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
4.12 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 ∈ Ω2
correspondence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.13 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 ∈ Ω3
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
7.7 Case 2: Stability map on h1 − h2 domain for fixed αi = 0.4 1/weeks,
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
7.9 Case 3: Stability map on h1 − h3 domain for fixed αi = 0.4 1/weeks,
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
CHAPTER 1. MOTIVATION OF THE RESEARCH
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
CHAPTER 1. MOTIVATION OF THE RESEARCH
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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
(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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
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
CHAPTER 2. PROBLEM STATEMENTS AND PRELIMINARIES
transitions, one should analyze the characteristic function on the imaginary
axis, <(s) = 0, by setting s equals to jω, ω ∈ R.
 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
CHAPTER 3. OPPORTUNITIES
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
CHAPTER 3. OPPORTUNITIES
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
CHAPTER 3. OPPORTUNITIES
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
CHAPTER 3. OPPORTUNITIES
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
CHAPTER 3. OPPORTUNITIES
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
CHAPTER 3. OPPORTUNITIES
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
CHAPTER 3. OPPORTUNITIES
® 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
CHAPTER 3. OPPORTUNITIES
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
(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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
[¯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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
(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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
(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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
Figure 4.12: 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 ∈ Ω2 correspondence.
Figure 4.13: 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 ∈ Ω3 correspondence.
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
CHAPTER 4. DELAY-DEPENDENT STABILITY ANALYSIS OF MTDS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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 ,
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CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
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
CHAPTER 5. DELAY-INDEPENDENT STABILITY ANALYSIS
‐ + 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
CHAPTER 6. TIME-DELAY SYSTEMS IN SUPPLY CHAIN MANAGEMENT
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
CHAPTER 6. TIME-DELAY SYSTEMS IN SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 6. TIME-DELAY SYSTEMS IN SUPPLY CHAIN MANAGEMENT
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.
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CHAPTER 6. TIME-DELAY SYSTEMS IN SUPPLY CHAIN MANAGEMENT
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.
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CHAPTER 6. TIME-DELAY SYSTEMS IN SUPPLY CHAIN MANAGEMENT
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.,
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CHAPTER 6. TIME-DELAY SYSTEMS IN SUPPLY CHAIN MANAGEMENT
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).
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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
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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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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)
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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)
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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)
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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.
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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.
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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.
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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.
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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)
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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)
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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)
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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.
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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.
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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.
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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.
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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.
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
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CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
CHAPTER 7. CONTRIBUTION TO SUPPLY CHAIN MANAGEMENT
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
CHAPTER 8. CONCLUSIONS AND FUTURE WORK
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
CHAPTER 8. CONCLUSIONS AND FUTURE WORK
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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