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Robust Synchronization in Networks of CompartmentalSystems
by
Milad Alekajbaf
A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science
Graduate Department of Electrical and Computer EngineeringUniversity of Toronto
© Copyright 2014 by Milad Alekajbaf
Abstract
Robust Synchronization in Networks of Compartmental Systems
Milad Alekajbaf
Master of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
2014
This thesis studies robust synchronization in networks of compartmental systems moti-
vated by cellular networks. These networks consist of compartments each of which is
composed by interconnected subunits. We investigate conditions on the dynamics of the
subunits and on the interconnection topology that guarantee robust synchronization in
the presence of external disturbances and possibly nonlinear perturbations on the net-
work. The results are illustrated with several examples, including a network of genetic
oscillators.
ii
Dedication
To my dearest parents and sister
iii
Acknowledgements
I would like to express my sincere thanks and gratitude to my supervisors, Professor Luca
Scardovi and Professor Bruce Francis for their guidance, patience, and support during
the past two years. I gratefully acknowledge Professor Scardovi for his trust, continued
assistance, and creative ideas. Furthermore, I thank him for the significant time that he
spent to thoroughly review my thesis. I truly appreciate Professor Francis for patiently
listening to my ideas and providing his feedback which has improved my research work.
Moreover, our discussions inside and outside the office, have been fundamental to my
growth as a researcher.
I would also like to express my appreciation to all of my friends in the Systems Control
Group for their great help and support. My particular thanks go to Farzad and Ashkan
for sharing their precious experience and countless stimulating discussions.
Last but not least, I am deeply indebted to my parents and sister for their unfailing
love and understanding. This work would not have been possible without their continued
unconditional support.
iv
Contents
1 Introduction 1
1.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Statement of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Background 10
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Connectivity of Graphs . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Laplacian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.3 Algebraic Connectivity . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Diagonal Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.2 Input-Output Stability and Passivity in L2 Space . . . . . . . . . 19
3 Coupled Harmonic Oscillators 22
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Synchronization in The Nominal Network . . . . . . . . . . . . . . . . . . 23
3.2.1 Balanced Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 General Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . 30
v
3.3 Robustness Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Robustness Analysis in The Presence of Disturbances . . . . . . . . . . . 46
3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5.1 A More General Example . . . . . . . . . . . . . . . . . . . . . . 54
4 Nonlinear Netwroks: An Input-Output Approach 60
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Example: Network of Goodwin Oscillators . . . . . . . . . . . . . . . . . 72
5 Conclusions and Future Work 79
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
A Goodwin Oscillator 82
Bibliography 83
vi
List of Tables
4.1 Robust synchronization conditions for a network in which only the first
species are allowed to diffuse . . . . . . . . . . . . . . . . . . . . . . . . . 78
vii
List of Figures
1.1 Network of four coupled genetic oscillators . . . . . . . . . . . . . . . . . 4
1.2 Block diagram of network of four coupled Goodwin oscillators . . . . . . 4
2.1 Some examples for different kind of graphs . . . . . . . . . . . . . . . . . 13
3.1 Block diagram of an isolated harmonic oscillator . . . . . . . . . . . . . . 23
3.2 Block diagram of the nominal network of harmonic oscillators . . . . . . 24
3.3 Block diagram of the perturbed network of harmonic oscillators . . . . . 41
3.4 Block diagram of the perturbed network of harmonic oscillators with dis-
turbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5 Graph representation of Gx and Gy and their union for Example 3.1 . . . 49
3.6 Asymptotic synchronization of the x components of the states for Example
3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.7 Asymptotic synchronization of the y components of the states for Example
3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.8 Graph representation of Gx and Gy and their union for Example 3.2 . . . 51
3.9 Asymptotic synchronization of the x components of the states for Example
3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.10 Asymptotic synchronization of the y components of the states for Example
3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.11 Graph representation of Gx = Gy and Gx ∪ Gy for Example 3.3 . . . . . . 52
viii
3.12 Asymptotic synchronization of the x components of the states for Example
3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.13 Asymptotic synchronization of the y components of the states for Example
3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.14 Asymptotic synchronization of the x components of the states for Example
3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.15 Asymptotic synchronization of the y components of the states for Example
3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.16 Block diagram of an isolated chain of integrators with a negative feedback
loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.17 Block diagram of N coupled chain of integrators. . . . . . . . . . . . . . . 58
4.1 The input, output, and external input of species k in compartment j . . 62
4.2 Schematic representation of the network. . . . . . . . . . . . . . . . . . . 63
4.3 An example of the network with cyclic species coupling. . . . . . . . . . . 64
4.4 (I+∆kjj) output for the input z = 0.5 sin(t) and constants r = 0.1,m = 0.25. 75
4.5 (I+∆kjj) output for the input z = 0.5 sin(t) and constants r = 0.1,m = 0.25. 76
4.6 Robust synchronization of six interconnected Goodwin oscillators with
“non-vanishing” external signals . . . . . . . . . . . . . . . . . . . . . . . 77
4.7 Robust synchronization of six interconnected Goodwin oscillators with
“vanishing” external signals . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.8 Comparison of ‖y‖T and ρ1‖v‖T +ρ2 for the “non-vanishing” external signals 78
ix
Chapter 1
Introduction
“At the heart of the universe is a
steady, insistent beat: the sound of
cycles in sync” [55]
Steven Strogatz
Synchronization phenomena arise in a variety of scenarios in natural sciences, engi-
neering, and social life [39]. Synchronization is ubiquitous in biological systems. For
instance, in heart’s natural pacemaker, about 10,000 cells work together synchronously
with no leader or outside instruction to autonomously produce a rhythmic heartbeat [55].
Abnormal synchronization can also cause critical problems in biological systems, e.g., it
is recognized that an anomalous synchronized neural activity can lead to epilepsy and
other neurological dysfunctions [29]. Therefore, it is of great interest to understand the
conditions under which synchronization can occur in such networks. Synchronization is
a dynamical phenomenon and therefore we make use of dynamical systems theory to
study it. As we will see in the following, for two or more coupled dynamical systems,
their degree of synchrony is quantified as a “distance” between systems’ trajectories (or
outputs).
We will focus on compartmental network models, popular in cellular biochemical
1
Chapter 1. Introduction 2
networks, where the compartments are diffusively coupled and the network is subjected
to external disturbances. A typical example of a compartmental model is a network
of circadian oscillators. By means of circadian networks, cells coordinate and share
information (via chemical diffusion) in order to obtain a synchronous behaviour.
Synchronization in cellular networks is currently an active research topic in distinct
research areas such as systems biology, mathematics, physics, engineering, etc. System
and control theory is instrumental to study synchronization as it allows the recursive
verification of important properties through the use of standard analysis tools. The ma-
jority of the existing approaches in systems and control theory assume perfect knowledge
of the mathematical model. However, in nature, dealing with uncertainties is crucial as
accurate mathematical models are virtually impossible to obtain.
Our main goal is to investigate under what conditions synchronization can be proven
to be a robust feature of compartmental networks where possibly nonlinear perturbations
on the diffusive coupling are present. We gradually approach our main problem in three
steps as follows.
We first focus our attention on a network of coupled linear oscillators (harmonic
oscillators). Our first step (Section 3.2) is to study synchronization in the nominal
network, i.e., we assume that the mathematical model is not uncertain.
Our second step is to include in the analysis the effect of possibly nonlinear and
time-varying perturbations on the coupling and the presence of external disturbances.
Sections 3.3 and 3.4 deal with these two problems.
Finally, in Chapter 4, we address the general problem. We consider a class of com-
partmental models where each component of the network (referred to as compartment)
consists of interconnected subsystems (referred to as species). Recently, [50] provides
sufficient conditions for this type of network to achieve synchronization; however, the
coupling is supposed to be linear and certain. We generalize this result by including
possibly nonlinear perturbations on the coupling.
Chapter 1. Introduction 3
1.1 Motivating Example
In a genetic network, DNA information is transcribed letter by letter into RNA language.
This process is called transcription. The RNA transcribed from a protein-coding gene
is called messenger RNA or mRNA. After transcription, mRNA travels to a specific
location of the cell where translation into protein occurs. Finally, the protein can act
as an inhibitor or activator for a specific gene. When the protein represses its own
production the resulting negative feedback loop can give rise to oscillations.
In Fig. 1.1, a graphical representation of a network of four coupled genetic oscilla-
tors is depicted where the negative feedback represents repression. In this network, each
cell is associated to a compartment and each compartment is composed of several sub-
units called species. There are two types of couplings in the network, namely, a species
coupling and a compartmental coupling. The species coupling is due to the chemical re-
actions that occur between two distinct species inside the cells, while the compartmental
coupling takes into account the chemical diffusion among identical species in different
compartments.
One of the most popular mathematical models of genetic oscillators is the Goodwin
oscillator [17, 18] (see the Appendix for more details). The Goodwin oscillator is the re-
sult of a chain of three linear time-invariant first order systems and a static nonlinearity
interconnected in negative feedback loop. In contrast with the usual modeling assump-
tions, in our framework, each compartmental coupling is allowed to be uncertain and
can include nonlinear time-varying terms. The block diagram of the network is depicted
in Fig. 1.2, where Σkj for k, j = 1, . . . , 4 denote the mathematical model for species
k in compartment j, and ∆ denotes the uncertainty operator. Although the cells are
identical, they may have distinct initial conditions so the operators of the corresponding
species are considered to be different. Our goal is to find conditions under which robust
synchronization is achieved in perturbed networks such as the one depicted in Fig. 1.2.
Chapter 1. Introduction 4
Figure 1.1: Network of four coupled genetic oscillators
Σ11 Σ21 Σ31 Σ41− Σ12 Σ22 Σ32 Σ42−
Σ14 Σ24 Σ34 Σ44− Σ13 Σ23 Σ33 Σ43−
I + ∆1
I + ∆2
I + ∆3
I + ∆4
I + ∆5
I + ∆6
Figure 1.2: Block diagram of network of four coupled Goodwin oscillators
Chapter 1. Introduction 5
1.2 Literature Review
This section reviews some of the existing work reported in the literature. We outline the
approaches to synchronization in networks of linear and nonlinear systems. In particular,
we review the topics in this area that are more related to our work, including synchro-
nization in second order linear systems, synchronization in nonlinear networks (with an
emphasis on the passivity approach), and robust synchronization.
Single Integrators
When the network is composed by interconnected single integrators, synchronization
is mostly referred to as consensus. The idea of relating the communication topology of
interconnected systems with graph theory was initiated in [60, 61]. In the last decade, the
consensus problem in networks of single integrators became of great interest in systems
and control with noteworthy contributions such as [24, 38, 34, 46]. In particular, the
authors in [38] studied three network structures, namely, directed balanced networks
with fixed topologies, directed balanced networks with switching topology, and undirected
networks with a communication time delay. Later, the results were extended to directed
graphs having a spanning tree [46]. There are massive developments in analyzing the
synchronization problem in networks of first order systems with dynamic communication
topologies in different setups such as continuous time, discrete time, and quantized data
communications. (see for instance [45, 30, 31, 44, 37, 69] for related works, and [7] for a
recent survey in this research area).
Second Order Linear Systems
The generalization to second order linear systems is not trivial and it is still an active
research topic. The main classes of networks considered in the literature are networks of
double integrators and networks of harmonic oscillators.
The first developments have been reported in [28, 49, 59], where the communication
Chapter 1. Introduction 6
among the agents is assumed to be bidirectional. Later, [44] extended the results to uni-
directional communication and showed that both the graph topology and the coupling
strength are essential in order to achieve synchronization. The paper [1] dealt with the
synchronization problem for a network of double integrators where the velocity measure-
ment is not available and the control input has a saturation constraint. The paper [68]
investigated some necessary and sufficient conditions for synchronization in a network of
double integrators with and without time delay where the coupling among the two com-
ponents of the states has the same structure but different gains. Recently, [15] provided
a necessary and sufficient condition for synchronization in a network of double integra-
tors where both the state components are coupled through two heterogeneous undirected
graph topologies.
Synchronization of coupled harmonic oscillators under directed fixed and switching
topologies was studied in [42], where the oscillators are diffusively coupled only through
the second component of the states. The paper [56] dealt with synchronization of cou-
pled harmonic oscillators in a dynamic proximity network popular in the study of flocking
behaviour [36]. The papers [6] and [8] made use of average theory and Lyanpunov sta-
bility theory on dynamical systems, respectively, to study nonlinearly coupled harmonic
oscillators over undirected graph topologies.
To the best of our knowledge, synchronization in networks of harmonic oscillators
where both components of the states are diffusively coupled with two different communi-
cation topologies has not been studied yet. We will present our main results for networks
of coupled harmonic oscillators in Chapter 3.
High Order Linear Systems
In the case of high-order linear systems, [47] dealt with the synchronization problem in
the special case of single input linear systems where the graph topologies associated to
the components of the states are identical. In this case, it is shown that synchronization
Chapter 1. Introduction 7
is achieved if and only if the directed graph has a spanning tree. Later, by exploiting
passivity properties, sufficient conditions are investigated for a more general class of lin-
ear systems to achieve synchronization under a possibly time-varying and directed graph
topology; however, the graph topologies are assumed to be identical for all the compo-
nents of the states [51]. In the case of heterogeneous linear systems, [66, 26] adopted an
internal model principle approach (see [13]) to take heterogeneities into account in the
network, thus generalizing the results in [51]. Lastly, [32] used an integral quadratic con-
straints approach to address the robust synchronization problem in networks of identical
linear systems with nonlinear perturbations.
Nonlinear Systems
Given a network of nonlinear dynamical systems, synchronization depends on both the
dynamics of the subunits and the interconnection topology [40, 21]. Recently, passivity-
based techniques proved to be instrumental for the analysis of synchronization in inter-
connected dynamical systems [20, 2, 9]. In this context, both a state space approach
[22, 19] and an input-output approach [50, 12] have been considered and sufficient condi-
tions, based on incremental passivity properties of the subunits, have been investigated.
The synchronization conditions in [50] can be seen as a generalization of stability con-
ditions proposed in [3] and [4] which, in turn, build on classical results on the stability
of large-scale interconnected systems [35, 65]. The paper [58] studied robust synchro-
nization in networks of identical nonlinear systems with a gain-bounded multiplicative
uncertainty in the inputs. The papers [33] and [70] used a state-space approach to derive
a bound for the synchronization error in networks of homogeneous systems with linear
additive uncertainties and in networks of heterogeneous systems, respectively.
Chapter 1. Introduction 8
1.3 Thesis Outline
The thesis is organized as follows. In the remaining of this chapter we outline the main
contributions of the thesis. In Chapter 2, we briefly summarize the notions of graph
theory and stability theory that we need throughout the thesis.
Chapter 3 focuses on synchronization of coupled harmonic oscillators in both the
nominal and the perturbed scenarios. In our framework both components of the harmonic
oscillators are diffusively coupled through possibly distinct communication topologies.
We claim that our network structure has not been studied before in the literature. We
first investigate necessary and sufficient conditions for the nominal network to achieve
synchronization. Once the synchronization conditions are established for the nominal
network, classical results in stability theory [25] can be used to address robustness of
synchronization.We illustrate the main results with several examples.
Chapter 4 deals with synchronization in nonlinear networks. Our contribution is an
extension of the work in [50]. In particular, [50] adopted an input-output framework
to study synchronization in the nominal network, while our main contribution is to
investigate under what conditions synchronization is a robust feature of the network
with respect to possibly nonlinear perturbations on the diffusive coupling. The result
is applied to derive sufficient conditions for synchronization in a network of Goodwin
oscillators.
Finally, in Chapter 5, we review the main results of the thesis and suggest possible
future research.
1.4 Statement of Contributions
The main contributions are summarized as follows.
1. Lemma 3.2 and Lemma 3.4
The lemmae address two useful properties of the kernel of Laplacian matrices as-
Chapter 1. Introduction 9
sociated with directed balanced and general directed graphs. These properties are
not only exploited in the main results of Chapter 3, but they are of independent
interest.
2. Theorem 3.1 and Theorem 3.2
Theorem 3.1 provides a necessary and sufficient condition for synchronization in
a network of multi-input harmonic oscillators where both the components of the
states are diffusively coupled with possibly distinct directed balanced communi-
cation topologies. In Theorem 3.2, the balanced condition for graphs is relaxed;
however, only sufficient conditions are provided.
3. Theorem 3.3
This theorem provides a necessary and sufficient condition, based on the spectral
properties of our model, for the harmonic oscillators to asymptotically synchronize
in the presence of directed communication topologies. Two corollaries provide
necessary and sufficient conditions for synchronization.
4. Theorem 3.4 and Theorem 3.5
In Theorem 3.4, a sufficient condition is achieved for robust synchronization of
coupled harmonic oscillators, and in Theorem 3.5, an upper bound for the synchro-
nization error of the network in the presence of disturbance signals is obtained.
5. Theorem 4.1
This Theorem is a generalization of [50, Theorem 1] where the coupling is nonlin-
early perturbed. Theorem 4.1 shows that under certain conditions, the network
“almost” synchronizes depending on the level of synchrony of external inputs. As a
special case, if the external inputs are in L2 then, under some technical assumptions,
asymptotic synchronization is achieved.
Chapter 2
Background
In this chapter, we review preliminary mathematical definitions and introduce the back-
ground ideas and foundations that are needed for the remaining chapters. Section 2.1
introduces the notation throughout the thesis. In Section 2.2, we summarize some no-
tions from graph theory. Finally, in Section 2.3, we review some concepts of stability
theory including diagonal stability, input-output stability and passivity.
2.1 Notation
The notation used in this thesis is fairly standard. We represent matrices with capital
roman letters, while scalars and vectors are represented with lower case letters. Calli-
graphic fonts (e.g. V ,W , etc.) are used to represent sets and in particular graphs and
subspaces. Let R and C denote the field of real and complex numbers, respectively. The
set of non-negative real numbers is denoted by R+. The notation Rn denotes the set of
n-dimensional real vector space, and Rn×m denotes the set of n ×m matrices with real
numbers. Similarly, Cn denotes the set of n-dimensional complex vector space. We write
f ∈ C1 to show that f is a continuously differentiable function.
For x1, . . . , xm ∈ Rn, col(x1, . . . , xm) ∈ Rmn denotes the stacked (column) vector. Let
‖x‖ denote the euclidean 2-norm of x ∈ Rn. Let In ∈ Rn×n be the n-dimensional identity
10
Chapter 2. Background 11
matrix, and 0n, 1n ∈ Rn be the column vector of all zeros and ones, respectively. We
write A = [aij] ∈ Rm×n to show that the elements of matrix A are aij. The transpose
of a matrix A is denoted by AT . A positive (negative) definite matrix A is denoted by
A 0 (A ≺ 0). Similarly, a positive (negative) semi-definite matrix A is denoted by
A 0 (A 0). A (block) diagonal matrix B ∈ Rn×n with (block) diagonal elements
b1, . . . , bn is denoted by B = diag(b1, . . . , bn). The (right) kernel and rank of matrix A
are respectively denoted by Ker(A) and rank(A).
At the end of this section, we briefly introduce the Kronecker product. Given two
matrices A := [aij] ∈ Rm×n and B ∈ Rp×q, the Kronecker product of A and B is defined
as
A⊗B =
a11B . . . a1nB
.... . .
...
am1B . . . amnB
.
The Kronecker product has the following properties [27]:
A⊗B ⊗ C = A⊗ (B ⊗ C) = (A⊗B)⊗ C (2.1a)
A⊗ (B + C) = A⊗B + A⊗ C (2.1b)
AB ⊗ CD = (A⊗ C)(B ⊗D). (2.1c)
Furthermore, when A and B are square matrix, the spectrum of square matrix A⊗B can
be obtained as follows. Let α1, . . . , αm be the eigenvalues of A ∈ Rm×m and β1, . . . , βn
be the eigenvalues of B ∈ Rn×n. The eigenvalues of A⊗B are αiβj for i = 1, . . . ,m and
j = 1, . . . , n.
Chapter 2. Background 12
2.2 Graph Theory
Graph theory is an important tool in the analysis of interconnected systems. We can
represent the communication topology of a network with a graph. If the information
flow is bidirectional, i.e., both systems receive information from each other, then the
corresponding graph is undirected. While, if the information flow is unidirectional then
the corresponding graph is directed. In this section we review some of the basic notions
of graph theory with emphasis on algebraic graph theory. For the reader interested in
more details about this section, we recommend [5], [14], [11], [67], [53].
A weighted directed graph (or simply digraph) G = V , E , A consists of a non-empty
finite set V := 1, . . . , n of elements called nodes, a finite set of edges (or arcs) E ∈ V×V ,
and an associated weighted adjacency matrix A := [aij] ∈ Rn×n where
aij > 0, (j, i) ∈ E
aij = 0, (j, i) /∈ E .
Note that the edge (j, i) is graphically represented by an arrow with tail node j and head
node i. This means node i receives information from node j, so node i is neighbour (or
out-neighbour) of node j. A graph is called undirected if all the pairs in E are unordered,
or equivalently, the adjacency matrix is symmetric. Throughout the thesis we assume
that the graphs are time invariant, that is, A is a constant matrix. Also, we assume that
for i = 1, . . . , n the pair (i, i) /∈ E , i.e., the graphs do not have self-edges.
The in-degree and out-degree of node i ∈ V are defined as
din(i) :=n∑j=1
aij, dout(i) :=n∑j=1
aji.
A directed graph is said to be balanced if for every node the in-degree and the out-degree
are equal. So every undirected graph is balanced but not vice versa.
Many operations such as union, converse, and different kind of products on directed
Chapter 2. Background 13
graphs are defined. Here we only introduce the union operation on weighted directed
graphs that will be used in Chapter 3. Let Gi = V , Ei, Ai for i = 1, . . . ,m have the
same node set. The union of Gi is defined as G := ∪iGi = V ,∪iEi,m∑i=1
Ai.
Different kind of weighted graphs with four nodes are depicted in Fig. 2.1. In this
figure, G2 is an undirected graph, while all the other graphs are directed. Note that graph
G1 is a balanced, but G3, G4, and G5 are not balanced.
1
2
3
4
2
2 2
2
(a) G1
1
2
3
4
11
0.20.2
33
0.50.5
(b) G2
1
2
3
4
1
1
0.2
(c) G3
1
2
3
4
3
1
0.5
2
(d) G4
1
2
3
4
3
4
5
(e) G5
Figure 2.1: Some examples for different kind of graphs
2.2.1 Connectivity of Graphs
A strong path (or directed chain) in a directed graph is a sequence of distinct nodes
v0, . . . , vk ∈ V such that for every i = 1, . . . , k, (vi, vi−1) ∈ E . A weak path in a directed
Chapter 2. Background 14
graph is a sequence of distinct nodes v0, . . . vk ∈ V such that for every i = 1, . . . , k, either
(vi, vi−1) ∈ E or (vi−1, vi) ∈ E . A directed graph is strongly connected if there exists a
strong path from every node to every other node. A directed graph has a spanning tree
if there exists at least one node, called the root node, that is connected to all the other
nodes by a strong path. So a spanning tree has exactly n nodes and n − 1 edges, and
every node, except the root node, has exactly one incoming edge. In some literature, a
graph having a spanning tree is called quasi-strongly connected. A strongly connected
graph has a spanning tree but not vice versa. A directed graph is connected (or weakly
connected) if there exists a weak path between every two nodes. It can be shown that a
connected balanced graph is strongly connected.
In Fig. 2.1, G1 and G2 are strongly connected, G3 is not strongly connected; however,
it has a spanning tree with root node 1. The graph G4 is connected, whereas G5 is not
connected. Node 4 in G5 is called isolated node.
2.2.2 Laplacian Matrix
Given a weighted directed graph G = V , E , A with set node V = 1, . . . , n and adja-
cency matrix A = [aij]. Let D := diag(din(1), . . . , din(n)) be the in-degree matrix. The
Laplacian matrix associated with graph G is defined as
L := D − A. (2.2)
Let L := [lij] ∈ Rn×n. It follows from (2.2) that
lij =
n∑j=1
aij, i = j
−aij, i 6= j.
(2.3)
Chapter 2. Background 15
To illustrate, consider Fig. 2.1. Let Lk be the Laplacian matrix associated with Gk for
k = 1, . . . , 5. Then the Laplacian matrices are as follows:
L1 =
2 0 0 −2
−2 2 0 0
0 −2 2 0
0 0 −2 2
, L2 =
4 −1 −3 0
−1 1.2 −0.2 0
−3 −0.2 3.7 −0.5
0 0 −0.5 0.5
,
L3 =
0 0 0 0
−1 1 0 0
−1 0 1 0
0 0 −0.2 0.2
, L4 =
4 −3 −1 0
0 0 0 0
0 0 0.5 −0.5
0 0 −2 2
, L5 =
5 0 −5 0
−3 3 0 0
0 −4 4 0
0 0 0 0
.
(2.4)
In general, a Laplacian matrix L associated with the graph G has the following properties:
1. Matrix L is diagonally dominant. So by the Gershgorin disk theorem [23], all the
eigenvalues of L are in the closed right half-plane (CRHP).
2. If G is undirected then L is symmetric, and hence it is positive semi-definite.
3. The Laplacian matrix L is a zero sum matrix, i.e., the vector 1n is in the right
kernel of L, i.e., L1n = 0n. Moreover 1n is also in the left kernel of L if and only if
G is balanced.
4. If G is balanced, then L = L+ LT is a symmetric Laplacian matrix [38].
Note that in (2.4), L2 is symmetric due to the fact that G2 is undirected. Also, since G1
is balanced, we have 1TnL1 = 0Tn . The following well-known lemma is a useful tool for
analyzing synchronization problem in interconnected systems.
Lemma 2.1. [46] Let G be a weighted directed graph and L be the associated Laplacian
matrix. Then L has a simple zero eigenvalue and all the other eigenvalues are in the
Chapter 2. Background 16
open right half-plane if and only if G has a spanning tree.
Lemma 2.1 implies that rank(L) = n− 1 if and only if G has a spanning tree. in Fig.
2.1, since G1, G2, and G3 have a spanning tree, we have rank(L1) = rank(L2) = rank(L3) =
3. In contrast, G4 and G5 do not have a spanning tree and rank(L4) = rank(L5) = 2.
2.2.3 Algebraic Connectivity
The connectivity properties of graphs are related to the algebraic properties of the cor-
responding Laplacian matrices and, in particular, to the notion of algebraic connectivity.
Fiedler [10] introduced the algebraic connectivity of an undirected graph as the second
smallest eigenvalue of its Laplacian matrix. Lately, this concept was extended to general
directed graphs as follows [67]:
Definition 2.1. Let G be a directed graph with Laplacian matrix L. Let also P := z ∈
Rn : z ⊥ 1n, ‖z‖ = 1. The algebraic connectivity of G is defined as
λ := minz∈P
zTLz. (2.5)
The algebraic connectivity of directed graph G with Laplacian L has the following
properties:
1. Let G and H have the same vertex set, and λ and µ be their algebraic connectivity,
respectively. Let I := G ∪ H with algebraic connectivity γ. Then γ ≥ λ+ µ.
2. Let λ1 and λ2 be the smallest and second smallest eigenvalue of L := L + LT ,
respectively. Then λ in (2.5) satisfies
1
2λ1(L) ≤ λ ≤ 1
2λ2(L).
3. If G is not connected, then λ ≤ 0.
Chapter 2. Background 17
4. For a balanced G, λ > 0 if and only if G is connected.
2.3 Stability Theory
This section is a review of some standard concepts in stability theory including LaSalle’s
invariance principle and diagonal stability, input-output stability, and passivity.
In Chapter 3 we use LaSalle’s invariance principle to study synchronization in net-
works of harmonic oscillators. Since the network under study in Chapter 3 is linear we
make use of the properties of invariant subspace in linear systems. We start with the
definition of an invariant subspace under the dynamics of a linear system
x = Ax, x ∈ RN , x(0) = x0. (2.6)
Definition 2.2. The subspace V ⊂ Rn is said to be invariant under (the dynamics) (2.6)
if x0 ∈ V =⇒ x(t) ∈ V, for every t ≥ 0.
Theorem 2.1. Consider linear system (2.6). Let V ⊂ Rn. Then V is invariant under
2.6 if and only if V is A-invariant, that is, AV ⊂ V.
Now we state the LaSalle’s invariance principle for the dynamical system
x = f(x), x ∈ X ⊂ Rn, (2.7)
where f is C1 and the origin is contained in X .
Theorem 2.2. Given the nonlinear system (2.7). Let B ⊂ X be a domain and Ω ⊂ B be
a compact positively invariant set. Let V : D → R be a C1 function such that for every
x ∈ Ω, V (x) ≤ 0. Let E := x ∈ Ω : V (x) = 0 and M be the largest invariant subset of
E. Then, for every x0 ∈ Ω, the solution φ(x, t)→M as t→∞.
Chapter 2. Background 18
In some cases, such as time-varying systems, the LaSalle invariance principle cannot
be applied. An alternative way to characterize the asymptotic behaviour of the solutions
is to use the Barbalat lemma:
Lemma 2.2. Let f : R+ → R be a uniformly continuous function for t ≥ 0. If
limt→∞
∫ t
0
|f(t)| exists and is finite, then limt→∞
f(t) = 0.
The Barbalat lemma implies that if a uniformly continuous signal is in L2, then the
signal converges to zero. We use this fact in Chapter 4.
2.3.1 Diagonal Stability
Diagonal stability plays an important role in studying stability of network made up
passive subsystems. Some of the well-known classical works in this area can be found in
[57], [65], and [35]. The definition of diagonal stability is as follows [4]:
Definition 2.3. A real square matrix A is said to be diagonally stable if there exists a
diagonal matrix D 0 such that
ATD +DA ≺ 0.
According to the above definition, a matrix is diagonally stable if it satisfies the
Lyapunov equation with a diagonal matrix. A necessary and sufficient condition for the
matrix
A =
−α1 0 . . . −β1
β2 1 . . . 0
.... . . . . . 0
0 . . . βN −αN
(2.8)
to be diagonally stable is the so called secant condition, as stated in the following lemma
[4]:
Chapter 2. Background 19
Lemma 2.3. The matrix (2.8) is diagonally stable if and only if
α1
β1. . .
αNβN
<[sec( πN
)]N.
Lemma 2.3 was extended to other matrix structures in [54].
2.3.2 Input-Output Stability and Passivity in L2 Space
This section provides the basic notions of input-output stability and passivity in L2 space.
The material in this section is mostly taken from [52] with some modifications.
Definition 2.4. The set Lm2 [0,∞) = Lm2 consists of all functions f : R+ → Rm which
are measurable and satisfy∫∞0‖f(t)‖2dt <∞.
Definition 2.5. Let f : R+ → Rm. Then for every T ≥ 0, the truncation of f to the
interval [0, T ] is denoted by fT : R+ → Rm and is defined as
fT (t) :=
f(t), 0 ≤ t < T
0, t ≥ T.
We denote by Lm2e the extended space of signals f : R+ → Rm which have the
property that fT ∈ Lm2 for every T ≥ 0. The set Lm2e is called the extended Lm2 space or
the extension of Lm2 . Given two signals v, w ∈ Lm2e and any finite T > 0, we define the
inner product of v, w as 〈v, w〉T :=∫ T0v(t)Tw(t)dt, and ‖w‖T :=
√〈w,w〉T .
Definition 2.6. An operator F : Lm2e → Lm2e is said to have unbiased finite L2 gain if
there exists δc ≥ 0 such that for every z ∈ Lm2e and for every T ≥ 0,
‖Fz‖T ≤ δc‖z‖T .
The L2 gain of F is defined as δ := infδc.
Chapter 2. Background 20
Now we introduce the notion of passivity in L2 space.
Definition 2.7. An operator F : Lm2e → Lm2e is passive if there exists constant β ∈ R
such that
〈Fu, u〉T ≥ β, ∀u ∈ Lm2e, ∀T ≥ 0.
The operator F is input strictly passive if there exist constants β ∈ R and δ > 0 such
that
δ〈Fu, u〉T ≥ ‖u‖2T + β, ∀u ∈ Lm2e, ∀T ≥ 0.
The operator F is output strictly passive if there exist constants β ∈ R and ε > 0 such
that
ε〈Fu, u〉T ≥ ‖Fu‖2T + β, ∀u ∈ Lm2e, ∀T ≥ 0.
Before concluding this chapter, we introduce the notion of relaxed co-coercivity as
follows [50]:
Definition 2.8. An operator F : Lm2e → Lm2e is relaxed co-coercive if there exists constant
γ ∈ R such that
γ‖Fu1 − Fu2‖2T ≤ 〈u1 − u2, Fu1 − Fu2〉T , ∀u1, u2 ∈ Lm2e, ∀T ≥ 0. (2.9)
If (2.8) holds for γ ≥ 0, then F is said to be monotone, and if (2.8) holds for γ > 0,
then F is said to be co-coercive. A definition similar to Definition 2.8 can be found
in [64]. In the literature, sometimes monotone, co-coercive, and relaxed co-coercive are
respectively referred to as incrementally passive, incrementally output strictly passive,
and incrementally output feedback passive [20, 22]. In Chapter 4, we define a slightly
Chapter 2. Background 21
adapted version of Definition 2.8.
Chapter 3
Coupled Harmonic Oscillators
3.1 Introduction
In this chapter, we deal with robust synchronization of diffusively coupled harmonic
oscillators. Our framework in this chapter is the simplified version of a nonlinear com-
partmental network that will be discussed in detail in Chapter 4; however, the results in
this chapter are of independent interest in synchronization of multi-input linear systems
where many problems have remained open.
The purpose of this chapter is to investigate the conditions under which synchroniza-
tion is achieved in the nominal and perturbed network of harmonic oscillators. Unlike
the results in the literature [42, 51], in our framework both state components are coupled
through possibly distinct directed communications topologies.
This chapter is organized as follows. In Section 3.2, we provide a necessary and
sufficient condition for synchronization of the nominal diffusively coupled harmonic os-
cillators. In Section 3.3, we show that synchronization can be achieved in the network,
in the presence of a possibly nonlinear time-varying perturbations on the coupling. In
Section 3.4, the effect of disturbance on the network is studied and an upper bound on
the norm of the synchronization error is obtained. Finally, the results of this chapter are
22
Chapter 3. Coupled Harmonic Oscillators 23
illustrated with several examples in Section 3.5.
3.2 Synchronization in The Nominal Network
Consider n coupled harmonic oscillators where each isolated one is modelled as
xi = −yi xi(0) = xi0
yi = xi yi(0) = yi0, xi, yi ∈ R, i = 1, . . . , n.(3.1)
The block diagram of system (3.1) is depicted in Fig. 3.1. Without loss of generality,
throughout the thesis, we fix the natural frequency of the harmonic oscillators to one.
1s
1s
xi yi−
Figure 3.1: Block diagram of an isolated harmonic oscillator
Assume that the first components of the states (x), as well as the second components
of the states (y), are diffusively coupled as follows:
xi = −yi +
n∑j=1
aij(xj − xi),
yi = xi +n∑j=1
bij(yj − yi), i = 1, . . . , n.
(3.2)
We represent the diffusive coupling terms by Laplacian matrices as follows. Let Lx :=
[lxij] ∈ Rn×n and Ly := [lyij] ∈ Rn×n, where
lxij :=
n∑j=1
aij, i = j
−aij, i 6= j,
lyij :=
n∑j=1
bij, i = j
−bij, i 6= j.
Chapter 3. Coupled Harmonic Oscillators 24
Let Gx and Gy be the corresponding directed weighted graphs associated with Laplacian
matrices Lx and Ly, respectively. Let x = [x1 . . . xn]T and y = [y1 . . . yn]T . In vector
form, we can rewrite (3.2) as
x = −y − Lxx x(0) = [x10 . . . xn0]T
y = x− Lyy y(0) = [y10 . . . yn0]T
(3.3)
or equivalently as
xy
︸︷︷︸z
=
−Lx −I
I −Ly
︸ ︷︷ ︸
A
xy
︸︷︷︸z
. (3.4)
The block diagram of the network is depicted in Fig. 3.2. Our first goal is to find
1s
. . .1s
1s
. . .1s
−Lx −Ly
x y
−
Figure 3.2: Block diagram of the nominal network of harmonic oscillators
conditions under which synchronization is achieved asymptotically in the network (3.4).
Then, we study robustness in synchronization with respect to nonlinear perturbations
on the diffusive coupling, and following that, we study the effect of external inputs on
synchronization in the network. In order to do so, first we give an explicit mathematical
definition for synchronization in network (3.4) as follows:
Definition 3.1. For the network of n coupled harmonic oscillators, the harmonic oscil-
Chapter 3. Coupled Harmonic Oscillators 25
lators asymptotically synchronize if for every initial condition
limt→∞‖xi(t)− xj(t)‖ = 0
limt→∞‖yi(t)− yj(t)‖ = 0, i, j = 1, . . . , n.
From Definition 3.1, the harmonic oscillators asymptotically synchronize if for every
initial condition, the solution of the network converges to the synchronization subspace
defined as
S := Span
1n
0n
,0n
1n
.
To find conditions under which synchronization occurs in the network, we need to study
the properties of graphs Gx and Gy. In this thesis, the weighted directed graphs are
taken into account that model the general class of the network structure. First, we
study balanced graphs as a particular class of directed graphs and then we relax this
condition to general directed graphs. For balanced graphs, we present a necessary and
sufficient condition for the network to achieve synchronization, while for the general
directed graphs, we only present a sufficient condition; however, we show a necessary and
sufficient condition for two special cases of the network structure, namely, when both
graphs are identical and when one of them is an empty graph.
3.2.1 Balanced Graphs
In this section we provide a necessary and sufficient condition under which synchroniza-
tion is achieved. We will use LaSalle’s invariance principle which has also been used for
the consensus problem in a network of single integrators [69]. We start with two lemmae
that are needed to prove the main result of this section. The first lemma [23] is a property
Chapter 3. Coupled Harmonic Oscillators 26
of symmetric positive definite matrices and the second lemma shows a useful property of
balanced graphs.
Lemma 3.1. [23] Let Q ∈ Rn×n be symmetric positive semi-definite and let x ∈ Rn.
Then xTQx = 0 if and only if Qx = 0.
Proof : The sufficiency is clear. For the necessity, let x 6= 0 and xTQx = 0. Consider
the following polynomial:
p(t) = (tx+Qx)TQ(tx+Qx) = t2xTQx+ 2txTQ2x+ xTQ3x = 2t‖Qx‖2 + xTQ3x.
Since Q is positive semi-definite, p(t) ≥ 0 for every t ∈ R. However, if ‖Qx‖ 6= 0 then
there exists a negative constant r such that for every t < r we have p(t) < 0. This implies
that ‖Qx‖ = 0 and hence Qx = 0.
Lemma 3.2. Let L ∈ Rn×n be a Laplacian matrix associated with a directed balanced
graph and L := L+ LT . Then Ker(L) = Ker(L).
Proof : First, we show that Ker(L) ⊂ Ker(L). Let v ∈ Ker(L), i.e., Lv = 0n. We have
2vTLv = vT (L+ LT )v = 0.
Since L is symmetric positive semi-definite (See [38, Theorem 7]), by Lemma 3.1, we
conclude that v ∈ Ker(L).
Now, we need to show that Ker(L) ⊂ Ker(L). Let w ∈ Ker(L), i.e., Lw = 0n and
wT Lw = 0. (3.5)
Chapter 3. Coupled Harmonic Oscillators 27
Let L := [lij] for i, j = 1, . . . , n where
lij :=
n∑j=1
cij, i = j
−cij, i 6= j.
(3.6)
We have
wTLw =n∑i=1
(w2i
n∑j=1
cij
)−
n∑i,j=1
cijwiwj =n∑
i,j=1
cij(w2i − wiwj), (3.7)
and similarly,
wTLTw =n∑
i,j=1
cji(w2i − wiwj) =
n∑i,j=1
cij(w2j − wjwi). (3.8)
It follows from (3.5), (3.7), and (3.8) that
wT Lw = wTLw + wTLTw =n∑
i,j=1
cij(w2i + w2
j − 2wiwj) =n∑
i,j=1
cij(wi − wj)2 = 0. (3.9)
So, if cij > 0, then wi = wj. Let wi := wj for every j such that cij > 0, i.e., the
corresponding elements of w for all neighbours of node i have the same value wi. Hence,
for every i = 1, . . . , n, the ith row of Lw, denoted by Liw, is
Liw =n∑j=1
lijwj = liiwi +n∑
j=1,j 6=i
lijwj = wilii + wi
n∑j=1,j 6=i
lij = wiLi1n = 0.
Therefore, w ∈ Ker(L), i.e., Ker(L) ⊂ Ker(L). This completes the proof.
Now, we are ready to state the main result of this section.
Theorem 3.1. Let Gx and Gy be balanced directed weighted graphs. All the harmonic
oscillators in network (3.4) asymptotically synchronize if and only if Gx∪Gy is connected.
Chapter 3. Coupled Harmonic Oscillators 28
Proof : (Sufficiency) We apply LaSalle’s invariance principle to prove the sufficiency.
Let V (x, y) = xTx+ yTy. The derivative of V along the trajectories of the network is as
follows:
V (x, y) = 2
[xT yT
]A
xy
(3.10)
=
[xT yT
](A+ AT )
xy
(3.11)
=
[xT yT
]−(Lx + LTx ) 0
0 −(Ly + LTy )
xy
(3.12)
= −xT Lxx− yT Lyy ≤ 0 (3.13)
Let c ≥ 0. Consider the compact set Ω := col(x, y) ∈ R2n : V (x, y) ≤ c. By Nagumo
theorem, we conclude that Ω is positively invariant. Let E := col(x, y) ∈ Ω : V (x, y) =
0 and M be the largest invariant subset of E . By LaSalle’s invariance principle we
conclude that for every initial condition z0 ∈ Ω, the solution Φ(t, z0)→M as t→∞.
Now, we need to show that if the union of the graphs is connected, then M = S.
First, we show that S ⊂ M, i.e., we want to show that the synchronization subspace as
defined in Definition 3.1 is invariant. It is easy to show that ±j are the eigenvalues of A
with the associated eigenvectors col(j1n, 1n) and col(−j1n, 1n), because they satisfy
−Lx −I
I −Ly
j1n
1n
=
−1n
j1n
= j
j1n1n
,−Lx −I
I −Ly
−j1n
1n
=
−1n
−j1n
= −j
−j1n1n
.(3.14)
Chapter 3. Coupled Harmonic Oscillators 29
So the subspace
Span
Re
j1n
1n
, Im
j1n
1n
= Span
0n
1n
,1n
0n
= S
is invariant.
Now, we need to show thatM⊂ S. Let D = Span1n. It is clear D ⊂ Ker(Lx) and
D ⊂ Ker(Ly). Hence, D ⊂ Ker(Lx) ∩Ker(Ly). Since Gx ∪ Gy is balanced and connected,
from Lemma 2.1, we have rank(Lx +Ly) = n− 1 and Ker(Lx +Ly) = D. Also, we know
that for every two matrices, the kernel of their sum contains the intersection of their
kernels, i.e., Ker(Lx) ∩ Ker(Ly) ⊂ D. Thus Ker(Lx) ∩ Ker(Ly) = D. From (3.13) and
Lemma 3.1 we get
E = col(v, w) ∈ R2n : v ∈ Ker(Lx), w ∈ Ker(Ly).
By Lemma 3.2, E can be written as
E = col(v, w) ∈ R2n : v ∈ Ker(Lx), w ∈ Ker(Ly).
Let col(v, w) ∈M ⊂ E . So we have v ∈ Ker(Lx) and w ∈ Ker(Ly). By Theorem 2.1, we
know that M is A-invariant, so
−Lx −I
I −Ly
vw
=
−wv
∈M ⊂ E .This implies that w ∈ Ker(Lx) and v ∈ Ker(Ly). Hence v, w ∈ Ker(Lx) ∩ Ker(Ly) = D
and thus col(v, w) ∈ S which means M⊂ S.
So we haveM = S, and we can conclude that all the harmonic oscillators asymptot-
ically synchronize.
Chapter 3. Coupled Harmonic Oscillators 30
(Necessity) We want to show that if all the harmonic oscillators asymptotically syn-
chronize then Gx ∪ Gy is connected, or equivalently rank(Lx + Ly) = n − 1. We argue
by contradiction. Suppose v := [v1 . . . vn]T ∈ Ker(Lx + Ly) and v /∈ D. It follows from
Lemma 3.1 and Lemma 3.2 that
vT (Lx + Ly)v = vT Lxv + vT Lyv = 0. (3.15)
Since Lx and Ly are positive semi-definite, it follows from (3.15) that vT Lxv = vT Lyv = 0,
and again by Lemma 3.1 and Lemma 3.2, one can conclude that v ∈ Ker(Lx) and
v ∈ Ker(Ly). Thus, Ker(Lx + Ly) ⊂ Ker(Lx) and Ker(Lx + Ly) ⊂ Ker(Ly). Similar to
(3.14) we can show that col(jv, v) and col(−jv, v) are eigenvectors of A associated with
eigenvalues ±j, i.e., the algebraic multiplicity and geometric multiplicity of ±j are at
least 2. So the subspace
V ′ := Span
Re
jvv
, Im
jvv
= Span
0n
v
, v
0n
is invariant, and hence for every nonzero initial condition in V ′ the solution will remain
in V ′ and does not converge to zero. From the fact that V ′ ∩ S = 0, we conclude that
there exists an initial condition such that all the harmonic oscillators do not asymp-
totically synchronize. This is a contradiction, therefore if all the harmonic oscillators
asymptotically synchronize then Gx ∪ Gy is connected. This completes the proof.
3.2.2 General Directed Graphs
In Section 3.2.1, it is shown that when the graphs are balanced, connectivity of their union
is the necessary and sufficient condition for the network to achieve synchronization. Now
we consider the general weighted directed graphs. In this case, our simulation results
show that, when the union of the graphs has a spanning tree, synchronization is achieved
Chapter 3. Coupled Harmonic Oscillators 31
in the network; however, this is only a conjecture. In this section, we investigate sufficient
conditions for the harmonic oscillators to asymptotically synchronize. Moreover, for two
special network structures, necessary and sufficient conditions for synchronization are
provided. We make use of the following two lemmae. These lemmae characterize the
left eigenvector of a Laplacian matrix corresponding to the zero eigenvalue. Lemma 3.3
is only for the case where the graph is strongly connected, whereas Lemma 3.4 can be
applied to general directed graphs.
Lemma 3.3. Let L be a Laplacian matrix associated with a strongly connected digraph.
Let p = [p1 . . . pn]T ∈ Rn be the left eigenvector of L associated with the zero eigenvalue,
and let P = diag(pi), i = 1 . . . , n. Then P 0 and PL+ LTP 0.
Proof : See [69, Lemma 6] and [41, Theorem 4.31]. Note that in Lemma 3.3, when
the graph is balanced, then p = 1n, P = In, and L 0.
Lemma 3.4. Let L ∈ Rn×n be a Laplacian matrix associated with a digraph. Suppose
there exists p = [p1 . . . pn]T ∈ Rn such that pi > 0 for i = 1, . . . n and pTL = 0Tn . Let
P = diag(Pi), i = 1, . . . , n. Then, Ker(LTP + PL) ⊂ Ker(L).
Proof : Let v ∈ Ker(LTP + PL), i.e., (LTP + PL)v = 0n and
vT (LTP + PL)v = vTPLv + vTLTPv = 2vTPLv = 0 (3.16)
W need to show that v ∈ Ker(L). Let L be defined as (3.6). The quadratic form vTPLv
can be computed as follows:
vTPLv =n∑i=1
(v2i pi
n∑j=1
cij
)−
n∑i,j=1
picijvivj =n∑
i,j=1
picij(v2i − vivj). (3.17)
Since pTL = 0Tn we have
pi
n∑j=1
cij =n∑j=1
pjcji (3.18)
Chapter 3. Coupled Harmonic Oscillators 32
It follows from (3.17) and (3.18) that
vTPLv =n∑i=1
v2i
(pi
n∑j=1
cij
)−
n∑i,j=1
picijvivj
=n∑i=1
v2i
( n∑j=1
pjcji
)−
n∑i,j=1
picijvivj
=n∑j=1
v2j
( n∑i=1
picij
)−
n∑i,j=1
picijvivj
=n∑
i,j=1
picij(v2j − vjvi).
(3.19)
From (3.16), (3.17), and (3.18) we obtain
2vTPLv = 2
[n∑i=1
v2i
(pi
n∑j=1
cij
)−
n∑i,j=1
picijvivj
]
=n∑
i,j=1
picij(v2i − vivj) +
n∑i,j=1
picij(v2j − vjvi)
=n∑
i,j=1
picij(v2i + v2j − 2vivj)
=n∑
i,j=1
picij(vi − vj)2 = 0
(3.20)
Since pi > 0 for i = 1, . . . , n, we can find the possible values of vi similar to Lemma 3.2,
that is, if cij > 0, then vi = vj and conclude that for i = 1, . . . n, Liv = 0. Thus, Lv = 0n
which implies that Ker(LTP + PL) ⊂ Ker(L). This completes the proof.
Now, we are ready to state our first theorem for network of diffusively coupled harmonic
oscillators with the general directed graph topologies.
Theorem 3.2. Let Gx and Gy be directed graphs, and Lx and Ly be the associated Lapla-
cian matrices. Assume the following:
1. Gx ∪ Gy is strongly connected.
2. the intersection of the left kernels of Lx and Ly is not zero.
Chapter 3. Coupled Harmonic Oscillators 33
Then, all the harmonic oscillators in network (3.4) asymptotically synchronize.
Proof : Since Gx ∪ Gy is strongly connected, it follows from Lemma 3.3 that there
exists p = [p1 . . . pn]T such that pi > 0, i = 1, . . . , n and p(Lx + Ly) = 0Tn . Note that
rank(Lx + Ly) = n− 1, so the left kernel of Lx + Ly has dimension 1, and it is Spanp.
Also, the intersection of the left kernels of Lx and Ly is contained in the left kernels
Lx +Ly, and according to our assumption the intersection of the left kernels is not zero,
so it is Spanp.
Let P = diag(pi), i = 1 . . . , n. Consider
V (x, y) =
[xT yT
]P 0
0 P
xy
The derivative of V along the trajectories of the network is as follows:
V =
[xT yT
](ATP + PA)
xy
=
[xT yT
](−LTx I
−I −LTy
P 0
0 P
+
−Lx −I
I −Ly
P 0
0 P
)xy
=
[xT yT
]−LTxP − PLx 0
0 −LTy P − PLy
xy
We can compute V by using (3.20) as follows:
V = −
(n∑
i,j=1
piaij(xj − xi)2 +n∑
i,j=1
pibij(yj − yi)2)≤ 0.
Similar to the proof of Theorem 3.1 consider the compact set Ω := col(x, y) ∈ R2n :
V (x, y) ≤ c where c ≥ 0. By Nagumo theorem, we conclude that Ω is positively
Chapter 3. Coupled Harmonic Oscillators 34
invariant. Let Ed := col(x, y) ∈ Ω : V (x, y) = 0, i.e.,
Ed = col(v, w) ∈ R2n : v ∈ Ker(LTx + PLx), w ∈ Ker(LTy P + PLy),
and let Md be the largest invariant subset of Ed. By LaSalle’s invariance principle we
conclude that for every initial condition z0 ∈ Ω, the solution Φ(t, z0) →Md as t → ∞.
One can follow the same steps in the proof of Theorem 3.1 and using Lemma 3.4 to
complete the proof.
We observe that for some special cases, the conditions in Theorem 3.2 are conser-
vative. For instance, [42] shows that when Gx is an empty graph, the network achieve
synchronization if and only if Gy has a spanning tree. In this situation the first condition
in Theorem 3.2 does not hold. This shows that we may be able to relax the condition
of strong connectivity to the existence of a spanning tree in the union of the graphs. In
order to do so, we study the properties of the eigenvalues of A in the network (3.4). We
know from (3.14) that ±j are in the spectrum of A. The next theorem, characterizes the
other eigenvalues of A to achieve synchronization.
Theorem 3.3. Consider the network of harmonic oscillators defined in (3.4). The har-
monic oscillators asymptotically synchronize if and only if the eigenvalues of A are in
the open left half-plane (OLHP) except a simple complex conjugate pair ±j.
Proof : (Sufficiency) Let βi, i = 1, . . . , 2n be the eigenvalues of A with β1,2 = ±j, and
ri, li, i = 1, . . . , 2n be respectively the associated right and left eigenvectors or generalized
eigenvectors with r1 =
j1n1n
, l1 =
lx1ly1
, r2 =
−j1n1n
, l2 =
lx2ly2
. Since β1,2 = ±j
the associated left eigenvectors are complex conjugate. We define Relx := Relx1 =
Relx2 and Imlx := Imlx1 = −Imlx2, and similarly we can define ly. We can write
Chapter 3. Coupled Harmonic Oscillators 35
A in Jordan canonical form as
A = [r1, r2, . . . , r2n]︸ ︷︷ ︸P
J1︷ ︸︸ ︷j 0
0 −j
0
0 J2
︸ ︷︷ ︸
J
lT1
lT2...
lT2n
︸ ︷︷ ︸P−1
, (3.21)
where J2 is a block diagonal matrix consists of the Jordan blocks associated with eigen-
values βi, i = 3, . . . , 2n. So we obtain
eAt = P
eJ1t 0
0 eJ2t
P−1. (3.22)
Since Reβi < 0, i = 3, . . . , 2n, we have limt→∞
eJ2t = 0. It follows that
limt→∞
∥∥∥∥∥∥∥eAt − P
eJ1t 0
0 0
P−1∥∥∥∥∥∥∥ = 0
Also, we have
P
eJ1t 0
0 0
P−1= ejtr1l
T1 + e−jtr2l
T2
= ejt
j1n1n
(Re
[lxT ly
T
]+ Im
[lxT ly
T
])
+ e−jt
−j1n1n
(Re
[lxT ly
T
]− Im
[lxT ly
T
])
Chapter 3. Coupled Harmonic Oscillators 36
=
(ejt
−1
j1n
1n
+ e−jt
1
j1n
1n
)Re
[lxT ly
T
]
+
(ejt
−1n−1
j1n
+ e−jt
−1n1
j1n
)Im
[lxT ly
T
]
=
−ejt − e−jt
j1nRelTx − (ejt + e−jt)1nImlTx −
ejt − e−jt
j1nRelTy − (ejt + e−jt)1nImlTy
(ejt + e−jt)1nRelTx −ejt − e−jt
j1nImlTx (ejt + e−jt)1nRelTy −
ejt − ejt
j1nImlTy
=
−2.(1n)(
sin(t)RelTx + cos(t)ImlTx )−2.(1n)
(sin(t)RelTy + cos(t)ImlTy
)2.(1n)
(cos(t)RelTx − sin(t)ImlTx
)2.(1n)
(cos(t)RelTy − sin(t)ImlTy
) .
(3.23)
The solution of the network (3.4) is given by
x(t)
y(t)
= eAt
x(0)
y(0)
. It follows from (3.22)
and (3.23) that for each oscillator
limt→∞
∣∣∣xi(t)− 2(
sin(t)RelTx + cos(t)ImlTx )x(0)− 2
(sin(t)RelTy + cos(t)ImlTy
)y(0)
∣∣∣ = 0
limt→∞
∣∣∣yi(t)− 2(
cos(t)RelTx − sin(t)ImlTx )x(0) + 2
(cos(t)RelTy − sin(t)ImlTy
)y(0)
∣∣∣ = 0
(3.24)
which implies that all the oscillators asymptotically synchronize.
(Necessity) We argue by contradiction. Similar argument can be found in [44], [43],
[68]. Let β1,2 = ±j and suppose ∃β ∈ βii=3,...,2n such that <β ≥ 0. Without
loss of generality, let β3 = β. Similar to the proof of the sufficiency, let J := [jik] for
i, k = 1, . . . , 2n be the Jordan canonical form of A so that jii = βi for i = 1, . . . , 2n.
This implies that limt→∞
ejiit 6= 0 for i = 1, 2, 3. Hene, we can conclude that the first three
rows of limt→∞
eJt are linearly independent so the rank of limt→∞
eJt as well as the rank of
limt→∞ eAt is at least three. According to our assumption, all the harmonic oscillators
Chapter 3. Coupled Harmonic Oscillators 37
asymptotically synchronize which means that
limt→∞
∥∥∥∥∥∥∥eAt −1nr
T
1nsT
∥∥∥∥∥∥∥ = 0,
where r, s ∈ Rn. So when synchronization occurs, the rank of limt→∞
eAt is at most two. This
is a contradiction. Therefore, if all the harmonic oscillators asymptotically synchronize,
then A has exactly two imaginary eigenvalues and all the other eigenvalues are in OLHP.
This concludes the proof.
In view of Theorem 3.3, we now study two special cases:
1. The case where the interconnection topologies are identical.
2. The case where only one interconnection is present.
Consider the first case, i.e.,
xy
=
−L −I
I −L
︸ ︷︷ ︸
A1
xy
, (3.25)
where L is the laplacian matrix associated with graph Gx = Gy. A necessary and sufficient
condition for the network (3.25) to achieve synchronization is stated in the following
corollary. Note that from [51, Lemma 1], the authors have already given a sufficient
condition for the harmonic oscillators in the network (3.25) to be synchronized. However,
we provide a necessary and sufficient condition in the following corollary.
Corollary 3.1. Consider the network (3.25). The harmonic oscillators in the network
asymptotically synchronize if and only if the graph has a spanning tree.
Proof : (Sufficiency) Let λi and vi for i = 1, . . . , n be the eigenvalues and the associ-
ated eigenvectors of L. It follows from Lemma 2.1 that the zero eigenvalue of L is simple.
Chapter 3. Coupled Harmonic Oscillators 38
We have−L −I
I −L
jvivi
=
−jλivi − vijvi − λivi
=
−(jλi + 1)vi
(j − λi)vi
=
−(λi − j)jvi
−(λi − j)vi
= −(λi − j)
jvivi
.(3.26)
similarly, we can show that
−L −I
I −L
−jvi
vi
= −(λi + j)
jvivi
. (3.27)
It follows from (3.26) and (3.27) that −λi ± j for i = 1, . . . , n are the eigenvalues of
A1. For the simple zero eigenvalue of L the corresponding eigenvalues of A1 are ±j,
and for the other eigenvalues of L, the corresponding eigenvalues are in the OLHP. From
Theorem 3.3 we conclude that all the harmonic oscillators asymptotically synchronize.
(Necessity) Assume all the harmonic oscillators asymptotically synchronize. By The-
orem 3.3, matrix A1 has two imaginary eigenvalues and all the other eigenvalues of A1 are
in the OLHP. It follows from (3.26) and (3.27) that, for every zero eigenvalue of L, the
matrix A1 has corresponding ±j eigenvalues. So L must have a simple zero eigenvalue.
By Lemma 2.1, we conclude that L must have a spanning tree. This concludes the proof.
Now consider the second special case where only one interconnection is present. With-
out loss of generality, assume that only y’s are diffusively coupled and Lx = 0. So we
get
xy
=
0 −I
I −Ly
︸ ︷︷ ︸
A2
xy
. (3.28)
The following corollary states a necessary and sufficient condition for the harmonic oscil-
Chapter 3. Coupled Harmonic Oscillators 39
lators in network (3.28) to asymptotically synchronize. Similar argument can be found
in [42].
Corollary 3.2. The harmonic oscillators in network (3.28) asymptotically synchronize
if and only if the directed graph Gy has a spanning tree.
Proof : (sufficiency)Let βi for i = 1, . . . , 2n be the eigenvalues of A2 and ri :=
col(rxi , ryi) ∈ C2n be the associated right eigenvectors. Then,
0 −I
I −Ly
rxiryi
= βi
rxiryi
. (3.29)
From (3.29) we get
−ryi = βirxi (3.30)
rxi − Lyryi = βiryi . (3.31)
Plugging (3.30) in (3.31) yields
Lyryi = −(ryiβi
+ βiryi
)= −
(βi +
1
βi
)ryi . (3.32)
From (3.32), λk := −βi− 1βi
for k = 1, . . . , n are the eigenvalues and ryi are the associated
right eigenvectors of Ly. For each λk, two eigenvalues of A2 satisfy λk = −βi − 1βi
. Since
Gy has a spanning tree, from Lemma 2.1 Ly has a simple zero eigenvalue and all the other
eigenvalues have positive real part. Let βi = ai + jbi, then 1βi
= aiai2+bi
2 + j −biai2+bi
2 so the
real parts of βi and 1βi
have the same sign. Without loss of generality, let λ1 = 0. The
corresponding eigenvalues of A2 can be easily find as β1,2 = ±j. Since the real part of
λk for k = 2, . . . n, are positive, we conclude that the real part of βi for i = 3, . . . , 2n,
are negative. Therefore, A2 has two imaginary eigenvalues ±j and all other 2n − 2
eigenvalues are in the OLHP. It follows from Theorem 3.3 that all the harmonic oscillators
Chapter 3. Coupled Harmonic Oscillators 40
asymptotically synchronize.
(Necessity) The proof is similar to the proof of Corollary 3.1. Assume all the harmonic
oscillators in the network (3.28) asymptotically synchronize. By Theorem 3.3, the matrix
A2 has two imaginary eigenvalues and all the other eigenvalues are in the OLHP. It follows
from (3.32) that the zero eigenvalue of Ly is simple. Therefore by Lemma 2.1, the directed
graph Gy has a spanning tree. This completes the proof.
Remark 3.1. We can find the solutions of the harmonic oscillators in network (3.28)
as follows. Let βi for i = 1, . . . , 2n be the eigenvalues of A2 and ri, li be the right and
left associated eigenvectors. it follows from (3.30), (3.32) that ri = col(− 1
βiryi , ryi).
Similarly, we can show that li = col(1
βilyi , lyi). In particular, for β1,2 = ±j we have
r1 =
j1n1n
, l1 =
−jlyly
, r2 =
−j1n1n
, l2 =
jlyly
. (3.33)
Substituting (3.33) in (3.24) gives
limt→∞|xi(t)− cos(t)1nl
Ty x(0)− sin(t)1nl
Ty y(0)| = 0,
limt→∞|yi(t)− sin(t)1nq
Tx(0) + cos(t)1nqTy(0)| = 0.
for i = 1, . . . , n.
3.3 Robustness Analysis
In this section, we deal with the perturbed diffusively coupled harmonic oscillators.
Namely, we study multiplicative uncertainty in the output of Laplacian matrices as it
is shown Fig. 3.3. We assume that ∆x,∆y : R × Rn → Rn are Lipschitz functions
and ∆x(t, 0) = 0, ∆y(t, 0) = 0. Our result allows the uncertainties to be nonlinear and
time-varying. In the new framework the network is modelled as
Chapter 3. Coupled Harmonic Oscillators 41
1s
. . .1s
1s
. . .1s
−Lx −Ly
∆x ∆y
x y
−
Figure 3.3: Block diagram of the perturbed network of harmonic oscillators
xy
=
−Lx −I
I −Ly
xy
+
∆x(t,−Lxx)
∆y(t,−Lyy)
. (3.34)
A useful tool to study synchronization is to project the dynamics onto the orthogonal
complement of the synchronization subspace. The projected model is called the reduced
model. So instead of studying the attractivity of the synchronization subspace, we can
study to asymptotic stability of the origin of the reduced model.
Recall that D = Span1n. Let Q ∈ R(n−1)×n be the projector onto the orthogo-
nal complement of D. The columns of QT ∈ Rn×(n−1) form an orthonormal basis of
orthogonal complement of D, so a possible choice of Q is
Q =
−1 + (n− 1)v 1− v −v · · · −v
−1 + (n− 1)v −v 1− v · · · ...
......
. . . . . . −v
−1 + (n− 1)v −v · · · −v 1− v
(n−1)×n
, (3.35)
where v = n−√n
n(n−1) . Note that the matrix Q has the following properties:
i)Q1n = 0, ii)QQT = In−1, iii)QTQ = In −1
n1n1Tn . (3.36)
Chapter 3. Coupled Harmonic Oscillators 42
For r ∈ Rn, let r denote Qr. From property iii) in (3.36) and the fact that Lx1n = 0n,
we have
LxQTQx = Lx(In −
1
n1n1Tn )x = Lx(Inx)− Lx(
1
n1n1Tn )x = Lxx.
This implies
QLxx = QLxQTQx = QLxQ
T x = Lxx, (3.37)
where Lx := QLxQT . By projecting the dynamics (3.34) we obtain
Qx = ˙x = Q(− y − Lxx+ ∆x(t,−Lxx)
)= −y −QLxx+Q∆x(t,−Lxx)
= −y − Lxx+Q∆x(t,−LxQT x).
Similarly, ˙y = x− Lyy+Q∆y(t,−LyQT y). Therefore, if we multiply (3.34) by diag(Q,Q)
we obtain ˙x
˙y
︸︷︷︸
˙z
=
−Lx −I
I −Ly
︸ ︷︷ ︸
A
xy
︸︷︷︸z
+
gx(t, x)
gy(t, y)
︸ ︷︷ ︸
g(t,z)
, (3.38)
where gx(t, x) := Q∆x(t,−LxQT x) and gy(t, y) := Q∆y(t,−LyQT y). The vector z is
called the synchronization error. So studying the synchronization problem in the network
(3.34), is equivalent to studying the stability of the origin in (3.38).
Remark 3.2. When the harmonic oscillators in the nominal model (3.4) asymptotically
synchronize, the origin in the reduced model is exponentially stable. By the converse
Lyapunov theorem (see e.g. [25]), for every symmetric positive definite matrix S, there
exists a unique symmetric positive definite matrix P that satifies the Lyapunov equation
Chapter 3. Coupled Harmonic Oscillators 43
ATP + PA = −S. So we construct the Lyapunov function for the reduced model of the
nominal network as V (z) = zTP z.
Our goal is to study robustness of synchronization with respect to nonlinear pertur-
bations on the diffusive coupling. We will provide conditions on the perturbed network
(3.34) such that all the harmonic oscillators asymptotically synchronize. Equivalently, we
will provide conditions on the reduced model (3.38) such that the origin be asymptotically
stable.
Assume that there exists δ ≥ 0 such that for every r ∈ Rn and every t > 0,
‖∆x(t, r)‖ ≤ δ‖r‖ and ‖∆y(t, r)‖ ≤ δ‖r‖. (For the sake of simplicity, we assume that δ
is the same for both ∆x and ∆y). Let σx denotes the largest singular value of Lx, i.e.,
‖Lx‖ = σx. Then we have
‖Q∆x(t,−Lxx)‖ = ‖Q∆x(t,−LxQT x)‖
≤ ‖Q‖‖∆x(t,−LxQT x)‖
≤ δ‖Q‖‖LxQT x‖
≤ δ‖Q‖‖Lx‖‖QT‖‖x‖
≤ δσx‖x‖
(3.39)
Note that in (3.39) we use the fact that ‖Q‖ = ‖QT‖ = 1. Similarly we have
‖Q∆y(t,−Lyy)‖ ≤ δσy‖y‖, (3.40)
where σy is the largest singular value of Ly. It follows from (3.39) and (3.40) that
‖g(t, z)‖ =
∥∥∥∥∥∥∥gx(t, x)
gy(t, y)
∥∥∥∥∥∥∥ ≤ δ ·maxσx, σy
∥∥∥∥∥∥∥x
y
∥∥∥∥∥∥∥ = δ ·maxσx, σy‖z‖. (3.41)
In order to determine when synchronization is a robust feature with respect to nonlinear
Chapter 3. Coupled Harmonic Oscillators 44
perturbations on the diffusive coupling, we make use of Lyapunov stability theory as
follows.
Theorem 3.4. Consider the network (3.34). Assume that all the harmonic oscillators
in the nominal network (3.4) asymptotically synchronize. Let S, P be symmetric positive
definite matrices that satisfy the Lyapunov equation ATP + PA = −S. Suppose that
there exists δ ≥ 0 such that for every r ∈ Rn and every t > 0, ‖∆x(t, r)‖ ≤ δ‖r‖ and
‖∆y(t, r)‖ ≤ δ‖r‖. Then all the harmonic oscillators in the perturbed network (3.34)
asymptotically synchronize if δ <λmin(S)
2λmax(P ) maxσx, σy.
Proof : Let V (z) = zTP z be the Lyapunov function. The derivative of V (z) along
the trajectories of the nominal system (i.e., ˙z = Az) is bounded above by
∂V
∂zAz = −zTSz ≤ −λmin(S)‖z‖2. (3.42)
Also, the norm of∂V
∂zis bounded above by
∥∥∥∥∂V∂z∥∥∥∥ = ‖2zTP‖ ≤ 2‖P‖‖z‖ = 2λmax(P )‖z‖. (3.43)
From (3.42) and (3.43), the derivative of V (z) along the trajectories of the perturbed
network (i.e., ˙z = Az + g(t, z).) satisfies
V (z) =∂V
∂zAz +
∂V
∂zg(t, z)
≤ −zTSz +
∥∥∥∥∂V∂z∥∥∥∥ ‖g(t, z)‖
(3.44)
It follows from (3.41), (3.43), and (3.44) that
V (z) ≤ −λmin(S)‖z‖2 + 2λmax(P ) δ ·maxσx, σy‖z‖2.
Thus, if δ <λmin(S)
2λmax(P ).maxσx, σy, then V (z) < 0 and so the origin in (3.38) is globally
Chapter 3. Coupled Harmonic Oscillators 45
exponentially stable. Therefore, all the harmonic oscillators in the perturbed network
(3.34) asymptotically synchronize. This concludes the proof.
Remark 3.3. The upper boundλmin(S)
2λmax(P )is maximized when S = I.
Proof : Let S be any matrix such that S = ST 0, k > 0, and µ(S) =λmin(S)
2λmax(P ).
Note that µ(S) is unchanged if we scale S by k, i.e.,
µ(kS) =λmin(kS)
2λmax(kP )=
λmin(S)
2λmax(P )= µ(S).
We choose k =1
λmin(S)so that λmin(kS) = 1. In the following, we want to show that
µ(I) ≥ µ(kS) = µ(S). Let P1 and P2 be the solutions of the Lyapunov equations
ATP1 + P1A = −I and ATP2 + P2A = −kS, respectively. The matrices P1 and P2 can
be written as
P1 =
∫ ∞0
eAT teAtdt, P2 =
∫ ∞0
eAT t(kS)eAtdt.
Then
P1 − P2 =
∫ ∞0
eAT t(I − kS)eAtdt.
Since λmin(S) = 1, we have kS I which implies
I − kS 0 =⇒ P1 − P2 0 =⇒ P2 P1 =⇒ λmax(P2) ≥ λmax(P1). (3.45)
It follows From (3.45) and the definition of µ that
µ(I) ≥ µ(kS) = µ(S)
Therefore, with the choice S = I, the maximum of ratioλmin(S)
2λmax(P )is achieved. This
Chapter 3. Coupled Harmonic Oscillators 46
concludes the proof of Remark 3.3.
3.4 Robustness Analysis in The Presence of Distur-
bances
In this case, we study synchronization in the presence of norm bound disturbance signals
vx(t), vy(t) as it is shown in Fig. So the state space model is
xy
=
−Lx −I
I −Ly
xy
+
∆x(t,−Lxx)
∆y(t,−Lyy)
+
vxvy
. (3.46)
By applying the projection diag(Q,Q), the reduced model of the network can be obtained
vx vy1s
. . .1s
1s
. . .1s
−Lx −Ly
∆x ∆y
x y
−
Figure 3.4: Block diagram of the perturbed network of harmonic oscillators with distur-bances
as ˙x
˙y
=
−Lx −I
I −Ly
xy
+
gx(t, x)
gy(t, y)
+
vxvy
︸ ︷︷ ︸
v
. (3.47)
Chapter 3. Coupled Harmonic Oscillators 47
So we can write (3.47) as
˙z = Az + g(z) + v(t). (3.48)
We use the following lemma to show that the synchronization error z is globally uniformly
bounded. The next lemma is the simplified version of Theorem 4.18 in [25].
Lemma 3.5. Consider the system x = f(t, x). Let α1 and α2 be class K functions and
W (x) be a continuous positive definite function. Let V : Rn → R be a continuously
differentiable function such that, ∀x ∈ Rn,
α1(‖x‖) ≤ V (x) ≤ α2(‖x‖)
V (t, x) ≤ −W (x), ∀‖x‖ ≥ µ > 0.
Then, there exists T > 0 such that
‖x(t)‖ ≤ b, ∀t ≥ T,
where b = α−11 (α2(µ)).
Theorem 3.5. Consider network (3.46) and its reduced model (3.47). Assume all the
harmonic oscillators in the nominal network (3.4) asymptotically synchronize. Let S, P be
symmetric positive definite matrices, that satisfy the Lyapunov equation ATP+PA = −S.
Suppose that there exists δ ≥ 0 and δv > 0 such that for every r ∈ Rn and every t > 0,
‖∆x(t, r)‖ ≤ δ‖r‖, ‖∆y(t, r)‖ ≤ δ‖r‖, and ‖v‖ ≤ δv. If δ <1
2λmax(P ) maxσx, σy, then
there exists T > 0 such that
‖z(t)‖ ≤ 2λmax(P )δv1− 2λmax(P ) δ ·maxσx, σy
√λmax(P )
λmin(P ), ∀t ≥ T. (3.49)
Chapter 3. Coupled Harmonic Oscillators 48
Proof : Let V (z) = zTP z be the Lyapunov function for (3.47). We have
λmin(P )‖z‖2 ≤ V (z) ≤ λmax(P )‖z‖2
So from Lemma 3.5,
α1(‖z‖) = λmin(P )‖z‖2, α2(‖z‖) = λmax(P )‖z‖2.
The derivative of V (z) = zTP z along the trajectories of (3.48) satisfies
V (z) ≤ −‖z‖2 +
∥∥∥∥∂V∂z∥∥∥∥ ‖g(z)‖+
∥∥∥∥∂V∂z∥∥∥∥ ‖v‖
≤ −‖z‖2 + 2λmax(P ) δ ·maxσx, σy‖z‖2 + 2λmax(P )δv‖z‖.(3.50)
Let c := 1− 2λmax(P ).δ.maxσx, σy. Then from (3.50)
V (z) ≤ −c‖z‖2 + 2λmax(P )δv‖z‖
= −‖z‖(c‖z‖ − 2λmax(P )δv
)
Let θ := c‖z‖ − 2λmax(P )δv. For ‖z‖ > 2λmax(P )δvc
, we have θ > 0, so for ‖z‖ >2λmax(P )δv
cwe obtain
V (z) ≤ −θ‖z‖, θ > 0.
Let µ :=2λmax(P )δv
c. From Lemma 3.5, we conclude that ‖z‖ is globally uniformly
bounded by
b = α−11 (α2(µ)) =
√α2(µ)
λmin(P )= µ
√λmax(P )
λmin(P )=
2λmax(P )δv1− 2λmax(P ) δ ·maxσx, σy
√λmax(P )
λmin(P ).
This completes the proof.
Chapter 3. Coupled Harmonic Oscillators 49
3.5 Examples
Example 3.1. Consider a network of four diffusively coupled harmonic oscillators. Let
Gx and Gy be balanced graphs as depicted in Fig. 3.5, and Lx and Ly be the associated
Laplacian matrices as follows:
Lx =
0.2 0 −0.2 0
−0.2 0.2 0 0
0 −0.2 0.2 0
0 0 0 0
, Ly =
0 0 0 0
0 0 0 0
0 0 0.1 −0.1
0 0 −0.1 0.1
.
Since Gx ∪ Gy is connected, by Theorem 3.1, we conclude that all 4 harmonic oscillators
1
2
3
4
0.2
0.2
0.2
(a) Gx
1
2
3
4
0.1
0.1
(b) Gy
1
2
3
4
0.2
0.2
0.2
0.1
0.1
(c) Gx ∪ Gy
Figure 3.5: Graph representation of Gx and Gy and their union for Example 3.1
asymptotically synchronize. The simulation results for this example with arbitrary initial
conditions are shown in Fig. 3.6 and 3.7.
Example 3.2. Consider again a network of four diffusively coupled harmonic oscilla-
tors, but now assume that Gx and Gy be directed weighted graphs that are not balanced.
Fig. 3.8 show both graphs and its union. The associated Laplacian matrices are
Lx =
0.3 −0.3 0 0
−0.1 0.1 0 0
0 0 0.1 −0.1
0 0 0.1 −0.1
, Ly =
0 0 0 0
0 0.2 −0.2 0
0 −0.1 0.1 0
0 0 0 0
.
Chapter 3. Coupled Harmonic Oscillators 50
0 10 20 30 40 50 60 70 80−10
−8
−6
−4
−2
0
2
4
6
8
10
time (se c )
x
x1x2x3x4
Figure 3.6: Asymptotic synchronization of the x components of the states for Example3.1
0 10 20 30 40 50 60 70 80−10
−8
−6
−4
−2
0
2
4
6
8
10
time (se c )
y
y1y2y3y4
Figure 3.7: Asymptotic synchronization of the y components of the states for Example3.1
Chapter 3. Coupled Harmonic Oscillators 51
1
2
3
40.1
0.3
0.1
0.1
(a) Gx
1
2
3
4
0.1
0.2
(b) Gy
1
2
3
40.1
0.3
0.1
0.10.1
0.2
(c) Gx ∪ Gy
Figure 3.8: Graph representation of Gx and Gy and their union for Example 3.2
It is easy to show that p =
[1 3 6 6
]Tis in the left kernel of both Lx and Ly.
Also, the union of the graphs is strongly connected. So, by Theorem 3.2 we conclude
that all four harmonic oscillators asymptotically synchronize. The simulation results are
depicted is Fig. 3.9 and 3.10. Note that the initial conditions are chosen arbitrarily.
0 10 20 30 40 50 60 70 80−5
−4
−3
−2
−1
0
1
2
3
4
5
time (se c )
x
x1x2x3x4
Figure 3.9: Asymptotic synchronization of the x components of the states for Example3.2
Example 3.3. Consider again a network of four diffusively coupled harmonic oscilla-
Chapter 3. Coupled Harmonic Oscillators 52
0 10 20 30 40 50 60 70 80−6
−4
−2
0
2
4
6
time (se c )
y
y1y2y3y4
Figure 3.10: Asymptotic synchronization of the y components of the states for Example3.2
tors, but now assume that Gx = Gy be directed weighted graphs that are not balanced
and their union is not strongly connected. The structure of the graph is depicted in Fig.
3.11.
1
2
3
4
1
1
1
(a) Gx
1
2
3
4
1
1
1
(b) Gy
1
2
3
4
2
2
2
(c) Gx ∪ Gy
Figure 3.11: Graph representation of Gx = Gy and Gx ∪ Gy for Example 3.3
Chapter 3. Coupled Harmonic Oscillators 53
The corresponding Laplacian matrices are
Lx = Ly =
0 0 0 0
−1 1 0 0
−1 0 1 0
−1 0 0 1
,
Since Lx has a spanning tree, by Corollary 3.1 we conclude that all four harmonic oscil-
lators asymptotically synchronize. This fact is illustrated in Fig. 3.12 and 3.13.
0 2 4 6 8 10 12 14 16 18 20−5
−4
−3
−2
−1
0
1
2
3
4
5
time (se c )
x
x1x2x3x4
Figure 3.12: Asymptotic synchronization of the x components of the states for Example3.3
Example 3.4. In this example we assume that everything is identical to Example
3.3 except that we assume Gy is a empty graph. So the coupling is only through the x
components of the states. It follows from Corollary 3.2 that all the harmonic oscillators
asymptotically synchronize. The simulation results are shown in Fig. 3.14 and 3.15.
In the simulations of examples 3.2.-3.4. we chose the same initial conditions. Com-
Chapter 3. Coupled Harmonic Oscillators 54
0 2 4 6 8 10 12 14 16 18 20−5
−4
−3
−2
−1
0
1
2
3
4
5
time (se c )
y
y1y2y3y4
Figure 3.13: Asymptotic synchronization of the y components of the states for Example3.3
paring all the simulation results in these examples shows that the convergence rate in
Example 3.3 is much faster than the others . This is reasonable since the weight of edges
is much stronger and also the graph topologies Gx and Gy are more connected.
3.5.1 A More General Example
So far we have discussed about synchronization of harmonic oscillators. As it is shown in
Fig. 3.1, an isolated harmonic oscillator is equivalent to two integrators interconnected
in negative feedback loop. At this point a reasonable question to ask is: can we extend
the results of this chapter to the network of linear identical systems each of which be a
chain of N integrators in feedback loop? (see Fig. 3.16.)
Let yki ∈ R for k = 1, . . . , N and i = 1, . . . , n be the output of integrator k in system
i and yk = col(yk1, . . . , ykn) be vector of all the outputs of kth integrators. The state
Chapter 3. Coupled Harmonic Oscillators 55
0 2 4 6 8 10 12 14 16 18 20−10
−8
−6
−4
−2
0
2
4
6
8
10
time (se c )
x
x1x2x3x4
Figure 3.14: Asymptotic synchronization of the x components of the states for Example3.4
0 2 4 6 8 10 12 14 16 18 20−10
−8
−6
−4
−2
0
2
4
6
8
10
time (se c )
y
y1y2y3y4
Figure 3.15: Asymptotic synchronization of the y components of the states for Example3.4
Chapter 3. Coupled Harmonic Oscillators 56
1s
1s
1s
y1i . . .y2i yNi
−
Figure 3.16: Block diagram of an isolated chain of integrators with a negative feedbackloop
space of the isolated system is given by
y1i...
yNi
=
01×(N−1) −1
IN−1 0(N−1)×1
︸ ︷︷ ︸
B
y1i...
yNi
,y1i(0)
...
yNi(0)
=
y01i...
y0Ni
. (3.51)
When N = 2 (i.e., the harmonic oscillator), the eigenvalues of B are on the imaginary
axis, and when N > 2 the system becomes unstable, because the system characteristic
polynomial and its roots are
λN + 1 = 0 =⇒
λk = e2kπjN , N is odd
λk = e(2kπjN
+πjN
), N is even.
So the eigenvalues are on the unit circle in the complex plane and when N > 2 we have
at least one eigenvalue in the open right half-plane. Let y := col(y1, . . . , yN), and let
L := diag(L1, . . . , LN), where each Lk ∈ Rn×n is the Lapacian matrix associated with
the diffusive coupling of the output yk. The entire network is modelled as
y =[(B ⊗ In)− L
]︸ ︷︷ ︸
A
y. (3.52)
Let β be an eigenvalue of B with the associated eigenvector v = col(v1, . . . , vN) ∈ RN .
Let v′ := col(v11n, . . . , vN1n) ∈ RnN . Since Lkvk1n = vkLk1n = 0n we get Lv′ = 0nN , and
Chapter 3. Coupled Harmonic Oscillators 57
hence
Av′ =[(B ⊗ In)− L
]v′ = (B ⊗ In)v′ − Lv′ = (B ⊗ In)v′ = βv′. (3.53)
This implies that β is also an eigenvector for matrix A. Therefore for N > 2, the matrix
B as well as the matrix A has at least an eigenvalue with positive real part, So the
network (3.52) is unstable.
Although the network (3.52) is unstable, the synchronization subspace for the network
can still be asymptotically stable as it will be shown in the following example.
Example 3.5. We consider the network (3.52) where each state component is inter-
connected with the same state component across the compartments as it is shown in Fig.
3.17. We assume that the interconnection topology is undirected and “complete” (or “all
to all”). Also, we assume that the weight of all edges for each graph are identical, i.e.,
for k = 1, . . . , N , the graph Gk has the associated Laplacian Lk = ak(In − 1n1n1Tn ) where
ak is the weight of its edges. The output of integrator k in system i is denoted by yki ∈ R
for k = 1, . . . , N and i = 1, . . . , n. Although all the graphs are assumed to be complete,
only some of the edges are drawn in the figure to avoid complexity.
Let Q ∈ R(n−1)×n be the projector onto the orthogonal complement of D = Span1n
as defined in (3.35) and (3.36). The reduced model of (3.52) is
˙y :=
˙y1
˙y2...
˙yN
=
Q
Q
. . .
Q
L1 0 . . . −In
In L2 . . . 0
.... . . . . . 0
0 . . . In LN
QT
QT
. . .
QT
y1
y2...
yN
(3.54)
According to property ii) in (3.36), we know QQT = In−1, and for complete graph Gk we
Chapter 3. Coupled Harmonic Oscillators 58
1s
1s
1s
y11 . . .y21 yN1
−
1s
1s
1s
y12 . . .y22 yN2
−
a1 a2 aN
a1 a2 aN
1s
1s
1s
y1n . . .y2n yNn
−
a1 a2 aN
Figure 3.17: Block diagram of N coupled chain of integrators.
have
QLkQT = Q
[ak(In −
1
n1n1Tn )
]QT = ak(QQ
T − 1
nQ(1n1Tn )QT ) = akIn−1. (3.55)
By using (3.55), (3.36), and Kronecker product properties we can simplify (3.54) to
˙y =
[(IN ⊗Q)
[(B ⊗ In)− L
](IN ⊗QT )
]y
=[(IN ⊗Q)(B ⊗ In)(IN ⊗QT )− (IN ⊗Q)(L)(IN ⊗QT )
]y
=[(B ⊗Q)(IN ⊗QT )− diag(QL1Q
T , . . . , QLNQT )]y
=[B ⊗QQT − diag(a1In−1, . . . , aNIn−1)
]y
=[B ⊗ In−1 − diag(a1, . . . , aN)⊗ In−1
]y
Chapter 3. Coupled Harmonic Oscillators 59
=[(B − diag(a1, . . . , aN))⊗ In−1
]︸ ︷︷ ︸
A
y.
All the compartments asymptotically synchronize if and only if all the eigenvalues of A
are in the OLHP. Thus, all the compartments asymptotically synchronize if and only if
all the eigenvalues of A := B − diag(a1, . . . , aN) are in the OLHP. For k = 1, . . . , N , we
multiply the kth row of A by1
akand denote the new matrix by
A0 :=
−1 0 . . . − 1
a11
a2−1 . . . 0
.... . . . . . 0
0 . . .1
aN−1
.
we want to find conditions on the weights a1, . . . , aN such that, A0 be Hurwitz. A similar
matrix structure has been studied in [63], and it is shown that the necessary condition
for A0 to be Hurwitz is
1
a1a2 · · · · aN<[sec(
π
N)]N. (3.56)
Also, [4, Lemma 1] shows that (3.56) is necessary and sufficient for A0 to be diagonally
stable. Since diagonal stability implies that the matrix is Hurwitz, we conclude that
(3.56) is the necessary and sufficient for A0 to be Hurwitz. Thus, we conclude that all
the compartments asymptotically synchronize if and only if (3.56) holds.
Although all the graphs are complete and they have the maximum connectivity, syn-
chronization does not occur if the weight of the edges are small. This shows that the idea
of connectivity of the union of the graphs cannot be applied to general linear systems.
Chapter 4
Nonlinear Netwroks: An
Input-Output Approach
4.1 Introduction
In this chapter, we adopt an input-output framework to provide sufficient conditions
for robust output synchronization in a network of compartmental models. Our net-
work model is motivated by cellular networks where signalling occurs both internally and
externally through intracellular interaction (chemical reaction) and intercellular interac-
tion (chemical diffusion), respectively. It is assumed that each component of the network
(referred to as compartment) is composed of interconnected subsystems (referred to as
species) each of which is represented as an operator in the extended L2 space.
With respect to the majority of existing approaches, the distinctive features of our
framework are as follows: i) following the lines of [50] we adopt an input-output frame-
work. The input-output framework requires minimal knowledge of the physical laws gov-
erning the system and of the interconnections within the black box. This, in principle,
results in a reduction of the quantitative data (such as accurate parameters estimates)
required to describe the system. This feature is particularly appealing in the study of
60
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 61
biological systems; ii) our modeling approach explicitly takes into account two nested
interconnections and the resulting synchronization conditions admit biological interpre-
tations on cellular network structures; iii) a distinctive feature of our approach with
respect to [50], is to take into account nonlinear perturbations on the diffusive coupling
interaction among the compartments. The main result is a sufficient condition for robust
synchronization in the form of an L2 estimate of the synchronization error over an arbi-
trarily long observation period. Our contribution shows that in the presence of finite L2
gain perturbations on the diffusive coupling, the maximum singular values of Laplacian
matrices (which characterize the diffusive coupling network) play an important role in
synchronization, namely, large singular values has negative effects on synchronization.
This chapter is organized as follows. In Section 4.2, the compartmental network under
study is introduced. The main result is stated in Section 4.3, and it is illustrated with
an example of the network of Goodwin oscillators in Section 4.4.
4.2 Problem Statement
Consider n compartments each being composed of N species. In the following, sub-
scripts k, j stand for, respectively, species and compartment. Let ukj, ykj, vkj ∈ L2e,
k = 1, . . . , N, j = 1, . . . , n be the input, output and external input of the k-th species
in the j-th compartment, respectively. Species k in compartment j is described by a
(possibly nonlinear) input-output operator Hkj : L2e → L2e. The interconnected system
under study is given by
ykj = Hkjukj, k = 1, . . . , N, j = 1, . . . , n, (4.1)
where
ukj = vkj +N∑m=1
ekmymj +n∑i=1
(I + ∆kji)[akji(yki − ykj)
]. (4.2)
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 62
Here akji ≥ 0, akii = 0 (no self-loops), ekm are real numbers, I is the single input-single
output identity operator, and ∆kji : L2e → L2e are (possibly nonlinear) uncertainty
operators.
The species interconnection matrix, representing the coupling among species in each
compartment, is defined as E := [ekm] ∈ RN×N . It is assumed that E is identical
in different compartments, i.e., the structure of the compartments is identical. The
compartmental interconnection matrix representing the coupling among corresponding
species in different compartments is defined as Lk :=[lkji]∈ Rn×n for k = 1, . . . , N ,
where
lkji :=
n∑i=1
akji, i = j
−akji, i 6= j.
Note that the compartmental interconnection matrices Lk, k = 1, . . . , N are Laplacian
matrices. If for every k we associate a node to each interconnected species and draw an
edge between j and i whenever akji > 0 we can define for each Lk a graph Gk. The operator
∆kji in (4.2) takes into account possible uncertainties in the compartmental coupling. The
block diagram for the k-th species in the j-th compartment is depicted in Figure 4.1.
m = 1, . . . , N ekmymj
vkj
Hkj
I + ∆kji
akji(yki − ykj)
ukj ykj
i = 1, . . . , n
Figure 4.1: The input, output, and external input of species k in compartment j
Now we vectorize all the signals and operators in order to describe the network in a
concise form. We denote by uk = [uk1 . . . ukn]T for k = 1, . . . , N the vectors of inputs
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 63
of the corresponding species in different compartments and by u = col(u1, . . . , uN) the
stacked vector of all inputs. Similarly, we denote by yk, vk the vectors of outputs and
external inputs of species k in different compartments and also by y and v the vectors
of all outputs and external inputs. Let ∆k :=[∆kji
]: Ln2e → Ln2e for k = 1, . . . , N ,
be the matrices of uncertainty operators for the compartmental coupling. We assume
that ∆k has unbiased finite L2 gain as defined in Definition 2.6. We define Hk :=
diag(Hk1, . . . , Hkn), H := diag(H1, . . . , HN), E := E ⊗ In, L := diag(L1, . . . , LN), and
∆ := diag(∆1, . . . ,∆N).
By making use of the above notations, the network (4.1), (4.2) can be rewritten as
y = Hu (4.3)
u = v + (I + ∆)(−Ly) + Ey. (4.4)
A schematic representation of (4.3), (4.4) is depicted in Fig. 4.2. A graphical representa-
tion of the network is depicted in Fig. 4.3, where it is assumed that the species coupling
is cyclic with negative feedback, i.e.,
E =
01×(N−1) −1
IN−1 0(N−1)×1
. (4.5)
v
H
∆
−L
E
u y
Figure 4.2: Schematic representation of the network.
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 64
v1
H1
v2
y1
−L1
∆1
w1 u1H2
−L2
∆2
w2 u2 y2 . . .
vN
HN
−LN
∆N
wN uN yN−
Figure 4.3: An example of the network with cyclic species coupling.
We recall now an operator property that will be extensively used in this chapter (the
next definition is a slightly adapted version of Definition 2.8 and [12, Definition 1]).
Definition 4.1. Let Fi : L2e → L2e for i = 1, 2 be possibly nonlinear operators. The
operators F1 and F2 are said to be mutually relaxed co-coercive (MRCC) if there exist
γc12 , β12 ∈ R such that for every u1, u2 ∈ L2e and for every T ≥ 0,
γc12‖F1u1 − F2u2‖2T ≤ 〈u1 − u2, F1u1 − F2u2〉T + β12. (4.6)
The constant β12 is called an MRCC bias. The MRCC gain, denoted by γ12, is defined
as γ12 := supγc12 : (∃β12)(∀u1, u2 ∈ L2e)(∀T ≥ 0) (4.6) holds. We assume that MRCC
gain exists for a pair of MRCC operators.
In the following, we show that two operators associated with identical linear scalar
systems and different initial conditions are MRCC.
Example 4.1. Consider the operators
F1 :
x1 = −ax1 + bu1
y1 = x1, x1(0) = x10,
F2 :
x2 = −ax2 + bu2
y2 = x2, x2(0) = x20,
where xi, ui ∈ R, b > 0, and a ∈ R. Let x := [x1 x2]T and define V (x) = 1
2(x1 − x2)2.
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 65
Then
V (x) = (x1 − x2)(x1 − x2)
= (x1 − x2) [−a(x1 − x2) + b(u1 − u2)]
= −a(x1 − x2)2 + b(x1 − x2)(u1 − u2).
(4.7)
Substituting in y = x and integrating both sides of (4.7) from zero to T yields
V (y(T ))− V (y(0)) =− a∫ T
0
(y1 − y2)2dt+ b
∫ T
0
(u1 − u2)(y1 − y2)dt
=− a‖y1 − y2‖2T + b〈u1 − u2, y1 − y2〉T .
Thus,
0 ≤ V (y(T )) = −a‖y1 − y2‖2T + b〈u1 − u2, y1 − y2〉T + V (y(0)).
This implies
a‖y1 − y2‖2T ≤ b〈u1 − u2, y1 − y2〉T + V (y(0)). (4.8)
Therefore, by choosing γc12 = ab
and β12 = 1bV (y(0)) = 1
2b(x10 − x20)2, we obtain (4.6)
and F1, F2 are MRCC.
Let Q ∈ R(n−1)×n be the projector onto the orthogonal complement of the subspace
D = Span1n, as it is explained in (3.35) and (3.36). We define yk = Qyk and y =
col(y1, . . . , yN) and similarly we define uk, u, vk, v.
Remark 4.1. Note that the norm ‖QTQyk‖T measures the synchronization error for the
k-th species with time horizon T ≥ 0. Notice that
‖QTQyk‖2T = ‖QT yk‖2T = 〈QT yk, QT yk〉T = 〈yk, QQT yk〉T = 〈yk, yk〉T = ‖yk‖2T .
Therefore, ‖yk‖T is the (L2) norm of the synchronization error for the k-th species and,
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 66
similarly, ‖y‖T is the norm of the synchronization error for the network with time horizon
T ≥ 0.
Lemma 4.1. Let Fi : L2e → L2e for i = 1, . . . , n with the input-output relation zi = Firi.
Suppose that Fi and Fj are pairwise MRCC with MRCC gains γij and MRCC biases βij
for i, j = 1, . . . , n. Let r = [r1 . . . rn]T , z = [z1 . . . zn]T , r = Qr, and z = Qz where Q
satisfies (3.36). Then for every T ≥ 0,
γ‖z‖2T ≤ 〈r, z〉T + β,
where γ = min γij and β = 12n
n∑i,j=1
βij.
Proof : We choose γ = min γij. With this choice of γ, (4.6) holds for each pair of
Fi and Fj. We define qi = ri − γzi for i = 1, . . . , n. If we vectorize all the signals, we
have q = r − γz. Let r = Qr, then for every T ≥ 0,
〈z, r〉T = 〈z, q〉T + γ〈z, z〉T . (4.9)
We first claim that there exists β such that 〈q, z〉T + β ≥ 0 for every T ≥ 0. From the
MRCC assumption we have
〈qi − qj, zi − zj〉T + βij = 〈ri − rj, zi − zj〉T − γ〈zi − zj, zi − zj〉T + βij ≥ 0, (4.10)
for i, j = 1, . . . , n. By summing (4.10) over i, j = 1, . . . , n and dividing by a normalization
constant we obtain
1
2n
n∑i,j=1
(〈qi − qj, zi − zj〉T + βij)
=1
2n
n∑i,j=1
(〈qi, zi〉T + 〈qj, zj〉T − 2〈qi, zj〉T + βij)
=1
2n
(n〈q, z〉T + n〈q, z〉T − 2
⟨n∑i=1
qi,n∑j=1
zj
⟩T
+n∑
i,j=1
βij
)≥ 0.
(4.11)
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 67
Let q, z be the averages of all qi, zi for i = 1, . . . , n, respectively. Then, from (4.11)
〈q, z〉T − n〈q, z〉T +1
2n
n∑inj=1
βij ≥ 0. (4.12)
Let β = 12n
n∑i,j=1
βij, then from (4.12) we can write
〈q, z〉T − n〈q, z〉T + β ≥ 0, ∀T ≥ 0.
Now consider 〈q, z〉T + β. We have
〈q, z〉T + β = 〈Qq,Qz〉T + β = 〈q,QTQz〉T + β. (4.13)
By substituting QTQ = In − 1n1n1Tn in (4.13) we obtain
〈q,QTQz〉T + β = 〈q, z − 1nz〉T + β = 〈q, z〉T − n〈q, z〉T + β ≥ 0. (4.14)
This concludes our first claim. Now from (4.9) and (4.14)
〈r, z〉T + β = (〈q, z〉T + β) + γ〈z, z〉T ≥ γ〈z, z〉T . (4.15)
Therefore, from (4.15),
γ‖z‖2T ≤ 〈r, z〉T + β ∀T ≥ 0.
This completes the proof.
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 68
4.3 Main Result
In this section, we present our main result. We will show that, under certain conditions on
the network, the closed loop system has the property that to inputs with a “high” level of
synchrony (as implied by a small ‖v‖T ) correspond outputs with the same property (small
‖y‖T ). Because of the nonlinear perturbations, the conditions on the compartmental
graphs will involve not only the algebraic connectivity but also the largest singular values
of Lk, k = 1, . . . , N .
We are now ready to state our main result.
Theorem 4.1. Consider the closed-loop system defined by (4.3) and (4.4). Assume the
following:
1. Each pair of operators Hki and Hkj are mutually relaxed co-coercive (MRCC) with
gain γkij as in Definition 4.1 for i, j = 1, . . . , n and k = 1, . . . , N .
2. The operators ∆k, k = 1, ..., N , have unbiased finite L2 gain δ.
3. For k = 1, . . . , N , γk = γk + λk − δσk > 0 where γk = minγkij
, σk is the largest
singular value of Lk, and λk is the algebraic connectivity of the corresponding graph
Gk.
4. The dissipativity matrix defined as Eγ = E − Γγ with Γγ = diag(γ1, . . . , γN) is
diagonally stable.
Then there exists ρ1, ρ2 ≥ 0 such that, for every v ∈ LNn2e and for every T ≥ 0,
‖y‖T ≤ ρ1‖v‖T + ρ2. (4.16)
Proof : Since Hki and Hkj are MRCC, by Lemma 4.1, there exists γk, βk ∈ R such
that for every u ∈ Ln2e and for every T ≥ 0,
γk‖yk‖2T ≤ 〈yk, uk〉T + βk. (4.17)
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 69
Let wk be the summation of species coupling and external input for each species. Then,
uk = wk + (I + ∆k)(−Lkyk). (4.18)
From (4.17) and (4.18),
γk‖yk‖2T ≤ 〈yk, wk〉T − 〈yk, QLkyk〉T + 〈yk, Q∆k(−Lkyk)〉T + βk. (4.19)
Now, we find a lower bound for the second term and an upper bound for the third term
of the right-hand side of (4.19). For the second term, we use the fact that Lk1n = 0 and
from property iii) in (3.36) we can write
Lkyk = LkQTQyk = LkQ
T yk (4.20)
Thus,
〈yk, QLkyk〉T = 〈yk, QLkQT yk〉T
=1
2〈yk, Q(Lk + LTk )QT yk〉T
=1
2
∫ T
0
yTk (t)Q(Lk + LTk )QT yk(t)dt
≥ λk
∫ T
0
yTk (t)yk(t)dt = λk‖yk‖2T .
(4.21)
where λk is the smallest eigenvalue of 12Q(Lk+LTk )QT for each k. By using the properties
of Q, it can be shown that λk in the algebraic connectivity of Lk as defined in Definition
2.1 [67]. For the third term of the right-hand side of (4.19) we again use (4.20)
〈yk, Q∆k(−Lkyk)〉T = 〈yk, Q∆k(−LkQT yk)〉T
= 〈yk, Q∆k(−LkQT yk)〉T
≤ ‖yk‖T‖Q‖‖∆k(−LkQT yk)‖T
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 70
where ‖.‖ denotes the induced 2-norm of a matrix. According to the second assumption,
we obtain
〈yk, Q∆k(−Lkyk)〉T ≤ ‖yk‖T‖∆k(−LkQT yk)‖T
≤ δ‖yk‖T‖LkQT yk‖T
≤ δ‖LkQT‖‖yk‖2T
≤ δσk‖yk‖2T .
(4.22)
From (4.19), (4.21), and (4.22),
(γk + λk − δσk)‖yk‖2T ≤ 〈yk, wk〉T + βk. (4.23)
From assumption (4), Eγ is diagonally stable. So there exists a diagonal matrix D 0
and a scalar α1 > 0 such that,
DEγ + ETγ D ≺ −2α1IN .
Therefore, for every input z ∈ LN2e we get
〈Dz,Eγz〉T =1
2
∫ T
0
zT (DEγ + ETγ D)zdt
≤ −α∫ T
0
zT zdt = −α1‖z‖2T .(4.24)
From assumption 3, let γk = γk+λk−δσk > 0 and let dk > 0 be the diagonal components
of matrix D. From (4.23),
〈dkyk, wk − γkyk〉T + dkβk ≥ 0. (4.25)
Let w = v + Ey, QN = IN ⊗ Q, y = QNy, v = QNv, w = QNw, and β =N∑k=1
dkβk. By
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 71
summing (4.25) for all species we have,
〈(D ⊗ In−1)y, w − (Γγ ⊗ In−1)y〉T + β
= 〈(D ⊗ In−1)y, v + (Eγ ⊗ In−1)y〉T + β
= 〈(D ⊗ In−1)y, v〉T + 〈(D ⊗ In−1)y, (Eγ ⊗ In−1)y〉T + β ≥ 0
(4.26)
From (4.24) and (4.26), we have
〈(D ⊗ In−1)y, v〉T + β ≥ −〈(D ⊗ In−1y, (Eγ ⊗ In−1)y〉T ≥ α1‖y‖2T (4.27)
Let ‖D‖ = maxdk = α2. Then From (4.27),
α1‖y‖2T ≤〈(D ⊗ In−1)y, v〉T + β +1
2
∥∥∥∥ α2√α1
v −√α1
α2
(D ⊗ In−1)y∥∥∥∥2T
=α22
2α1
‖v‖2T +α1
2α22
‖(D ⊗ In−1)y‖2T + β
=α22
2α1
‖v‖2T +α1
2α22
‖D‖2‖y‖2T + β
=α22
2α1
‖v‖2T +α1
2‖y‖2T + β
This implies
‖y‖2T ≤ (α2
α1
)2‖v‖2T +2β
α1
. (4.28)
By taking the square root of (4.28) and using the fact that√a2 ± b2 ≤ |a|+ |b| we have
‖y‖T ≤ (α2
α1
)‖v‖T +
√2|β|α1
. (4.29)
Therefore, by choosing ρ1 = α2
α1and ρ2 =
√2|β|α1
we obtain (4.16). This completes the
proof.
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 72
Remark 4.2. If the external signal v ∈ LNn2 (which implies v ∈ LN(n−1)2 ) and all con-
ditions of Theorem 4.1 are satisfied, from (4.16) we can conclude that y ∈ LN(n−1)2 .
Therefore, if the outputs are uniformly continuous, by invoking Barbalat’s lemma, we
can conclude that the network asymptotically synchronizes, i.e., limt→∞
(yki − ykj) = 0 for
i, j = 1, . . . , n and k = 1, . . . , N .
Remark 4.3. Theorem 4.1 requires the matrix Eγ to be diagonally stable. This condition
reduces to algebraic conditions for a number interconnection structures (see [3, 50]). In
particular, when the species coupling is a negative cyclic interconnection as in (4.5), a
necessary and sufficient condition for Eγ to be diagonally stable is the secant condition
(see section 2.3 for more details)
1
γ1 · · · γn<[sec( πN
)]N. (4.30)
4.4 Example: Network of Goodwin Oscillators
Consider a network of n identical Goodwin oscillators that are coupled through a com-
partmental coupling described by Laplacian matrices Lk, k = 1, 2, 3 in the presence of
uncertainties ∆k. Given the initial conditions, the Goodwin oscillator is described by the
following ODE model (see the Appendix and [62] for more details):
x1j = −b1x1j + c1
(1
1 + xp3j+ v1j
),
x2j = −b2x2j + c2(x1j + v2j), (4.31)
x3j = −b3x3j + c3(x2j + v3j), j = 1, . . . , n,
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 73
where vkj ∈ L2e for k = 1, 2, 3 are external signals, the scalars bk, ck > 0, and p > 1. We
can decompose (4.31) into four different species as follows:
Hkj :
xkj = −bkxkj + ckukj
ykj = xkj, k = 1, 2, 3,
H4j : y4j = − 1
1 + up4jj = 1, . . . , n.
(4.32)
From Example 4.1., the corresponding operators Hki and Hkj are MRCC with the esti-
mated gain γkij = bkck
and bias βkij = 12ck
(xki(0) − xkj(0))2. Also, the static nonlinearity
H4j is a monotonically increasing function on ukj with bounded slope and as it is shown
in [50], H4i and H4j are MRCC for i, j = 1, . . . , n with the gain
γ4ij =
(p−1
√(p−1p+1
)p+ 1
)2
(p+ 1)
p(p− 1). (4.33)
Suppose that the uncertainty operators ∆k for k = 1, 2, 3 have unbiased finite L2 gain δ.
According to (4.31) the species are coupled through the cyclic interconnection matrix
E =
0 0 0 −1
1 0 0 0
0 1 0 0
0 0 1 0
.
Since the species coupling is cyclic we can use the secant condition (4.30) to verify the
diagonal stability of the dissipativity matrix. If the secant condition holds, namely,
(γ1 + λ1 − δσ1)(γ2 + λ2 − δσ2)(γ3 + λ3 − δσ3) > c, c =1
γ4[sec(π
4)]4 ,
then all the conditions of Theorem 4.1 are satisfied. Therefore, Theorem 4.1 guarantees
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 74
the robust synchronization and (4.16) holds for some ρ1 > 0, ρ2 ∈ R.
We will present now some numerical results. Consider a network of 6 interconnected
Goodwin oscillators. The oscillators are coupled through the compartmental intercon-
nection matrices Lk, k = 1, 2, 3. We assume that only the first and the third species in
each compartment are allowed to diffuse, i.e., L2 = 0. Suppose that all the edges of graph
Gk with corresponding Laplacian matrix Lk have the same weight qk. Let G1 and G3 be
complete graphs with q1 = 1 and q3 = 2, i.e., the strength of the diffusive coupling of the
third species is twice that of the coupling strength of the first species. Let zk = −Lkyk.
Let (I + ∆kjj), j = 1, . . . , 6, be quasi-saturation operators defined by
(I + ∆kjj)(t, zkj) =
zkj, |zkj| ≤ r
zkj|zkj|
r + (1−m+m sin 10t)
(zkj(|zkj| − r)|zkj|
), |zkj| > r,
(4.34)
where r is a threshold value and m takes into account the maximum perturbation the
input zkj is subjected to outside the linear region [32]. As an example, Fig. 4.4 depicts the
action of the operator (I+∆kjj) on an input zkj = 0.5 sin(t). Let the off-diagonal elements
of ∆k be zero. This means that the uncertainty of each compartment is independent of
the other compartments. According to (4.34), one can find the uncertainty operator
∆kjj, k = 1, . . . , 4, j = 1, . . . , 6, as follows:
∆kjj(t, zkj) =
0, |zkj| ≤ r
(−m+m sin 10t)
(zkj(|zkj| − r)|zkj|
), |zkj| > r.
(4.35)
An example of quasi-saturation operator is depicted in Fig. 4.5 .The L2 gain of ∆kjj is
‖∆kjj‖ = sup
zkj ,T 6=0
‖∆kjjzkj‖T‖zkj‖T
≤ supzkj ,T 6=0
∥∥∥∥(−m+m sin(10t))
(zkj(|zkj| − r)|zkj|
)∥∥∥∥T
‖zkj‖T
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 75
≤ supzkj ,T 6=0
∥∥∥∥(2m)
(zkj(|zkj| − r)|zkj|
)∥∥∥∥T
‖zkj‖T= 2m.
Let r = 0.1, and m = 0.25. Since ∆k is diagonal, we have δ = ‖∆k‖ = ‖∆kjj‖ ≤ 2m = 0.5.
0 1 2 3 4 5 6−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(I+
∆k jj)(t,zkj)
t ime (sec )
Figure 4.4: (I+ ∆kjj) output for the input z = 0.5 sin(t) and constants r = 0.1,m = 0.25.
Let p = 19, b1 = 0.5, b2 = b3 = c1 = c2 = c3 = 1. According to Example 4.1., the MRCC
gain for the first three species are γ1 = 0.5, γ2 = γ3 = 1 and from (4.33), γ4 = 0.21. Thus,
the secant condition (4.30) is
(0.5 + λ1 − δσ1)(1 + λ3 − δσ3) > c, c =1
γ4[sec(π
4)]4 ∼= 1.19. (4.36)
In this example, the left-hand side of (4.36) is 24.5, hence there exists a diagonal
matrix D such that ETγ D + DEγ < 0. A possible choice for matrix D can be com-
puted by the method in [3] as D = diag(0.2857, 0.4409, 1.3605, 0.8817) which yields
ρ1 = 9.72. This is not the minimum possible value for ρ1, for example for the choice
of D = diag(0.2857, 0.4409, 0.95, 0.95) the value of ρ1 becomes 6.79. In simulations, we
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 76
0 1 2 3 4 5 6−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
∆k jj(t,zkj)
t ime(sec )
Figure 4.5: (I+ ∆kjj) output for the input z = 0.5 sin(t) and constants r = 0.1,m = 0.25.
consider both “vanishing” and “non-vanishing” external signals. In the former case, we
consider different constants signals in the interval [−0.6, 0.6], while in the latter we con-
sider different exponential signals in L2. The simulation results are shown in Fig. 4.6
and Fig. 4.7. In Fig. 4.8 the norm of the synchronization error is compared to the norm
of the external inputs for the simulation in Fig. 4.6.
It is of interest to apply Theorem 4.1 find conditions for robust synchronization with
some typical undirected graph topologies. We consider a network of n compartments
where only the first species are allowed to diffuse with coupling strength q1 = q. Table
4.1 shows the conditions under which robust synchronization of the network is guaranteed
for different graphs.
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 77
0.5
1
1.5
2
2.5
0
0.5
1
1.5
2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x1x2
x3
Figure 4.6: Robust synchronization of six interconnected Goodwin oscillators with “non-vanishing” external signals
0
0.5
1
1.5
2
2.5
0
0.5
1
1.5
2
2.5
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x1x2
x3
Figure 4.7: Robust synchronization of six interconnected Goodwin oscillators with “van-ishing” external signals
Chapter 4. Nonlinear Netwroks: An Input-Output Approach 78
0 5 10 15 20 25 30 35 400
5
10
15
20
25
30
35
40
45
t ime (se c )
L2enorm
‖ y‖T
ρ 1‖ v‖T + ρ 2
Figure 4.8: Comparison of ‖y‖T and ρ1‖v‖T +ρ2 for the “non-vanishing” external signals
Table 4.1: Robust synchronization conditions for a network in which only the first speciesare allowed to diffuse
Graph λ1 σ1 Robust synchronization Condi-tion
Complete nq nq q >c− 0.5
n(1− δ)Star q nq q >
c− 0.5
1− δn2q(1− cos(
2π
n)) 2q(1 + cos(
π
n)) (n : odd) q >
c− 0.5
2(1− δ − cos(2π
n)− cos(
π
n))
Cyclic 4q (n: even) q >c− 0.5
2(1− cos(2π
n))− 4δ
Chapter 5
Conclusions and Future Work
5.1 Conclusions
This thesis studies robust synchronization in networks of compartmental systems. In the
following, we address the main contributions of the thesis.
• In Chapter 3, we studied the network of harmonic oscillators where each oscillator
has two control inputs and both components of the states are allowed to diffusive
with possibly distinct directed communication topologies. This network model is an
extension of the model in [42] where the harmonic oscillators have a single control
input and only the second components of the states are diffusively coupled. We
showed that a necessary and sufficient condition for the harmonic oscillators to
asymptotically synchronize in a directed balanced network, is connectivity of the
union of the graphs associated with the communication topologies. Furthermore,
for the case where the graphs are not necessarily balanced, we showed that if their
union is strongly connected and the intersection of the left kernels of their Laplacian
matrices is not an empty subspace, then all the harmonic oscillators asymptotically
synchronize. Finally, we investigated a necessary and sufficient condition for the
network of harmonic oscillators to achieve synchronization, based on the eigenvalues
79
Chapter 5. Conclusions and Future Work 80
associated with the network model.
• In Sections 3.4 and 3.5, we focused on robustness analysis in the network of har-
monic oscillators. In particular, in Section 3.4, we studied the case where the
diffusive coupling is nonlinearly perturbed. We showed that if the harmonic oscil-
lators asymptotically synchronize in the nominal network and the perturbation is
sufficiently small, then synchronization occurs in the perturbed network. In Section
3.5, we studied the effect of disturbances. In this context, perfect synchronization
is not achieved. Thus, we derived a bound on the norm of the synchronization error
satisfying the degree of synchronization.
• In Chapter 4, we presented an input-output framework for the analysis of output
synchronization in a network of nonlinear compartmental models. Our result is a
generalization of [50] where synchronization is studied in a nominal network. We
derived a sufficient condition for robust synchronization in the presence of finite L2
gain perturbations on the diffusive coupling. The resulting conditions show that
the largest singular value of Laplacian matrices as well as the algebraic connectivity
of the graphs play important roles when nonlinear perturbations on the diffusive
coupling are taken into account.
5.2 Future Work
The results of this thesis could be extended is the following areas:
• Since in a wide variety of applications, the communication topologies are not fixed,
solving the synchronization problem for time-varying graphs could be a useful ex-
tension.
• We prove only sufficiency in Theorems 3.2, 3.4, 3.5, and 4.1. The necessity for these
theorems can be addressed as future work.
Chapter 5. Conclusions and Future Work 81
• Other classes of uncertainties such as additive uncertainty in the diffusive couplings
or uncertainty in the dynamics of the species could be considered.
• In our framework in Chapter 4, the compartmental coupling is assumed to be only
through identical species in different compartments. A possible extension is to take
into account the general class of compartmental coupling where distinct species in
different compartments can be diffusively coupled.
Appendix A
Goodwin Oscillator
The Goodwin oscillator was proposed by Brian Goodwin in 1965 [17] and modified by
Griffith in 1968 [18]. The Goodwin oscillator models physiological oscillation where a
protein inhibits the transcription of its own gene. About 30 years later, this model
turned out to be an important part of circadian rhythms in various organisms such as
mammalian circadian rhythms [48, 16].
Now we describe the model in more detail [62]. Let [X1], [X2], and [X3] be the
concentrations of mRNA, protein, and end product. The Goodwin oscillator is described
by the following kinetic equations
d[X1]
dt=
v0
1 + ([X3]
Km
)p− k1[X1],
d[X2]
dt= v1[X1]− k2[X2],
d[X3]
dt= v2[X2]− k3[X3].
(A.1)
where
v0 := rate of transcription,
v1 := rate of translation,
82
Appendix A. Goodwin Oscillator 83
v2 := rate of catalysis,
k1, k2, k3 := rate constants for degradation of each component,
1
Km
:= binding constant of end product to transcription factor,
p := Hill coefficient (that measures the cooperativity of end product repression).
Now, we change the variables as follows. We define the dimensionless variables
x1 :=v1v2[X1]
k2k3Km
, x2 :=v2[X2]
k3Km
, x3 :=[X3]
Km
, τ :=v0v1v2k2k3Km
t. (A.2)
From (A.1) and (A.2), we can rewrite the model in terms of the new variables as
dx1dτ
=1
1 + xp3− b1x1,
dx2dτ
= b2(x1 − x2),
dx3dτ
= b3(x2 − x3),
(A.3)
where bi :=kik2k3Km
v0v1v2for i = 1, 2, 3.
It is proved in [19] that for every initial condition the solution of (A.3) is bounded.
The oscillatory behaviour of Goodwin’s model highly depends on the parameters of the
model. For example, for b1 = b2 = b3, it can be shown that when p < 8, the origin is
asymptotically stable, and when p = 8 the system undergoes a Hopf bifurcation. The
bifurcation analysis in Goodwin oscillator shows that if the constants b1, b2 and b3 are not
equal, then the minimum value of p such that the system exhibits limit cycle becomes
larger [62]. For instance, if the coefficients be b1 = 0.5 and b2 = b3 = 1, as in Example
4.2, it can be shown that for p < 18 the origin is asymptotically stable, and for p = 18
the system undergoes a Hopf Bifurcation.
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