STABILITY ANALYSIS OF MULTIPLE TIME-DELAY SYSTEMS WITH 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 requirements for the degree of Doctor of Philosophy in the field of Mechanical Engineering Northeastern University Boston, Massachusetts May, 2011
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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
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
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.
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
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 ,
76
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.
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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.
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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)
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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 .
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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
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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
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
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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
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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).
90
Chapter 7
Contribution to Supply Chain
Management
7.1 Inventory Dynamics in Supply Chains with
Three Delays
7.1.1 Characteristic Equation of APIOBPCS with Three
Delays
In order to incorporate additional delays, one needs to carefully re-write the supply
chain model considering the two additional delays corresponding to decision-making
and transportation times. As motivated and justified earlier, transportation and
production delays are taken as pure delays, h2 and h3 (Figure 6.1) and decision
making is taken as a combination of a dead-time, h1, and a first-order smoothing
(Figure 6.1) which together becomes e−h s/(λ s+ 1) in Laplace domain.
Incorporation of these new terms (Figure 7.1) leads to the following homogeneous
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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
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
110
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
111
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
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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