Robust Synchronization in Networks of Compartmental Systems · This thesis studies robust synchronization in networks of compartmental systems moti-vated by cellular networks. These

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


3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.11 Graph representation of Gx = Gy and Gx ∪ Gy for Example 3.3 . . . . . . 52

viii


3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55


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.


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.


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


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


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


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.


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-


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.


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


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


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)


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


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.


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→∞.


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]:


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.


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


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.


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-


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


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)


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.


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)


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.


(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


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)


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.


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


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


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

])


=

(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


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.


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-


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


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


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)


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


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


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


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


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)


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)


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.


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

.


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


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


Example 3.3. Consider again a network of four diffusively coupled harmonic oscilla-


0 10 20 30 40 50 60 70 80−6

−4

−2

0

2

4

6

time (se c )

y

y1y2y3y4


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


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


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-


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


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


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


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



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


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


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


=[(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)


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


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.


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.


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,


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)


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.


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)


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


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


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.


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,


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


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| > 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))


)∥∥∥∥T

‖zkj‖T


≤ supzkj ,T 6=0

∥∥∥∥(2m)


)∥∥∥∥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


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.


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


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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