalls et al., 2000) present an appropriate example about how a logical

decision leads to oscillation in supply chain dynamics such as increasing manager’s

response. These two counter-intuitive arguments justify the need to study delay

effects to dynamic behavior of the SC (Croson et al., 2004; Riddalls et al., 2000;

Beamon, 1998; Hafeez et al., 1996; Sipahi and Delice, 2010).

We quest if there are ways to uncover the effects of delays to inventories and to

SCM. If these effects can be understood with respect to intrinsic parameters defin-

ing the SC, then it would be possible to come up with new management strategies

that can combat against undesirable effects of delays. This is exactly what forms

84


the main objective here and it is aligned with the earlier work in (Mak et al., 1976;

Sarimveis et al., 2008; Warburton, 2004; Ge et al., 2004; Riddalls and Bennett,

2002b; Sterman, 2000; Simchi-Levi et al., 2000; Sterman, 1989). By performing sta-

bility analysis of the SC, the aim is to reveal various dynamical behaviors of the

SC and inventory levels with respect to delays and the parameters pertaining to

management strategies. The stability/instability definitions used in this disserta-

tion are along the lines of for instance Riddalls and Bennett (2002a); Naim et al.

(2004). For various combinations of management strategies, we are particularly in-

terested in finding the delay values with which the inventories behave in a desirable

way where inventory perturbations damp out, “stability”, rather than exhibiting

oscillatory behavior, “instability”. It may be true that SC dynamics may eventu-

ally stabilize itself with the presence of bounds (such as capacity limits), however,

the long durations of inventory oscillations, which are known to have large periods,

may put the SC into large financial losses before such bounds and extremis may

take over and stabilize the SC. In this sense, the contribution of this research can

be seen as the characterization of delay effects to such persistent and undesirable

transient behaviors observed in the inventory levels. As a result of our analysis,

SC manager has a decision-making tool with which the SC can be operated in a

stable regime based on various strategies and delays. With the tools this research

provides, it is also possible in some cases to dictate desirable inventory behavior

by scheduling some of the activities with appropriate delays similar to the work in

(Lee and Feitzinger, 1995), and to choose appropriate ordering policy with which

the inventory levels are rendered insensitive to undesirable effects of delays. The

results of this dissertation bridge the gap between surfacing undesirable effects of

delays in SC and how to make proper decisions to avoid these effects in SCM.

The mathematical framework of the study is constructed on Laplace domain,

which is known to have been used first time in 1952 (Simon, 1952) for studying

85


the stability of supply chains by Nobel Prize winner Herbert Alexander Simon. In

1961, Jay Wright Forrester also derived differential equations for the same reason

(Forrester, 1961). Furthermore, Denis R. Towill, in 1982 deployed Laplace transform

for studying inventory and order based production control system (Towill, 1982). In

Table 4 of Disney et al. (2006), it was shown that continuous time domain studies

are more preferable due to various reasons except one, that is, the pure delays. The

work presented here removes this concern, making continuous time domain analysis

and its connection with Laplace transform a perfect platform to analyze SC and

SCM.

The particular SC problem studied in this research is along the lines of Towill

(1982); John et al. (1994); Riddalls and Bennett (2002b, 2003), where an Automatic

Pipeline Inventory and Order Based Production Control System (APIOBPCS) with

two intrinsic deterministic parameters regulating a single inventory of a single prod-

uct shipped via a single link transportation path is considered. APIOBPCS model is

analyzed in subsection 6.2.2 with details and interested readers are referred to Delice

and Sipahi (2009b); Sarimveis et al. (2008); Beamon (1998) for other dynamic supply

chain models and to Zhou et al. (2006); Ilgin and Gupta (2010) for reverse supply

chain models. This model is also used in simplified forms in Hafeez et al. (1996);

Lewis et al. (1995). Interestingly APIOBPCS is similar to the heuristic stock ac-

quisition strategy of John D. Sterman which Sterman obtained from experiments

involving multiple users playing a beer distribution game (Sterman, 1989). What

is different in this dissertation is that delay originates from five dissimilar physical

sources hence five different delays are considered. These delays emerge from (i)

decision-making, (ii) production and (iii) transportation time and (iv) information

lags due to the time needed for respectively reporting of inventory and pipeline

(products in shipment but not in the inventory yet) levels to the decision-maker.

Hitherto, effects of each one of the five delays together was not investigated within

86


a unified model, despite the fact that these delays are known to exist (Ge et al.,

2004; Riddalls and Bennett, 2002b; Sterman, 2000). With the analysis performed in

this dissertation, we wish to present a broader picture as to how each delay governs

the stability mechanisms of the SC.

6.2 Preliminaries

In this section, the mathematical model of the SC with delays is presented. For the

SC model, we follow the earlier work in Towill (1982); John et al. (1994); Riddalls and

Bennett (2002b, 2003) in which a delay accounting for lead time in manufacturing

is considered. For representing the delay effects, similar research lines as in Riddalls

and Bennett (2002a,b, 2003) are adapted.

6.2.1 Mathematical Modeling of Delays

In order to realistically create the SC model, we consider lead times and trans-

portation times as pure time translation blocks acting on production and product

transport, respectively. Choice of pure delay is aligned with the fact that transporta-

tion is on a single path with one target delivery point (the inventory from which the

customers buy), and distribution and production delays exhibit much smaller vari-

ance. These choices also align with the earlier work of Riddalls and Bennett (2002b)

from which we quote “Hence, pure time delays are more realistic ... Indeed, for dis-

tribution systems, a pure delay lead time is unquestionably the most appropriate.”

A schematic depiction of pure delay effect on an inflow is depicted in Figure 2.1.

It is remarked that pure delay is not the only option in representing delays as it is

the case with decision making delays which have been argued to be slowly adapting

rather than rapidly changing. Consistent with pg 432 of Sterman (2000), we will

model decision making using a first-order system mimicking such an adaptation.

87


The first-order system as a matter of fact smoothness the stimulus received, repre-

senting the adaptation behavior. Moreover, humans not only adapt, but they also

need to process the stimulus and make a meaning out of it. This has been seen in

vehicle driving where human driver was modeled by similar adaptation (smoothen-

ing) functions in series with pure delays representing a dead-time during which the

humans process the incoming stimulus (Sipahi et al., 2007). It was also discussed in

Sterman (2000) that beliefs begin to respond only after some time has passed. These

facts suggest that decision making will behave as shown schematically in Figure 6.1

upon receiving a stimulus in the form of a step function. It is worth to mention

that this pure delay (dead-time) in some cases can be a very short period time,

thus can be neglected, however, this parameter as a means of a tuning parameter is

maintained.

6.2.2 Mathematical Modeling of the Supply Chain

Mathematical model considered here is the well-known Automatic Pipeline Inven-

tory and Order Based Production Control System (Towill, 1982; John et al., 1994)

that was also investigated in Riddalls and Bennett (2003, 2002b) where respective

authors analyzed the stability of this model with respect to two parameters and a

delay. The details of the model can be found in Riddalls and Bennett (2002b) and

in slightly different forms in Sterman (2000); Warburton (2004).

What governs the changes in inventory levels i(t) is the difference between the

Pure delay plus first-order

delay model

Step outflowat time

Time Time

Step inflowat time 0

0

Figure 6.1: Combination of pure (dead-time) and first-order delay model and itseffect on step input. This type of model can represent decision making delay.

88


inflow (pc(t): the completed production rate) to and outflow (o(t): customer demand

rate) from the inventory,

di(t)

dt= pc(t)− o(t) . (6.1)

The lead-time h > 0, which is the production delay, determines the relation between

demanded (pd(t), the rate at which the orders are placed at the manufacturer)

and completed production rates (the rate at which the manufacturer completes the

orders),

pc(t) = pd(t− h) . (6.2)

Notice that h is constant and shifts pd along the time axis. This still maintains

the continuity of pc when t > h, and represents first-in first-out type transport

phenomenon in supply chains (Sterman, 2000).

The heuristic decision-making policy developed by Sterman (1989, 2000) de-

termines how the desired production rate pd(t) should be formed as orders to the

manufacturer. The order rate to be placed to the manufacturer is equal to the

summation of forecast of demand, inventory regulation policy and work-in-progress

(WIP) control. Mathematically, it is given by

pd(t) = L(t) + αi(i(t)− i(t)) + αWIP

(h L(t)−

∫ t

t−hpd(µ)dµ

), (6.3)

where the first term is the expected demand L(t),

L(t) =1

T

∫ t

t−To(µ)dµ , (6.4)

which is a trend detector formed by measuring customer demand o(t) during a

period of time T . Decision-making parameters αi and αWIP are positive constants

penalizing discrepancy of the inventory from the desired set-point inventory level

i and WIP, respectively, and h is the expected production delay (M.-Jones et al.,

89


1997). The third term in (6.3) considers the steady state production hL found from

Little’s Law, which is then adjusted based on what has been already placed in the

production line (accumulation of pd(t) during [t− h, t] time window).

Next, one differentiates (6.3) and combines it with (6.1), (6.2) and (6.4) to obtain

the following differential equation,

dpd(t)

dt= −αWIPpd(t)− (αi − αWIP )pd(t− h)

+1

T(αWIP h+ 1 + αi T )o(t)− 1

T(αWIP h+ 1)o(t− T ) . (6.5)

The decision-making dynamics (6.5) known as APIOBPCS is widely studied in the

literature for its stability with respect to β = αWIP/αi and the lead time-delay h

(John et al., 1994; Towill et al., 1997; Riddalls et al., 2000; Riddalls and Bennett,

2002b,a; Lalwani et al., 2006; Sarimveis et al., 2008). When new products arrive

to the inventory, inventory level changes, and generally, it does not return to its

desired level i if the estimation h of h is incorrect. In other words, when h 6= h, a

drift from the desired inventory levels i will occur, even if the inventory dynamics

in (6.5) is stable (Disney and Towill, 2005).

90

Chapter 7

Contribution to Supply Chain

Management

7.1 Inventory Dynamics in Supply Chains with

Three Delays

7.1.1 Characteristic Equation of APIOBPCS with Three

Delays

In order to incorporate additional delays, one needs to carefully re-write the supply

chain model considering the two additional delays corresponding to decision-making

and transportation times. As motivated and justified earlier, transportation and

production delays are taken as pure delays, h2 and h3 (Figure 6.1) and decision

making is taken as a combination of a dead-time, h1, and a first-order smoothing

(Figure 6.1) which together becomes e−h s/(λ s+ 1) in Laplace domain.

Incorporation of these new terms (Figure 7.1) leads to the following homogeneous

91

CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT

1

s

1

s

- +

+

+ i

WIP

-

-

Inventory Desired

Inventory

pd pc

2h se

1

1

h se

s

3h se

pe

Figure 7.1: Block diagram representation of inventory dynamics displaying only theparts leading to homogeneous delay differential equation (7.1).

part of the governing dynamics

λd2pe(t)

dt2+dpe(t)

dt= −αWIP

(pe(t− τ1)− pe(t− τ2)

)− αipe(t− τ3) , (7.1)

where h in (6.5) is denoted by h2, τ1 = h1, τ2 = h1 + h2, τ3 = h1 + h2 + h3, λ

is a smoothing parameter of the decision making adaptation. The characteristic

equation of (7.1) is given by

f(s, ~τ) = λs2 + s+ αWIP (e−τ1 s − e−τ2 s) + αie−τ3 s = 0 , (7.2)

where ~τ = (τ1, τ2, τ3). Clearly when λ = h1 = h3 = 0, one recovers the homogeneous

part of (6.5). The block diagram representation of (7.1) which shows the homoge-

neous part of the SC dynamics is given in Figure 7.1 and an example simulation is

presented in Figure 7.2 to visualize the time-shifting and smoothing effects of de-

lays, how these delays affect the flows and how inventory changes after h1 + h2 + h3

amount of time elapses.

In the following, the methodology for analyzing the stability of (7.2) with respect

to ~τ is presented. Once this analysis is established, it is straightforward to express

92


Figure 7.2: Simulation of block diagram in Figure 7.1 for λ = 1.0, h1 = 1, h2 = 4,h3 = 3 weeks.

the results in the domain of ~h = (h1, h2, h3) via the obvious mapping defined between

~τ and ~h parameters.

7.1.2 Stability Analysis of a Supply Chain with Three De-

lays

For the complete stability analysis in 3D delay space ~τ , it is necessary and suffi-

cient to identify all the hypersurfaces, denote them by ℘, defining the locations of

delays that impart imaginary roots, s = jω, in the characteristic function (7.2).

Mathematically, ℘ hypersurfaces are defined as

℘ = ~τ ∈ R3+ | f(s, ~τ)

∣∣∣s=jω

= 0, ∀ω ∈ Ω . (7.3)

In order to facilitate easier depiction of stability, we present the cross sectional

views of the stability maps for any given fixed τ3 values, without loss of generality.

This leads to the display of curves on the τ1− τ2 plane for some non-zero τ3. This is

a logical approach, but the identification of these curves is not trivial. For a given

non-zero τ3, let us denote the cross-sections of ℘ with the curve ℘. In order to

93


obtain ℘, one needs to solve for all (τ1, τ2) ∈ R2+ and ω ∈ R+ from the complex

function f(jω, ~τ) = 0 in (7.2). This equation in terms of β and αi 6= 0 becomes

f(jω, ~τ) =1

αi(jω − λω2) + β e−jωτ1 − β e−jωτ2 + e−jωτ3 = 0 , (7.4)

where the first term is always non-vanishing. Although τ3 is assumed to be fixed here,

it is important that the stability analysis methodology is versatile to accommodate

any given τ3. The main challenge is that the characteristic equation couples all the

parameters ~τ as well as ω, therefore it is not trivial to solve all τ1, τ2 and ω from

(7.4). We will show that this challenge can be tackled by adapting the procedure

presented next.

7.1.3 Extracting Stability Switching Curves

In the sequel, the technique in Section 4.2 is applied to inventory regulation problem.

Without loss of generality, the stability for a fixed τ3 is investigated, however, the

technique allows us to fix any one of the delays and obtain the stability maps on

the plane of the remaining two delays. (7.4) is recast into

f(jω, ~τ) =3∑`=0

P`(jω)e−jωτ` = 0 , (7.5)

where τ0 = 0 and P`(jω) is self evident. ω is a given sweep parameter, thus it is

numerically known. Our procedure comprises the following steps for a given τ3:

Step i. For dummy variable ` = 1, 2, define the real and imaginary parts of P`(jω)

using (7.4) as:

P1<(ω) = β , P2<(ω) = −β , P1=(ω) = P2=(ω) = 0 . (7.6)

94


Step ii. For dummy variable ` = 0, 3, the terms correspond to a numerically known

complex number for a given sweep parameter ω. We distinguish them by defining

the following

χ(ω) + jγ(ω) = P0(jω) + P3(jω)e−jτ3ω , (7.7)

where χ(ω) = cos(τ3ω)− λω2

αiand γ(ω) = ω/αi − sin(τ3ω).

Step iii. Define

e−jτ`ω = x` + j y`, ` = 1, 2 , (7.8)

where (x`, y`) ∈ R2. Scalars x`, y` and τ` are the unknowns and the exponential

terms on the left-hand side of (7.8) define a unit circle in C,

|e−jτ`ω| = 1 ⇒ C`.= x2

` + y2` − 1 = 0, ` = 1, 2 . (7.9)

Following steps i-iii above, real and imaginary parts of equation (7.5) can be

expressed as

2∑`=1

M`

x`

y`

+

χ(ω)

γ(ω)

=

0

0

, (7.10)

where M1 = β I, M2 = −M1.

Step iv. The problem now reduces down to simultaneously solving (x`, y`) pairs

from the coupled equations (7.9) and (7.10). Since M1 is invertible, we have

x1

y1

= −M−11

M2

x2

y2

+

χ(ω)

γ(ω)

. (7.11)

95


Substituting M1, M2, χ(ω) and γ(ω), we obtain (x1, y1) as:

x1

y1

=

x2 − cos(τ3ω)β

+ λω2

αiβ

y2 − ωαiβ

+ sin(τ3ω)β

. (7.12)

Back substituting (x1, y1) solution from above into C1 yields a line equation on the

x2 − y2 plane,

L(x2, y2) = x2 Γ1(ω) + y2 Γ2(ω) + Γ0(ω) = 0 . (7.13)

The terms Γ`(ω), ` = 0, 1, 2 are frequency (ω) dependent coefficients,

Γ0(ω) =Γ

(αiβ)2, (7.14)

Γ1(ω) =2

αiβ

(λω2 − αi cos(τ3ω)

), (7.15)

Γ2(ω) =2

αiβ

(αi sin(τ3ω)− ω

), (7.16)

where Γ = 2αiω(1− sin(τ3ω) + λω cos(τ3ω)

)+ (αi − ω)2 + (λω2)2 .

Step v. For a given τ3 and ω, one can solve for (x2, y2) as follows. The (x2, y2) solu-

tions lie at the intersection points p1 and p2, as shown in Figure 7.3, between C2 and

the line equation in (7.13). By simultaneous solution, one obtains x2 components

of the points p1 and p2 as

x2 =−Γ0Γ1 ∓ Γ2

√∆

Γ21 + Γ2

2

, (7.17)

where ∆ = Γ21 + Γ2

2 − Γ20, and some arguments are suppressed for conciseness. If

x2 is a real number then proceed to next step, else go back to (7.14)-(7.16) and

numerically increment ω.

Step vi. The solutions x2 are used to obtain the corresponding y2 either from (7.9)

96


or (7.13). Then, one obtains the respective (x1, y1) pairs using (x2, y2) in (7.11).

Step vii. The corresponding (τ1, τ2) solutions are obtained from (7.8) by using

(x`, y`) found at Steps i-vi along with the sweep parameter ω,

τ` = − 1

ω

(/x` + j y` ∓ 2πη`

), η` ∈ N , ` = 1, 2 . (7.18)

Notice that delays (τ1, τ2) in (7.18) are the points that lie on the curve ℘. Once

℘ curves are found, it is trivial to express these curves in (h1, h2) by back transfor-

mation. We finally state that sweep parameter ω is proven to be upper bounded

by a finite ω (Hale and Verduyn Lunel, 1993; Sipahi and Delice, 2009). Hence one

needs to perform the procedural steps above by incrementing ω only from zero to

this upper bound.

7.1.4 Ordering-Policy Design for Delay-Independent Stabil-

ity

The procedure above only sweeps ω parameter from steps v-vii, and extracts the

stability maps on the τ1 − τ2 plane for any given τ3. This is a valuable tool for

the SCM and its repercussions to effective management will become clearer as we

2x

2

A

1A

2 22 2 1 0x y

2 2, 0L x y

2y

Figure 7.3: Intersection of unit circle and ω-dependent line equation as per (7.9)and (7.13), ω is fixed.

97


demonstrate the case studies in the next section. At this point, it is of interest to

develop some useful tools from ‘managerial point-of-view’. We quest if there can

be ways to choose ordering-policies with which inventory levels always behave as

desired (they oscillate with decreasing amplitudes and resume an equilibrium), no

matter what the detrimental effects of delays are. Obviously, the consequences of

achieving this would be quite desirable in designing delay-independent stability (DIS)

of SC. DIS can also be seen as finding the correct ordering-policy which renders the

SC insensitive to detrimental effects of delays. In what follows are the derivations

which lay out the management conditions under which delay-independent stability

of the inventory levels is possible. This can be seen as the continuation of the work

Riddalls and Bennett (2002b) which worked on DIS for h1 = 0, h2 6= 0, h3 = 0.

Theorem 8. Given αi and λ, for the inventory dynamics in (7.1) to exhibit stability

independent of any combination of τ1 and τ2,

1. τ3 satisfies,

0 ≤ τ3 < arctan(1/(λω))/ω , (7.19)

where ω =√−0.5 +

√0.25 + (αiλ)2 /λ ,

2. the inequality below is satisfied for all frequency values [0, ω] ,

4 < Γ0(ω) . (7.20)

Proof. Condition (1) guarantees the stability of the origin of τ1-τ2 plane. At the

origin, τ1 = τ2 = 0, the characteristic equation (7.2) becomes

f(s, τ3) = λs2 + s+ αie−τ3 s = 0 , (7.21)

stability of which holds if no s = jω solution exists for (7.21). The stability can be

easily analyzed; substitute s = jω in (7.21) and solve for ω and τ3. This would yield

98


only one solution for ω, which is ω =√−0.5 +

√0.25 + (αiλ)2/λ and corresponding

infinitely many τ3 solutions, which are τ3 = (arctan(1/(λω))∓2πη3)/ω. For stability

of (7.21), it is necessary and sufficient that 0 ≤ τ3 < arctan(1/(λω))/ω, which is the

case with counter η3 = 0. Thus, inequality (7.19) is obtained, see also Figure 7.4.

Condition (2) guarantees the stability in the remaining part of the τ1-τ2 delay plane.

To prove condition (2), it is important to notice that the existence of the solutions

in (7.18) indicates that these solutions lead to particular delay values separating

stability and instability behavior of the inventory dynamics. Hence, one should elicit

the cases when there exist no solutions of (7.18). This can be done by inspecting

the key formula (7.17) of our procedure. If the radicand in this equation is negative,

then there exist no real but complex solutions, which also implies that real delay

solutions do not exist. This requires that ∆ < 0 is satisfied. Substituting the

expressions from (7.14)-(7.16) leads to the inequality in (7.20).

Remark 13. One can obtain the necessary condition for delay-independent stability

on τ1 − τ2 domain of the SC by taking the limit ω → 0 in (7.20). The necessary

condition is found as β < 0.5. For single delay treatment in Riddalls and Bennett

(2002b), the necessary condition for delay-independent stability was found as β ≥

Figure 7.4: Given τ3 delay value, the maximum αi is computed for different λ valuesas part of the conditions guaranteeing the delay independent stability of the supplychain.

99


0.5. This condition is true only when h1 = h3 = 0, but the problem here is completely

different since h1 6= 0, h2 6= 0, h3 6= 0. It is also important to note that the procedure

we presented above can be developed for the special case of h1 = h3 = 0 and it can

be easily shown that β ≥ 0.5 condition found in Riddalls and Bennett (2002b) is

correct.

Approach for Policy Design

Notice that given τ3, it is straightforward to choose αi from Figure 7.4 or to satisfy

(7.19). This choice is also convenient as it does not depend on β. Next, one needs to

satisfy the inequality in (7.20). Since this inequality carries trigonometric terms, an

analytical result is not tractable, however, a simple computational approach exists.

Re-write this inequality as,

4β2 <1

(αi)2Γ(ω) , (7.22)

where the right hand side is only a function of ω. One can now sweep ω in a finite

range ω ∈ [0, ω] to find the infimum of the right hand side of the above inequality.

Using this infimum measure, admissible β range can be computed and formulated,

assuming β > 0,

0 < β < infω∈(0,ω]

√Γ(ω)

2αi. (7.23)

If the radicand in (7.23) is negative, then no admissible β values exist. In such a

case, stability independent of h1 and h2 cannot be possible for the given αi and τ3

values.

Managerial Repercussions

Rendering the inventory dynamics insensitive to delay effects is intriguing and it

was also an appealing theme in the aforementioned references. From control theory

perspective, a system that can maintain its stability independent from the values of

100


delays is feasible and elegant control design (similar to ordering-policy in SC) can

make this possible (Niculescu, 2001; Gu et al., 2003). From mathematical perspec-

tive, the realization of the idea makes logical sense if one assures no delay solutions

of the characteristic equation exist to distinguish stability from instability. From

practical point-of-view, however, these arguments may seem counter-intuitive. Let

us explain how stability may become insensitive to delays in the SC and discuss

the managerial repercussions of DIS. It is important to emphasize that larger delay

τ3 only allows smaller αi as can be seen from Figure 7.4. This observation nicely

ties with the well-known low gain control design or weak dynamic coupling where

controller is not aggressive, but it is chosen weak enough in order to avoid initiating

instability (Michiels and Niculescu, 2007). This is exactly what is happening in the

context of SCM. The DIS requires very small penalizing gain, αi, which weakly acts

on correcting the discrepancies in the inventories and in the WIP. The trade-off here

is between avoiding instability no matter what the (τ1, τ2) delays are and possibly

slower compensation of the inventory levels. In other words, to render the SC in-

sensitive to delays, the manager should choose less-aggressive policies, but he/she

should not expect rapid compensation of the inventory with the application of these

policies.

Policy design guaranteeing DIS can also be expressed on the parameter domain

of αi and β. For a given τ3, one chooses the αi on the curves in Figure 7.4. This

relationship can then be connected with β using equation (7.23). Mapping directly

on αi versus β plane reveals Figure 7.5 for different λ values. In this figure, any

point above each curve will guarantee DIS so long β < 0.5 as per Theorem 8.

101


Figure 7.5: Policy design for delay independent stability.

7.1.5 Repercussions to Supply Chain Management

Case 1: Stability for zero dead-time in decision making

The first case study considers that there exists no dead-time h1 in decision making.

When h1 = 0, the characteristic equation in (7.4) reduces to

f(jω, h2, h3) =1

αi(jω − λω2) + β − β e−jωτ1 + e−jωτ2 = 0 , (7.24)

where τ1 = h2, τ2 = h2 +h3. Notice that characteristic equation (7.24) is a sub-class

of the three delay characteristic equation (7.4). Our method reveals the correspond-

ing stability maps as shown in Figure 7.6. In this figure, the hatched side of the

stability boundaries is the region where inventory dynamics is stable, while delays

in the remaining regions lead to unstable inventory behavior. Figure 7.6 clearly

shows on h2− h3 plane that increasing β widens the stability regions offering larger

number of choices to SCM in rendering stability in the SC.

102


Figure 7.6: Case 1: Stability map on h2 − h3 domain for fixed αi = 0.4 1/weeks,λ = 2.5 weeks and dead-time h1 = 0 weeks.

Case 2: τ3 fixed

The second case study considers αi = 0.4 for various β values ranging from 0.5 to 1.

Delay τ3 = h1 + h2 + h3 is taken as 8 weeks. As mentioned earlier, this is the total

amount of delays it will take the new products to reach to the customer. In this

regard, the eight week delay time is fixed, but we shall see that the way each delay

shares this eight-week time will affect internally what happens with the inventory

dynamics. Our objective is to reveal the stability features of the inventory dynamics

with respect to αi, β and the other two delays h1 and h2. In Figure 7.7, stability

maps for β = 0.5, β = 0.7 and β = 1.0 cases are depicted. The hatched side of the

stability boundaries is the region where inventory dynamics is stable, while delays

in the remaining regions lead to unstable inventory behavior. For instance, when

β = 0.5, for h1 = 0.5 and h2 = 7 weeks (hence, h3 = 0.5 weeks), the inventories

exhibit stable behavior (similar to Point A in Figure 2.3c), while the delays h1 = 1

and h2 = 6 weeks (hence, h3 = 1 week) corresponds to unstable inventory behavior

(similar to Point C in Figure 2.3c).

Stability favoring effect of increasing β is again observed consistent with the

103


Figure 7.7: Case 2: Stability map on h1 − h2 domain for fixed αi = 0.4 1/weeks,λ = 2.5 and τ3 = 8 weeks.

earlier work, Warburton and Disney (2007); Warburton (2004); Riddalls and Bennett

(2003, 2002b). What is different in Figure 7.7 is that such a well-known fact is

confirmed in the presence of multiple delays.

Assume now that transportation time (h3) is 2 weeks. Since τ3 = 8 weeks, h1+h2

becomes 6 weeks. One can now exploit Figure 7.7 to decide which choice of β and

h1 + h2 = 6 weeks lead to stability of inventories. The parametric definition of

h1 + h2 = 6 weeks is a line on Figure 7.7 connecting the 6 weeks points on h1 and

h2 axis. A quick inspection reveals that most of this line lies in unstable regions

for β = 0.7, while it partially overlaps with the stable regions in the case when

β = 1.0. For β = 0.7, this stable region requires that decision making delay should

be less than 1.5 weeks and production delay should be in between 4.5 and 6 weeks.

A simulation of the inventories for this scenario where h1 = 0.5 and h2 = 5.5 is

shown in Figure 7.8 cross-validating the readings in Figure 7.7.

Case 3: τ2 fixed

In this case study, we change the domain of interest to the two delays h1 and

h3, while we fix τ2 = h1 + h2 to 5 weeks. By keeping αi = 0.4 1/weeks, we depict

104


Figure 7.8: Case 2: Inventory levels are adapting to a change of 10 units from aninitial 200 units to 210 units in Figure 7.1. h1 = 0.5, h2 = 5.5, h3 = 2 and λ = 2.5weeks.

Figure 7.9: Case 3: Stability map on h1 − h3 domain for fixed αi = 0.4 1/weeks,λ = 2.5 and τ2 = 5 weeks.

stability regions on h1 vs h3 plane for different choices of β, Figure 7.9. One extracts

observations similar to those from Figure 7.7: when β increases, effective stability

regions enlarge. For instance, when β = 1.0, the point h1 = 1.5 and h3 = 3 weeks

leads to stable inventory behavior while the same point causes instability when β

becomes 0.7. Simulation of a scenario is depicted in Figure 7.10 in order to compare

the effects of β at the point h1 = 1 and h3 = 3 weeks.

105


Figure 7.10: Case 3: Inventory levels are adapting to a change of 10 units from aninitial 200 units to 210 units in Figure 7.1. h1 = 1, h2 = 4, h3 = 3 and λ = 2.5weeks.

7.2 Generalized Supply Chain Model

7.2.1 Development of the Model

In this section, we derive the equations that incorporate five delays into the supply

chain model in (6.5), along with first order adaptation dynamics for decision-making,

production and transportation. Moreover, a PI controller is added to (6.5) in order

to eliminate the inventory drift, see in Figure 7.11 the schematic representation of

products and information flow in the supply chain considered.

A first-order adaptation dynamics in decision-making is reasonable to model how

pd(t) will relax to pe(t− h1) with a time-constant λ1 (Nise, 2004; Sipahi and Delice,

2010),

λ1dpd(t)

dt+ pd(t) = pe(t− h1) , (7.25)

where pd is the input rate to the manufacturer, pe is the decision-making error rate,

and h1 is decision-making delay. Similarly, between pd(t) and completed production

106


Management Transportation

3h

Information of products in shipment

with 5h

Manufacturing Inventory Customers

Information of inventory level

with 4h

+ +

New orders

2h 1h

Figure 7.11: Schematic representation of the flow of products and information. h1,h2, h3, h4 and h5 respectively denote human decision-making, production, trans-portation, information of inventory level and information of products in shipmentdelays.

rate pc(t) (modifying (6.2)), we have

λ2dpc(t)

dt+ pc(t) = pd(t− h2) , (7.26)

and the second-order dynamics between pc(t) and inventory level i(t) becomes

λ3d2i(t)

dt2+di(t)

dt+ λ3

do(t)

dt+ o(t) = pc(t− h3) , (7.27)

where h2 and h3 are production and transportation delays, respectively, and λ`,

` = 2, 3 are related to adaptation speeds (time-constants) in (7.26)-(7.27). By

selecting sufficiently small or large λ`, one can render fast and slow adaptations, as

desired. Moreover, one recovers (6.1)-(6.2) when λ` = 0, ` = 1, 2, 3 and h1 = h3 = 0,

h2 = h in (7.25)-(7.27). Furthermore, the heuristic decision-making policy is now

developed based on pe(t), which is the available information. Similar to (6.3), it is

formed by

pe(t) = L(t) + αi(i(t)− i(t− h4)) + αI∫ t

0

(i(t)− i(µ− h4))dµ

+ αWIP

(h L(t)−

∫ t

0

(pd(µ− h5)− pc(µ− h5))dµ

), (7.28)

107


where h4 and h5 are information delays due to the time needed for respectively

reporting of inventory and pipeline (products in shipment but not in the inventory

yet) levels to the decision-maker. We wish also eliminate inventory drift by utilizing

a PI controller, which brings an additional parameter αI to be considered in (7.28).

αI is known as the integral gain of the PI controller.

In order to convert the system of differential equations (7.25)-(7.28) into a single

equation, we first re-arrange (7.25) and (7.26) in terms of inventory level so that one

can substitute them into (7.28). Delaying (7.26) by h3 and using (7.27), equation

(7.26) becomes

pd(t− h2 − h3) = λ2 λ3d3i(t)

dt3+ (λ2 + λ3)

d2i(t)

dt2+di(t)

dt

+ λ2 λ3d2o(t)

dt2+ (λ2 + λ3)

do(t)

dt+ o(t) . (7.29)

Similarly, delaying (7.25) by h2 + h3 and using (7.29), (7.25) becomes

pe(t− h1 − h2 − h3) = λ1 λ2 λ3d4i(t)

dt4+ (λ1 λ2 + λ1 λ3 + λ2 λ3)

d3i(t)

dt3

+ (λ1 + λ2 + λ3)d2i(t)

dt2+di(t)

dt+ λ1 λ2 λ3

d3o(t)

dt3+ (λ1 λ2 + λ1 λ3 + λ2 λ3)

d2o(t)

dt2

+ (λ1 + λ2 + λ3)do(t)

dt+ o(t) . (7.30)

Secondly, differentiating (7.28) with respect to time and delaying by h3, one can

substitute (7.27) into (7.28),

dpe(t− h3)

dt= (1+αWIP h)

dL(t− h3)

dt−αi di(t− h3 − h4)

dt+αI (i− i(t−h3−h4))

−αWIP pd(t−h3−h5)+αWIP

(λ3d2i(t− h5)

dt2+di(t− h5)

dt+ λ3

do(t− h5)

dt+ o(t− h5)

),

(7.31)

108


where desired inventory level i(t) = i is assumed to be constant. Thirdly, delaying

(7.31) by h2, equation (7.29) can be substituted to yield

dpe(t− h2 − h3)

dt= (1 + αWIP h)

dL(t− h2 − h3)

dt− αi di(t− h2 − h3 − h4)

dt

+αI (i−i(t−h2−h3−h4))−αWIP

(λ2 λ3

d3i(t− h5)

dt3+ (λ2 + λ3)

d2i(t− h5)

dt2+di(t− h5)

dt

+λ2 λ3d2o(t− h5)

dt2+ (λ2 + λ3)

do(t− h5)

dt+ o(t− h5)

)

+αWIP

(λ3d2i(t− h2 − h5)

dt2+di(t− h2 − h5)

dt+ λ3

do(t− h2 − h5)

dt+ o(t− h2 − h5)

).

(7.32)

Finally, delaying (7.32) by h1, derivative of (7.30) and (6.4) can be substituted, and

we obtain the differential equation of the generalized supply chain model as

λ1 λ2 λ3d5i(t)

dt5+ (λ1 λ2 + λ1 λ3 + λ2 λ3)

d4i(t)

dt4+ (λ1 + λ2 + λ3)

d3i(t)

dt3+d2i(t)

dt2

+ αWIP

(λ2 λ3

d3i(t− h1 − h5)

dt3+ (λ2 + λ3)

d2i(t− h1 − h5)

dt2+di(t− h1 − h5)

dt

)

−αWIP

(λ3d2i(t− h1 − h2 − h5)

dt2+di(t− h1 − h2 − h5)

dt

)+αi

di(t− h1 − h2 − h3 − h4)

dt

+ αI i(t− h1 − h2 − h3 − h4) = −

(λ1 λ2 λ3

d4o(t)

dt4+ (λ1 λ2 + λ1 λ3 + λ2 λ3)

d3o(t)

dt3

+(λ1+λ2+λ3)do2(t)

dt2+do(t)

dt

)−αWIP

(λ2 λ3

d2o(t− h1 − h5)

dt2+(λ2+λ3)

do(t− h1 − h5)

dt

+ o(t− h1 − h5)

)+ αWIP

(λ3do(t− h1 − h2 − h5)

dt+ o(t− h1 − h2 − h5)

)

+1

T(1 + αWIP h)

(o(t− h1 − h2 − h3)− o(t− h1 − h2 − h3 − T )

)+ αI i . (7.33)

109


1s

1s

- +

+

+ ( )C s

WIP

-

-

Inventory Level

Desired Inventory Level

pd pc

5h se

1

1 1

h ses

4h se

pe 2

2 1

h ses

3

3 1

h ses

ˆ1 WIP hs

se

Customer Demand

+

+

-

-

WIP compensation

Decision-making

Production

Transportaton

Information delay

Information delay

Forecasting

Inventory regulation

Figure 7.12: Block diagram representation of the supply chain model (7.33). C(s) iseither αi for proportional control or αi +αI/s for proportional-integral (PI) control.

Block diagram representation of (7.33) using Laplace algebra is shown in Figure 7.12,

and the characteristic function of (7.33), which is the Laplace transform of the

homogeneous part of (7.33), is given by

f(s,~h) =s2

αi(λ1 s+ 1)(λ2 s+ 1)(λ3 s+ 1) + β s (λ2 s+ 1)(λ3 s+ 1) e−(h1+h5) s

− β s (λ3 s+ 1) e−(h1+h2+h5) s + (s+ αI/αi) e−(h1+h2+h3+h4) s , (7.34)

where s ∈ C is the Laplace variable in the complex plane C, and ~h = (h1, h2, h3, h4, h5).

We can now use the characteristic function of our model to analyze stability. We

next proceed to the stability analysis of the generalized supply chain model.

Remark 14. Steady-state analysis of block diagram in Figure 7.12 reveals that drift

110


in the generalized supply chain model equals to β (h − h2 − λ2) o(t). Notice that β

and o(t) are positive values, however, by setting h to h2 + λ2, drift problem can be

avoided if the decision-maker has the exact knowledge of summation of production

delay h2 and the time-constant of the manufacturer λ2. Hence, in order to prevent

drift without utilizing a PI controller, not only does the manager have to predict

production delay (as in the case of APIOBPCS model), but also he needs the pre-

cise value of time-constant of the manufacturer. To avoid the difficulty of these

estimations, PI controller can be utilized as we demonstrate in the next section.

7.2.2 ACFS Application to Inventory Regulation Problem

In the sequel, some key parts of the ACFS technique is presented succinctly for

inventory regulation problem. ACFS can extract stability maps of linear time-

invariant systems with multiple delays in any two-delay domain while there can be

arbitrarily large number of delays in these system. Without loss of generality, we

choose h1 − h2 domain as the domain of stability displays, and fix the remaining

delays as h3 = h3, h4 = h4 and h5 = h5 where • indicates a fixed value of the

variable •. The rationale behind fixing some delays in the context of supply chains

is as follows. In supply chains, there can be some delays that are more or less fixed

such as transportation and information transmission delays, while decision-making

and production delays could be tunable or more uncertain. Furthermore, in some

cases, several shipment or manufacturing options with different delays may exist.

Hence, by fixing the known delays one can study the effects of different options to

the inventory oscillations.

Due to the presence of transcendental terms, the characteristic functions similar

to (7.34) possess infinitely many roots for a given set of delays. Furthermore, these

functions are classified as ‘retarded’ systems whose roots exhibit continuity in C with

respect to delays (Datko, 1978). Since instability occurs when a characteristic root s

111


has a positive real part, it makes sense to focus on the imaginary root solutions that

transit from stability to instability, or vice versa. To study the critical roots s = jω

that lie on the imaginary axis, we deploy Rekasius substitution (3.1), which is exact

for s = jω, for the delays h1 and h2. In (3.1), T` can be seen as a parameter that

facilitates the necessary calculations without the overwhelm of exponential terms.

That is to say, substitution (3.1) simply creates an algebraic equation in terms of

(ω, T1, T2) by eliminating the exponential terms with (h1, h2). Consequently, the

characteristic function to be studied becomes

g(jω, T1, T2, e−jωh3 , e−jωh4 , e−jωh5) =

f(s,~h)

∣∣∣∣∣∣∣∣∣∣e−jωh` := 1−jωT`1+jωT`

,

` = 1, 2.

2∏`=1

(1 + jωT`) .

(7.35)

By means of frequency sweeping, (T1, T2) roots of (7.35) can be found, and

the corresponding delays h1, h2 can be solved from (3.1) using the frequency ω and

(T1, T2) pairs. That is, delays are found from (3.2). More importantly, characteristic

functions of ‘retarded’ type are guaranteed to exhibit ω solutions only within finite

ranges (Stepan, 1989). This property enables convenient sweeping of ω parameter

in the ACFS framework. For each ω = ω, the characteristic function (7.35) can be

decomposed into real and imaginary parts

g(T1, T2) = g<(T1, T2) + j g=(T1, T2) , (7.36)

where g< = <(h) and g= = =(h) are the real and imaginary parts of (7.35), respec-

tively. Notice that numerically known terms are dropped from the arguments for

clarity. If g = 0, then both g< and g= should be satisfied for some (T1, T2) pairs that

have a mapping in (h1, h2) via (3.2). We find that g< and g= are in the following

112


particular forms

g< = a1(T1)T2 + a0(T1) = 0 , (7.37)

and

g= = b1(T1)T2 + b0(T1) = 0 , (7.38)

where a0, a1, b0 and b1 are real polynomials in terms of T1.

As suggested above, the frequency ω can be swept and common solutions of

(7.37)-(7.38) can be computed. For this, a conservative upper bound of the frequency

can be selected first, ω. Then, for each ω ∈ (0, ω] with an appropriately chosen step

size, we can perform the following steps: (i) First, solve T1 and T2 from the linear

system of equations (7.37)-(7.38). (ii) Secondly, if T1 and T2 are real, proceed to

the third step, otherwise increase ω by an amount of the step size and return to the

first step. (iii) Thirdly, using the back transformation formula in (3.2), compute

the delay values (h1, h2) corresponding to (T1, T2) real pairs, and restart from the

first step after increasing ω by an amount of the step size chosen previously. When

frequency ω reaches to its upper bound, all delay values (h1, h2) that construct

the boundaries of the delay domain are extracted completely. These boundaries

decompose the h1−h2 space into regions where the inventory oscillations are either

stable or unstable.

The first step of the ACFS procedure has some intriguing properties. Notice

that common T1 solutions in (7.37)-(7.38) exist if the following matrix is singular

S =

a1(T1) a0(T1)

b1(T1) b0(T1)

. (7.39)

Moreover, existence of real T1 values depends on the discriminant of the quadratic

polynomial equation a1(T1)b0(T1)− b1(T1)a0(T1) = 0, which is the determinant of S.

If the discriminant is positive in the second step of the ACFS procedure, then there

113


exist two real (T1, T2) pairs satisfying (7.37)-(7.38). Similarly, if the discriminant

of the polynomial is zero, (T1, T2) pairs coalesce into one and a double root occurs.

Otherwise, there exist no real (T1, T2) common solutions.

7.2.3 Supply Chain Management in the Presence of Multi-

ple Time-Delays

In the sequel, we present the implementation of ACFS for the stability analysis

of inventory levels in the generalized supply chain model (7.33). The objective is

to extract the stability maps using ACFS. Recall that, for the PI controller case,

integral controller in Laplace domain is C(s) = αi +αI/s, whereas the proportional

controller is given by C(s) = αi in Laplace domain.

Case 1: Proportional Controller

First, we consider the case when C(s) = αi. The characteristic function becomes

f(s,~h) =s

αi(λ1 s+ 1)(λ2 s+ 1)(λ3 s+ 1) + β (λ2 s+ 1)(λ3 s+ 1) e−(h1+h5) s (7.40)

−β (λ3 s+ 1) e−(h1+h2+h5) s + e−(h1+h2+h3+h4) s .

In order to extract the stability map of (7.40) on h1 − h2 domain, the following

parameters are fixed as αi = 0.4 1/weeks, λ1 = 2.0 weeks, λ2 = 0.5 weeks, λ3 = 0.4

weeks, and delays h3 = 1 week, h4 = 0.15 weeks and h5 = 0.4 weeks. We then

proceed to extract three stability boundaries by means of ACFS. The results are

given in Figure 7.13 for three different β values. In this figure, blue, black and red

color stability curves correspond to β = 0.5, β = 0.7 and β = 1.0, respectively.

Delay values inside the closed regions that include the origin h1 = h2 = 0 reveal

asymptotic stability of the inventory levels in the supply chain. In the remaining

regions, inventory levels are unstable.

114


To give an example, when h1 = 1.0 week, h2 = 3.0 weeks, and β = 0.5, then the

inventories are not stable due to the way the corresponding blue stability boundary

partitions the delay space. However, for β = 0.7 and β = 1.0 values, inventories

exhibit stability for (h1, h2) = (1.0, 3.0) point. We also perform time-domain simula-

tions of the block diagram in Figure 7.12 for various β values for the fixed operation

point, αi = 0.4 1/weeks, αI = 0, o(t) = 5 step units, T = 40 weeks, λ1 = 2.0 weeks,

λ2 = 0.5 weeks, λ3 = 0.4 weeks, and delays h1 = 1 week, h2 = 3 weeks, h3 = 1

week, h4 = 0.15 weeks and h5 = 0.4 weeks and the incorrect estimation of h2 + λ2

is taken as h = 5.5 weeks in order to demonstrate how inventory drift occurs, see

Figure 7.14. Inspection of the inventory behaviors in Figure 7.14 concludes that

β = 1.0 can be a more suitable parameter when h1 = 1.0 week and h2 = 3.0 weeks.

When we choose β = 0.5, we have less choices of h1 and h2, since the stability

region is smaller compared to the cases with β = 0.7 and β = 1.0 values, see

Figure 7.13. By inspection of this figure, we state that conditions 0 < h1 < 0.5

and 0 < h2 < 4 can be selected for stable supply chain management. Moreover,

Figure 7.13: Case 1: Stability map on h1−h2 domain for different β values, β = 0.5(blue), β = 0.7 (black), β = 1.0 (red). Parameters αi = 0.4 1/weeks, λ1 = 2.0weeks, λ2 = 0.5 weeks, λ3 = 0.4 weeks and delays h3 = 1 week, h4 = 0.15 weeks,h5 = 0.4 weeks are fixed. For a given β value, delay values chosen from the regionsthat include the origin reveal stability.

115


if we have a supplier that manufactures the products in approximately 4 weeks,

Figure 7.13 implies that the maximum delay for decision-making should be smaller

than 0.5 weeks.

It is intriguing to observe that decision-making delay does not change signifi-

cantly when production delay is relatively small, h2 → 0, see Figure 7.13. Therefore,

if the manager needs more time to decide, he/she could prolong the production in

order to fall into a stable operation region. This example demonstrates how delays

create counter-intuitive scenarios, in which human comprehension may be limited.

This point is also highlighted in M.-Jones and Towill (1997) which reports that expe-

rienced decision-makers may perform badly in the presence of multiple time-delays.

Clearly, the availability of stability maps by means of ACFS prevents confusion and

enables variety of strategies in supply chain management for maintaining steady

inventory levels.

In this case study, we can also fix the production delay, and obtain the stability

maps on h1 − h3 domain. For example, we first let supplier production delay h2 =

4.0 weeks and obtain the corresponding stability map in Figure 7.15. One can

Figure 7.14: Case 1: Simulation of block diagram in Figure 7.12 for various β values.

116


now choose proper delay combinations that lead to stability in inventory levels.

For instance, h1 = 1.0 week and h3 = 2.0 weeks is a stable operation point for

both β = 0.7 and β = 1.0. However, if the manager chooses β = 0.5, then the

supply chain becomes unstable. Here β = 0.5 indicates that correcting the work-

in progress is half as important as correcting inventory discrepancy. Moreover, in

contrast to Figure 7.13, decision-making delay varies significantly in Figure 7.15 and

increases with increasing β value as h3 → 0. Similar observations can be done for

h1 → 0 in Figure 7.15. It can be inferred that by decreasing decision-making delay

(ordering with less delays), the manager of the supply chain is free to add delay to

transportation times with proper importance of WIP adjusted by β, and he/she can

still maintain the operation point inside of the stability boundaries.

With respect to the ratio β, we observe that increasing β in this case study en-

larges the stability regions. This can be utilized as another decision-making strategy.

It implies that penalizing the discrepancies between work-in-progress and inventory

levels with identical weights, that is, αi = αWIP and β = αWIP/αi = 1.0, the

Figure 7.15: Case 1: Stability map on h1−h3 domain for different β values, β = 0.5(blue), β = 0.7 (black), β = 1.0 (red). Parameters αi = 0.4 1/weeks, λ1 = 2.0weeks, λ2 = 0.5 weeks, λ3 = 0.4 weeks and delays h2 = 4 weeks, h4 = 0.15 weeks,h5 = 0.4 weeks are fixed. For a given β value, delay values chosen from the regionsthat include the origin reveal stability.

117


stability regions enlarge compared to the case when β = 0.5. This indicates that

when inventory discrepancies are penalized two times more than the discrepancies

in WIP, the stability regions shrink. This is an interesting conclusion in choosing

the ‘control parameters’, αi and αWIP , of the chain.

Enlargement of stability regions with increasing β has a limit. In h1 − h2 do-

main, stability of (3.0, 9.0) point is lost when one increases β from 0.7 to 1.0 though

(3.0, 5.0) point and its surrounding region change their stability property from insta-

bility to stability. In h1− h3 domain, it is observed in Figure 7.15 that increasing β

in the range of (0, 1] enlarges the stability regions. Moreover, it is also remarked that

an ellipsoidal shape (partially shown) penetrates through the stable region, which,

as a result, shrinks the stable region. Although we gain some new stable regions

when h1 → 0 or h3 → 0, we lose the center part of the stable region (see the curved

segment of the boundary), which can be more crucial for supply chain management

when delays are larger. The center part may become stable again by tuning the

previously fixed parameters, e.g., by decreasing αi or changing the decision-making

dynamics by increasing the adaptation constant λ1. Since the stability maps ren-

der the stability question transparent with the aid of ACFS, making some regions

stable/unstable is almost always under the control of the supply chain manager.

Case 2: Proportional-Integral Controller

As discussed in Section 7.2.1, the PI controller is known to eliminate the inventory

drift (steady-state errors) (Towill et al., 1997). We will now study how the stability

of inventories is affected when using a PI controller in our five-delay supply chain

model. In this regard, we analyze the effects of the integral gain αI of the PI

controller on the stability boundaries. For this purpose, β value is fixed as 1.0, and

all the remaining parameters are kept the same as with the proportional case study

(Section 7.2.3). It is of interest to investigate how the stability maps change as the

118


integral gain αI varies.

With the PI controller, that is, C(s) = αi + αI/s, the characteristic function

is given in (7.34). Implementing the ACFS algorithm, we extract three stability

boundaries as shown in Figure 7.16 for three different αI values. On this figure, red,

black and blue stability boundaries correspond to αI = 0.0 (proportional control

case studied in the previous subsection), αI = 0.02 and αI = 0.04, respectively. For

a given αi and β, delay values h1 and h2 inside the regions including the origin yield

stability of the inventories. For instance, the red stability boundary, representing the

proportional control case, does not encircle the operation point (4.5, 6.5), hence the

chain is not stable for αI = 0.0. On the other hand, designing the PI controller with

suitable integral gain, e.g., αI = 0.02, makes the same operation point stable while

eliminating the drift in the inventory levels, see the simulations in Figure 7.17. On

the other hand, increasing αI does not enlarge the stability region, on the contrary,

it shrinks the region significantly. We see this as a trade-off between eliminating

Figure 7.16: Case 2: Stability map on h1−h2 domain for different αI values, αI = 0.0(red), αI = 0.02 (black), αI = 0.04 (blue). Parameters β = 1.0, αi = 0.4 1/weeks,λ1 = 2.0 weeks, λ2 = 0.5 weeks, λ3 = 0.4 weeks and delays h3 = 1 week, h4 = 0.15weeks, h5 = 0.4 weeks are fixed. For a given αI , delay values chosen from closedregions which include the origin reveal stability.

119


Figure 7.17: Case 2: Simulation of block diagram in Figure 7.12 for various αI

values.

inventory drift and tolerating larger delays without destabilizing the chain.

To show the PI controller’s effect on the inventory response, we perform time-

domain simulations of the block diagram in Figure 7.12 for various αI values and

for the fixed operation point, αi = 0.4 1/weeks, β = 1.0, o(t) = 5 step units, T = 40

weeks, λ1 = 2.0 weeks, λ2 = 0.5 weeks, λ3 = 0.4 weeks, and delays h1 = 0.2 weeks,

h2 = 4 weeks, h3 = 1 week, h4 = 0.15 weeks, h5 = 0.4 weeks, h = 6.5 weeks. By

inspection of the stability map in Figure 7.16, this operation point should reveal

stability of the generalized supply chain model for all the three αI values. Our

simulation results in Figure 7.17 validate these analytical findings. Without the PI

controller, the response of the supply chain converges to another set point other than

the desired inventory level i = 200 units (red color). Adding an integral controller

with αI = 0.02 simply resolves this problem (black color). Furthermore, increasing

integral gain still eliminates the drift problem, however, as the stable region shrinks

(see Figure 7.16), larger αI induce more oscillatory response (blue color).

Next, in order to elicit the changes in stability maps with respect to decision-

making and transportation delays, production delay is fixed as h2 = 4.0. Then, we

120


extract the stability map on h1 − h3 domain for αI = 0.0 (red stability curve), 0.02

(black color), and 0.04 (blue color) values, see Figure 7.18. Inspection of Figure 7.18

reveals that increasing αI shrinks the stable region on h1 − h3 domain. Note that

small values of αI , e.g., 0.02, enlarge the stable region when h3 → 0. We note that,

the supply chain manager can decide the optimum αI gain based on the information

in Figures 7.16 and 7.18. It is noteworthy that if larger αI is needed, then larger

stable regions can still be created by properly tuning additional parameters such as

αi and β.

We so far extracted stability maps on h1 − h2 and h1 − h3 domains for demon-

stration purposes. The manager can actually fix any of the delay values and extract

stability maps on the plane of the remaining two delays of interest. For example, if

the goal is to find the relation between production and transportation delays, one

can extract the stability maps for various β and αI values on the h2 − h3 domain

after fixing h1 and the remaining parameters. In this study, αi was fixed, and with a

Figure 7.18: Case 2: Stability map on h1−h3 domain for different αI values, αI = 0.0(red), αI = 0.02 (black), αI = 0.04 (blue). Parameters β = 1.0, αi = 0.4 1/weeks,λ1 = 2.0 weeks, λ2 = 0.5 weeks, λ3 = 0.4 weeks and delays h2 = 4 weeks, h4 = 0.15weeks, h5 = 0.4 weeks are fixed. For a given αI , delay values chosen from closedregions which include the origin reveal stability.

121


similar approach, one can relax αi and study its effects on the stability boundaries.

7.3 Limitations

The limitations in our work are in parallel to the existing literature studying sim-

ilar types of problems. The analysis performed in this research is constructed on

linear system theory. In this sense, the analysis uncovers how a supply chain that

is perfectly in equilibrium gets perturbed by increased consumer demand or any

other perturbation causing deviations from that equilibrium. The equilibrium is

defined as ideal conditions where the demand, transportation, production and or-

dering rates are equal to each other, and the rate at which the products leave the

inventory is equal to the rate at which the products arrive to the inventory. Even

if there are delays in the pipelines, the pipelines are filled by precisely the same

amount of products produced at the manufacturer at precisely the same amount at

all times. This is obviously an ideal scenario which cannot occur in the presence

of perturbations and variable consumer demand. Therefore, the inventory behavior

will always oscillate around or deviate from this equilibrium.

Considering the discussions above, our analysis is limited to revealing the stabil-

ity mechanisms of deviations from an equilibrium under perturbations or changing

consumer demand. It is also noted that the ordering-policy gain αi and β are con-

stant parameters. Therefore, stability maps are extracted for a given (αi, β) pair

and they are drawn with respect to delays, which are assumed to be time-invariant

quantities.

122

Chapter 8

Conclusions and Future Work

8.1 Concluding Remarks

For delay-dependent stability analysis, a novel methodology, Advanced Clustering

with Frequency Sweeping (ACFS), is proposed for studying the asymptotic stability

of linear time-invariant multiple time delay systems in the parameter space of delays.

ACFS does not impose any restrictions on the number of delays and system order,

which is the main discrepancy from the existing methods. By means of ACFS,

potential stability switching curves on any 2D delay domain are extracted precisely

and completely. In addition to asymptotic stability analysis of linear time-invariant

multiple time delay systems, maximum number of kernel points is studied. We

derive a measure which delineates the complexity of the extracted stability maps

on 2D delay parameter space. This measure also signifies the computational effort

needed to extract the potential stability switching curves. Moreover, a new formula

which captures the precise lower and upper bounds of the crossing frequency set that

is crucial to ACFS implementation is proposed. Although ACFS methodology can

tackle the stability of general-class linear time-invariant multiple time delay systems,

it may not be very effective for some sub-class problems which do not carry any

123

CHAPTER 8. CONCLUSIONS AND FUTURE WORK

cross-talk and commensurate delay terms. For this reason, a new algorithm specific

to a sub-class of general-class linear time-invariant multiple time delay systems is

developed for computational efficiency. This algorithm also allows to choose the

system order and the number of delays arbitrarily large.

For delay-independent stability analysis, an approach for revealing the exact

positive lower and upper bounds of the crossing frequency set of the most general

linear time-invariant multiple time-delay system is firstly developed. Secondly, weak

delay-independent stability test is proposed. This test can also check robust stability

(well-posed) of single-delay systems directly, without sweeping any parameter or

using graphical display. To achieve this with necessary and sufficient conditions,

a connection between polynomials and transcendental functions is established for

the first time via the iterated discriminants in algebraic geometry. Thirdly, a new

approach is presented to synthesize control laws that render the most general multi-

input linear time-invariant multiple time-delay system delay-independent stable.

This is achieved with transformed characteristic function, which is algebraic, and

iterated discriminants. The approach leads to computationally efficient practical

tools to compute the set of controller gains with sufficient delay-independent stability

conditions.

Stability of inventory behavior controlled by a widely studied Automatic Pipeline

Inventory Order Based Production Control System (APIOBPCS) is investigated

with respect to delays originating from different physical sources; i.e. decision-

making, production lead-times, transportation times and information lags. Firstly,

APIOBPCS model is realistically generalized to cover multiple delays. Not only

does unstable inventory behavior cause ineffective supply chain management, but

also, inventory deficit may cause financial looses. Therefore, proportional-integral

controller in order to prevent the deficit in supply chain is added to the developed

generalized supply chain model. Secondly, analytical procedures are developed to

124


tackle the stability problem by following theories in Chapters 4 and 5. End results

of the stability analysis are the stability maps with respect to the delays, where on

these maps stable and unstable inventory behavior are classified. Stability maps are

supportive for managerial decision-making as they lay out which combinations of

delays give rise to desirable inventory behavior. With the developed tools, it also

becomes possible to extract generalizing rules in designing the ordering policy in a

way that undesirable effects of delays are mitigated and the supply chain becomes

insensitive to delays. The efficacy of the proposed approaches and interpretations

for managerial decisions are presented over some supply chains scenarios.

Additional constraints in the supply chains may exist and they will further nar-

row down the admissible stability regions found in the stability maps; for instance,

the production time can be less or greater than a specific time. These constraints

should be carefully superposed on the stability maps before judging on stability.

Upon having considered these constraints, the arising stability tableau may assist

the supply chain manager with the contractual agreements. Without availability of

such a tableau, the supply chain manager would not realize that seemingly innocuous

combination of decision-making, production, transportation and information delays

could eventually put the inventory behavior into extremis.

8.2 Future Works

¬ Supply chain ordering policies for delay-independent stability of inventory lev-

els will be proposed in the future.

The roots of the polynomial D(ω) is investigated via Descartes’s Rule of Signs.

This choice leads to sufficient conditions. Solving D(ω) for finding both nec-

essary and sufficient conditions are left to the future work.

® Investigation of structural control law which stabilizes SC with necessary and

125


sufficient conditions are left to the future work. Hence, DIS maps on αi−αWIP

(or αi − β) can be extracted.

¯ Stability (steady behavior of inventory levels) and instability (undesirable

growing amplitudes in the inventory levels) are not the only two mechanisms

observed in supply chains. Bullwhip effects and inventory drift are other major

issues that need attention. Inventory drift is eliminated utilizing PI controller

in this research and investigation of Bullwhip effect, which is known as the

amplification of demand pattern towards upstream in a supply chain across

multiple echelons, is left to the future work.

° Perturbation analysis in Section 4.1.4 can be utilized to determine the more

robust points of stability stemming from perturbations in fixed delay values.

126

Appendix A

Derivation of Line Equation in

Section 4.2

From equation (4.22), x1 and y1 are obtained as follows,

x1 = − 1

P 21< + P 2

1=

(P1<P2< + P1=P2=)x2 + (−P1<P2= + P1=P2<)y2

+ (P1<P3< + P1=P3=)x3 + (−P1<P3= + P1=P3<)y3 + P1< χ+ P1= γ

,

y1 = − 1

P 21< + P 2

1=

(−P1=P2< + P1<P2=)x2 + (P1=P2= + P1<P2<)y2

+ (−P1=P3< + P1<P3=)x3 + (P1=P3= + P1<P3<)y3 − P1= χ+ P1< γ

.

Then, substituting (x1, y1) into circle C1 and multiplying with P 21< + P 2

1= leads to,

127

APPENDIX A. DERIVATION OF LINE EQUATION

(x2

1+y21−1

)(P 2

1<+P 21=) = (P 2

2<+P 22=) x2

2+(P 22<+P 2

2=) y22+(P 2

3<+P 23=) x2

3+(P 23<+P 2

3=) y23

+

(2(P2< χ+ P2= γ) + 2 x3

(P2=P3= + P2<P3<

)+ 2 y3(P2=P3< − P2<P3=)

)x2

+

(2(P2< γ − P2= χ) + 2 x3

(P2<P3= − P2=P3<

)+ 2 y3(P2=P3= + P2<P3<)

)y2

+ 2 x3

(P3< χ+ P3= γ

)+ 2 y3

(− P3= χ+ P3< γ

)+ (χ2 + γ2)− (P 2

1< + P 21=).

Since x23 + y2

3 = 1 and x22 + y2

2 = 1, the above equation can be put in the form,

(x2

1 + y21 − 1

)(P 2

1< + P 21=) = (χ2 + γ2)− (P 2

1< + P 21=) + (P 2

2< + P 22=) + (P 2

3< + P 23=)

+ 2 x3

(P3< χ+ P3= γ

)+ 2 y3

(− P3= χ+ P3< γ

)+

(2(P2< χ+ P2= γ) + 2 x3

(P2=P3= + P2<P3<

)+ 2 y3(P2=P3< − P2<P3=)

)x2

+

(2(P2< γ − P2= χ) + 2 x3

(P2<P3= − P2=P3<

)+ 2 y3(P2=P3= + P2<P3<)

)y2.

Hence, equation (4.23) is obtained where arguments are omitted for brevity.

128

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