OBSERVERS AS A TOOL TO REDUCE INFORMATION EXCHANGE
AND INCREASE CONVERGENCE RATE IN MULTI-AGENT SYSTEMS

By

Dhrubajit Chowdhury

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Electrical Engineering – Doctor of Philosophy

2020

ABSTRACT

OBSERVERS AS A TOOL TO REDUCE INFORMATION EXCHANGE AND

INCREASE CONVERGENCE RATE IN MULTI-AGENT SYSTEMS

By

Dhrubajit Chowdhury

Observers form an integral part of output feedback control of linear and nonlinear sys-

tems. This dissertation investigates the use of observers in multi-agent systems to reduce

information exchange and increase the convergence rate. Multi-agent systems have been

immensely popular since the last two decades due to their broad applicability in practical

problems, some of them being distributed sensor networks, formation control, and cooper-

ative robotics. The controller for each agent is distributed in nature, which means that it

only depends on the local information available to it. The distributed approach has several

advantages such as less computational effort, reliability, etc., compared to the centralized one

where there is a central agent that does all the computations and then makes the decision.

The convergence rate of consensus algorithms is an important performance measure.

We show that by using observers, we can increase the convergence rate of the consensus

algorithm. The observer is used for estimating the missing links at each agent. We also

study the effect of increasing network size on the consensus algorithm. For networks without

a leader, the rate of convergence of the consensus protocol becomes slow for certain classes

of graphs, while for networks with a single leader, the convergence rate becomes slow for

undirected graphs. We design scalable consensus algorithms for first-order linear agents

and second-order nonlinear heterogeneous agents where the convergence rate remains almost

invariant of the network size.

We consider the case of reduced information exchange in a network of nonlinear hetero-

geneous agents having the same relative degree r. We use observers along with feedback

control to compensate for the heterogeneity at each agent. Finally, motivated by the practi-

cal application of multi-agent systems to power systems frequency synchronization, we fuse

dynamic consensus algorithms with observers to achieve practical frequency synchronization

under time-varying power-demand. We show that the frequency synchronization error can

be made arbitrarily small by tuning controller and observer parameters.

Copyright by
DHRUBAJIT CHOWDHURY
2020

This dissertation is dedicated to my beloved parents.

v

ACKNOWLEDGMENTS

First and foremost, I would like to thank one of the essential contributors to this thesis,

my advisor Dr. Hassan Khalil. I will always be indebted to him for first accepting me as

his student and then for his constant guidance and support throughout my Ph.D. journey.

During my time as a Ph.D. student, I was deeply moved by his ability to teach students as

well as maintain high-quality research successfully. His constant enthusiasm in answering

questions during a research meeting or when teaching in a class has been a real inspiration to

me. As I start my academic career, I hope to become not only a well-established researcher

but also an inspiring teacher. I also thank him for giving me the flexibility to pursue my

research ideas and also work on ‘side-projects’ over the years. I wish him the best for his

retirement life.

My committee members have provided excellent support over the years. I want to thank

Dr. Ranjan Mukherjee for allowing me to collaborate with him and work on the control of

mechanical systems. He has helped me understand the application of control to mechanical

systems. Dr. Xiaobo Tan has always been welcoming when discussing ideas and providing

insights into an academic career. The knowledge I have gained while working with him during

his class projects is immeasurable. I always enjoyed talking with Dr. Vaibhav Srivastava

about state of the art in network systems. He was always available for a quick chat and gave

me pointers about new research directions. He serves as a role model for what I want to

achieve in the next phase of my career.

I would especially like to thank Dr. Tim Hogan for being flexible with my teaching

assignments and all the instructors in my teaching assistantship.

I appreciate all ECE

department administrators and staff members for their help. I also want to acknowledge the

vi

initial funding I received from the National Science Foundation (ECCS - 1128476).

This journey could not have been completed without the support of friends and fam-

ily. I want to thank all my friends and colleagues at Michigan State University, specially

Pratap Bhanu Solanki, Yashesh Dhebar, Gaurav Chauda, Rahul Dey, Dr. Saran Karru-

puswami, Suhas Vidhate, Saikat Mondal, Jason Greenberg, Billy Stevers, and Rakshith

Lokesh. Thanks to my undergraduate friends Bikiran Guha and Archan Ray, who are also

pursuing their Ph.D. in U.S.A. for all the memorable trips. I am also thankful to my friends

Suraj, Sunny, Sudip, Shashank, Soumitra, Mukhesh, Tanaya, Monalisa, Sitangshu, Niladri

in India for their encouragement and support.

There is a trade-off in everything we do in life. Unfortunately, during this journey, I

lost my father, (late) Ajit Kumar Chowdhury, in 2017. I am sure he would be one of the

happiest people in the world watching me complete this journey. I will always be indebted

to my mother, Swapna Chowdhury, for her unconditional love and support for me. Her life’s

journey is a source of inspiration for me. I also thank my brother Debashish Chowdhury for

his help. Finally, I would like to thank my girlfriend, Paola, for all the love and support.

vii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Review on Algebraic Graph Theory . . . . . . . . . . . . . . . . . . .
1.1.2 Consensus and Synchronization Algorithms Review . . . . . . . . . .
1.1.3 Application of Consensus Algorithms to Power Systems Review . . .
1.2 Contribution and Organization . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Overview of Contribution . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
1
3
9
11
11
14

Chapter 2 Fast Consensus with Star Topology Using High Gain Observers 16
16
18
18
19
19
20
21
21
22
23
24
32
34

2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Consensus on Different Network Topologies . . . . . . . . . . . . . . . . . . .
2.2.1 Consensus with Complete Graph . . . . . . . . . . . . . . . . . . . .
2.2.2 Consensus with Star Graph . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Target System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Exploiting the Star Topology Structure . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Extended High Gain Observers
. . . . . . . . . . . . . . . . . . . . .
2.4.2 Peaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Observer Error Dynamics
. . . . . . . . . . . . . . . . . . . . . . . .
2.5 Simulations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Design Preliminaries

2.4 Controller Design and Analysis

Chapter 3

Scalable Consensus Using High Gain Observers . . . . . . . . .
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
State feedback Controller Design . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Output-Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Preliminaries
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Peaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Cyclic Pursuit on Plane
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Tree Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35
35
36
37
43
43
48
48
58
62
65

viii

Chapter 4 Practical Scalable Synchronization in Leader-Follower Networks 66
66
68
68
68
71
71
74
78
78
79
81
88
88
92
95

4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Class of Systems
4.2.2 Leader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Controller Design With Full Information . . . . . . . . . . . . . . . . . . . .
4.3.1 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Closed-loop System Analysis . . . . . . . . . . . . . . . . . . . . . . .
4.4 Output Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Peaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Analysis of the closed loop system . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IEEE 300-Bus System . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.1
4.5.2 Platooning of Vehicles

4.5 Examples

Chapter 5 Practical Synchronization in Networks of Nonlinear Heteroge-
nous Agents under Reduced Information . . . . . . . . . . . . .
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97
97
98
5.2.1 Leader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 Relative Dynamics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4 Synchronization under full information . . . . . . . . . . . . . . . . . . . . . 106
5.5 Synchronization under output feedback . . . . . . . . . . . . . . . . . . . . . 109
5.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.7 Case with Some Unbounded Leader States . . . . . . . . . . . . . . . . . . . 128
State feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . 131
. . . . . . . . . . . . . . . . . . . . . . . 131
5.8 Frequency Control of Power Systems
. . . . . . . . . . . . . . . . . . . . . . 136
5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.7.1
5.7.2 Output feedback Controller

6.1

Chapter 6 Power Systems Frequency Synchronization . . . . . . . . . . . . 141
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.1.1
System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.2 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.3 Controller Design Under known Time-Varying Power Demand . . . . . . . . 147
6.3.1 Communication Topology . . . . . . . . . . . . . . . . . . . . . . . . 147
6.3.2 Dynamic Average Consensus . . . . . . . . . . . . . . . . . . . . . . . 147
6.3.3 Analysis of the Closed-loop System . . . . . . . . . . . . . . . . . . . 149
6.4 Controller Design Under Unknown Time Varying Power Demand . . . . . . . 155
6.4.1 Extended High-Gain Observer . . . . . . . . . . . . . . . . . . . . . . 155
6.4.2 Analysis of the Closed-loop System . . . . . . . . . . . . . . . . . . . 157
6.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.5.1 Case I : Unknown Constant Power Demand . . . . . . . . . . . . . . 162

ix

6.5.2 Case II : Unknown Time-Varying Power Demand with variable Frequency164
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Chapter 7 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . 166
7.1 Overview of Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.2.1
Stability of Networks with Signed Laplacians . . . . . . . . . . . . . . 169
7.2.2
Scalable Formation Control of Mobile Robots
. . . . . . . . . . . . . 170
7.2.3 Frequency Synchronization in Higher Order Power System Models . . 171

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

x

LIST OF TABLES

Table 3.1: Laplacian Eigenvalues of Tree-like fractal graphs

. . . . . . . . . . . . . .

64

Table 3.2: Laplacian Eigenvalues of Vicsek fractal graphs

. . . . . . . . . . . . . . .

64

xi

LIST OF FIGURES

Figure 2.1: Star Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Figure 2.2: Consensus on Complete Graph . . . . . . . . . . . . . . . . . . . . . . .

33

Figure 2.3: Consensus on Star Graph with no Observers . . . . . . . . . . . . . . . .

33

Figure 2.4: Consensus on Star Graph with Observers and 
 = 0.001 . . . . . . . . . .

33

Figure 2.5: Deviation of agent 6 trajectory from target trajectory . . . . . . . . . . .

34

Figure 3.1: A directed circulant graph . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Figure 3.2: Agent 4 state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Figure 3.3: Agent 4 control signal

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Figure 3.4: Agent 4 control signal with µ = 0.1, 
 = 0.0001 . . . . . . . . . . . . . .

61

Figure 3.5: Tree-like Fractal graphs

. . . . . . . . . . . . . . . . . . . . . . . . . . .

62

Figure 3.6: Vicsek Fractal graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Figure 4.1: Diagram of node connections for the IEEE 300-Bus System . . . . . . .
Figure 4.2: State one synchronization error ex = x − x0 . . . . . . . . . . . . . . . .
Figure 4.3: State two synchronization error ev = v − v0

. . . . . . . . . . . . . . . .

89

90

90

Figure 4.4: Control signal of the oscillators . . . . . . . . . . . . . . . . . . . . . . .

91

Figure 4.5: Log-Linear plot of the contol signal for different value of k . . . . . . . .

91

Figure 4.6: Performance recovery with k = 300 . . . . . . . . . . . . . . . . . . . . .

92

Figure 4.7: Line graph representing platoon of vehicles . . . . . . . . . . . . . . . . .
Figure 4.8: Position error of vehicle ten, ˜x10 = x10 − x0d
Figure 4.9: Velocity error of vehicle ten, ˜ev10 = v10 − v0

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

10

93

94

94

Figure 4.10: Control signal of vehicle ten . . . . . . . . . . . . . . . . . . . . . . . . .

94

xii

Figure 5.1: Network interconnection between the agents . . . . . . . . . . . . . . . . 123

Figure 5.2: States of the agents under the proposed controller with 
 = 0.001 . . . . 126

Figure 5.3: Control of the agents with 
 = 0.001 . . . . . . . . . . . . . . . . . . . . 127

Figure 5.4: States of the agents under the proposed controller with 
 = 0.0001 . . . . 127

Figure 5.5: Control of the agents with 
 = 0.0001 . . . . . . . . . . . . . . . . . . . . 128

Figure 5.6: Phase of the buses

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 5.7: Frequency of the buses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 5.8: Control

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 6.1: Four area network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Figure 6.2: Frequency deviation (ω) under unknown constant power demand . . . . 163

Figure 6.3: Control (u) under unknown constant power demand . . . . . . . . . . . . 163

Figure 6.4: Frequency deviation (ω) under unknown time-varying power demand . . 164

Figure 6.5: Control (u) under unknown time-varying power demand . . . . . . . . . 165

xiii

Chapter 1

Introduction

Multi-agent systems are popular due to their wide applicability in practical problems, some

of them being distributed sensor networks [1], [2] and cooperative robotics [3], [4]. The

consensus problem in multi-agent systems has been well studied in the last decade. Consensus

problems were studied in the field of computer science in distributed computing in which

the computer processes are required to reach an agreement on certain data values [5]. The

controllers designed for multi-agent systems are distributed in nature and are based on the

local information available to the agents, which in turn depends on the communication

topology. The communication constraints and the convergence rate of the protocol are given

more emphasis while solving the consensus problem.

1.1 Literature Review

1.1.1 Review on Algebraic Graph Theory

The communication topology between the agents is defined by a time-invariant directed

communication graph which describes the information flow among the agents. A weighted
directed graph is denoted by G = (V,E,A), where V = {v1, v2, . . . , vN} is the set of nodes,
one for each of the N agents present in the network, E ⊆ V × V is the set of edges which
represent the interconnection between the agents, and A is a weighted adjacency matrix with

1

non-negative elements [A]kj = akj ≥ 0. If the edge weights akj are only zero or one, then
the graph is said to be unweighted, otherwise it is said to be weighted. An edge (vk, vj) ∈ E

implies that there is a directed edge from node vk to node vj and node vj receives information
from node vk. Moreover, vk is called an in−neighbor of vj, and vj is called an out−neighbor
of vk. If (vk, vj) ∈ E =⇒ (vj, vk) ∈ E, then the graph is said to be undirected, otherwise

it is said to be directed. A path is defined by a sequence of vertices such that for each of

its vertices vk the next vertex in the sequence is a neighbor of vk. Two nodes vk and vj are

connected if and only if there exists a directed path from node vk to node vj. We assume

that there are no self-loops which implies akk = 0 for k = 1, 2, . . . , N. A directed graph

contains a directed spanning tree if there exists a node called root such that there exists a

directed path from the root node to every other node in the graph. In other words, this

N(cid:80)

j=1,j(cid:54)=k

implies that the graph is connected.

The Laplacian matrix L ∈ RN×N is defined as [L]kj = lkj where lkk =

akj,
lkj = −akj for j (cid:54)= k. The Laplacian of an undirected graph can be defined as L = BΓLBT ,
where ΓL is a diagonal matrix with the diagonal elements being the weights of edge (i, j) ∈ E
for all (i, j) ∈ V, B ∈ Rn×m is the incidence matrix and m is the cardinality of the edge
set E, where cardinality is defined as the number of elements in a set. The incidence matrix
satisfies BT 1 = 0, where 1 ∈ Rn represents the vector of all 1’s. It can be also be defined as
L = D − A, where D is called the degree matrix of G.

The structure of the Laplacian matrix is such that the diagonal entries are non-negative,

and the off-diagonal entries are non-positive and, for each row, the sum of all the entries on
this row is zero. A matrix that satisfies these properties is called Metzler [6] or M matrix.

As a result it has an eigenvalue λ1(L) = 0 and the corresponding eigenvector associated with
it is a vector of all-ones 1 = col(1, 1, . . . , 1). We denote the other N − 1 eigenvalues of L

2

as λ2(L), . . . , λN (L). A time-invariant directed graph is connected if and only if L has one

simple zero eigenvalue λ1(L) = 0 and all other eigenvalues λ2(L), . . . , λN (L) have positive
real parts [7], [8]. If for each node, the number of in − neighbor is equal to the number of
out − neighbor, we say the graph is balanced, and it follows that 1T L = 0.

For leader-follower networks with a single leader in the network we define the expanded
graph as ¯G = ( ¯V, ¯E, ¯A), where ¯V = {v0, v1, v2, . . . , vN} and ¯E, ¯A contains the edges and edge
weights from the leader to the other agents in addition to the edges and edge weights from E
and A, respectively. In this case, the communication topology is described by the grounded

Laplacian matrix LG. The grounded Laplacian matrix is obtained by removing certain rows

and columns associated with the leader from the Laplacian matrix. It is also studied in the

context of agent stubbornness [9], where an agent can influence the network by exchanging

its information with the other agents. Still, in return, it is not influenced since it does not

use any information from the other agents in the network. It can also be defined with respect
to the Laplacian of the graph G by LG = L + D, where D = diag(d1, . . . , dN ), and dk > 0

if agent vk receives information from the leader v0 otherwise dk = 0. The leader does not
receive any information from the agents in the network. If the expanded graph ¯G contains a

directed spanning tree with the leader as the root node, then the grounded Laplacian matrix
LG is a nonsingular M-matrix [10].

1.1.2 Consensus and Synchronization Algorithms Review

Consensus and synchronization both describe the effect of reaching agreement in a group of

agents in some sense. Consensus-based approaches are used for solving cooperative control

problems in multi-agent systems which include formation control [11], [12], connectivity

maintenance [13] and flocking [14]. The survey papers [15], [16], and the books [17], [18]

3

provide an exhaustive material on the consensus problems in multi-agent systems.

Consensus in first-order agents was achieved in [19]. The convergence rate is an important

factor in the design of the consensus controller. The communication topology through which

the agents communicate is important as it decides the convergence rate of the consensus

algorithm. The communication topology is generally encoded in a matrix called the graph

Laplacian [20], denoted by L. The convergence rate of the consensus algorithm for an

undirected graph depends on the second smallest eigenvalue λ2(L) of the graph Laplacian,

also known as the algebraic connectivity [7] of the graph. For an undirected graph, the

second smallest eigenvalue of the graph Laplacian is non-decreasing if edges are added to the

graph [18]. The effect on the convergence rate by adding edges on a directed acyclic graph

has been studied in [21].

To increase the convergence rate, a simple solution would be to multiply the consensus

equation by a gain. However, if the second smallest eigenvalue of the graph Laplacian

of the graph tends to zero, then the gain needs to be very large. Another approach to

solving this problem is by redesigning the network topology [22], [23], to increase the second

smallest eigenvalue of the graph Laplacian. Redesign of the network topology may lead to

a high communication cost as nodes that are at a large geographical distance may need to

communicate with each other. Next, we mention other approaches where fast convergence

is achieved without using high-gain or redesigning the network topology. Fast consensus is

achieved in [24] by recording the past state of a selected agent and its neighbors. In [25], fast

convergence is achieved for a strongly connected network of agents with a single leader in the

discrete-time setting. The control of the leader is designed by making use of past values of

the leader’s state, which improves the convergence rate. Some of the other methods estimate

the second smallest eigenvalue of the graph Laplacian, which can then be used as a control

4

parameter to achieve fast convergence. Decentralized estimation of the second smallest

eigenvalue of the graph Laplacian was achieved for undirected graphs in [26], and strongly

connected graphs in [27]. However, these methods require each agent to communicate a

vector whose size increases linearly with the number of agents in the network. In [28], the

second smallest eigenvalue of the graph Laplacian is estimated, and adaptive control is used

to tune the edge weights for undirected graphs. However, the effect of change in edge weights

on the control signal is not shown, and the controller requires more information exchange.

The decrease of the second smallest eigenvalue of the graph Laplacian also affects finite-time

consensus algorithms [29], [30], as the upper bound on the finite time is inversely proportional

to the second smallest eigenvalue of the graph Laplacian.

The scalability of consensus laws is also an important factor as with an increase in the

network size, the second smallest eigenvalue of the graph Laplacian decreases for many

classes of graphs, including planar, lattice, and tree graphs; see [31]. These classes of graphs

are contained in the non-expander family [32], where the second smallest eigenvalue of the

graph Laplacian decreases towards zero with an increase in the network size. For example,

in a circulant undirected graph [17, Chapter 2], where all the agents are arranged in a circle

and each agent can communicate with its two adjacent neighbors, the eigenvalues of the

Laplacian for this undirected network has a closed-form given by

0, 2 − 2 cos

2π
N

, . . . , 2 − 2 cos

2(N − 1)π

N

Therefore as the number of agents (N ) increases we have

N→∞ 2 − 2 cos

lim

2π
N

= 0

5

which implies that the second smallest eigenvalue of the graph Laplacian is decreasing to-

wards zero, and the convergence rate of the consensus algorithm is becoming slow. In [18,

Chapter 10], a similar conclusion has been drawn for a directed circulant graph. On the

contrary, expander family contains classes of graphs where the second smallest eigenvalue of

the graph Laplacian is bounded away from zero, with an increase in network size [33], which

includes complete graphs, random regular graphs, small-world networks [34], etc. The recent

thesis [31] describes this effect. It covers some non-exhaustive classes of graphs where the

second smallest eigenvalue of the graph Laplacian decreases with an increase in network size.

This property has also been investigated in the platooning of vehicles, where the vehicles

are arranged in a line formation, and with an increase in the number of vehicles, the second

smallest eigenvalue of the graph Laplacian decreases and, as a result, the closed-loop system

becomes unstable [35].

Performance measures of the consensus algorithms are also studied for leader-follower

networks. The communication topology is defined by the grounded Laplacian matrix, which

is obtained by removing specific rows and columns associated with the leader from the

Laplacian matrix. These networks are used for applications such as platooning, where the

lead vehicle guides the platoon. The performance measure for these networks depends on

the smallest eigenvalue of the grounded Laplacian matrix [36]. For leader-follower networks,

the performance measure can degrade with an increase in network size, even if the graph

among the followers belongs to the expander family. The decrease in the smallest eigenvalue

of the grounded Laplacian matrix or the second smallest eigenvalue of the graph Laplacian
also affects other performance measures like the H2-norm, which is commonly used [37] for

studying the robustness of consensus algorithms in the context of linear systems [38], [39].
The H2-norm is calculated from the difference between the systems output and the average

6

output of the network to the disturbance input. This framework has been applied to analyze

the performance of the distributed controllers used for the platooning of vehicles [35]. It was
shown in [40], that for first-order systems, the H2-norm is O(N ), where N is the network

size, for the case when the underlying graph among the followers is a random regular graph,

which belongs to the expander family. This occurs due to the smallest eigenvalue of the

grounded Laplacian matrix approaching zero with an increase in network size. Similarly,

observations were made in [32], which also comments on the loss of stability for agents of

order greater or equal to three.

In synchronization problems, the emphasis is on the individual dynamics rather than on

the communication constraints; for example, in [41], an all to all communication topology

was considered for the synchronization of a group of Kuramoto oscillators. The synchroniza-

tion problem for general linear systems was also solved in [42] by a low gain approach. Later

on, a distributed observer-based protocol was designed in [43], which required the exchange

of the outputs of the internal states of the controller. However, the topology of the network

was restrictive as the edge weights could be only 0 or 1. The results of [43] were extended

in [44] where the extension was done for general time-invariant directed topologies. The

synchronization problem for a network of integrators connected through a time-varying com-

munication graph was first addressed in [19] and [45]. The extension to time-varying graphs

is generally carried out in the framework of switching. The topology switches among a finite

set of graphs and there is a dwell-time in between the switches; see [46], [47], [48], [49], [50].

In all of the works mentioned, the state synchronization was done for homogeneous agents,

which means that the agent models were identical. It was shown in [51] that an internal

model principle is necessary and sufficient for output synchronization of heterogeneous linear

systems. Some of the works in output synchronization of heterogeneous multi-agent systems

7

include [52], [53], [54].

The work of [51] was extended to uncertain heterogeneous nonlinear agents in [55] where

each agent has a copy of an exosystem and the exosystems were first synchronized and then

local controllers were designed at each agent to track the output of the agents respective

exosystem. But the design procedure requires an explicit solution of the output regulator

equation, which requires solving partial differential equations. The work in [55] was extended

in [56] to nonlinear agents in a leader-follower configuration with time-varying topology,

with the assumption that each agent knows its output. The work in [57] uses a distributed

internal model to convert the problem to a robust stabilization of the augmented system.

Then it solves the stabilization problem via distributed dynamic output control law by

utilizing and combining a backstepping, high gain feedback control, and distributed high

gain observer. An extension to time-varying topologies and non-identical relative degree was

recently proposed in [58]. However, similar to [55], the design procedure in [57] and [58]

requires an explicit solution of the output regulator equation and assumes that the solution

of the output regulator equation along the trajectory of the exosystem is polynomial. In [59]

optimal output synchronization in a network of heterogeneous agents was achieved. However,

the proposed approach requires the knowledge of leader dynamics to be known at each agent

or its approximation using neural networks, which requires the knowledge of basis functions.

Moreover, the HJB solution requires knowledge of the system data along the trajectories of

the augmented system. In the second part, an off-policy reinforcement learning is proposed,

which does not require the system data but requires that each agent knows its state and the

dimension of the leaders state. Synchronization in homogeneous nonlinear and heterogeneous

linear agents without the exchange of controller states was done in [53].

In [60], almost

regulated output synchronization for nonlinear heterogeneous agents with a time-varying

8

graph was achieved. Unlike [55], the paper [60] does not require knowledge of the local

output. However, the class of nonlinear systems is a special case of the class of [55] because

the input coefficient is taken to be one. In [61] practical synchronization was achieved in a

network of nonlinear heterogeneous agents where it was shown that the synchronization error

could be made small by increasing the coupling strength of the interconnection. However, the

agents were assumed to be semi-passive, and it was assumed that for the emergent dynamics

(the dynamics to which all the agents converge to), there exists a compact invariant set that

is asymptotically stable.

Distributed observers are used in multi-agent systems when only relative output infor-

mation is available to the agents. Synchronization was achieved in [43], [52], [62] using

distributed observers where the observers were required to exchange the estimates through

the communication topology. In [63] synchronization in a network of identical systems was

achieved with reduced information exchange using standard Luenberger observers; how-

ever, the exchange of controller states was required. In [53] distributed observers were used

without the exchange of observer estimates among the agents to achieve synchronization

in homogeneous linear and nonlinear agents and heterogeneous linear agents. In a recent

work [64], local observers were designed based on LMI approaches to achieve robust output

synchronization in linear heterogeneous multi-agent systems.

1.1.3 Application of Consensus Algorithms to Power Systems Re-

view

One of the major applications of the synchronization problem is in power systems, which

are studied with the help of swing equations [65]. The power grid is a large network of

9

subsystems called control areas, where each area produces, consumes, and transfers power

to adjacent areas to balance power supply and demand in real-time. The frequency of the

AC signal should be maintained very close to a nominal value (typically 50 or 60 Hz), to

avoid tripping of generators, degraded power quality, etc. Therefore, one of the control

objectives in power systems is to minimize the frequency deviation from the nominal value

in an economically efficient way in the presence of load fluctuations. The decentralized

control is computationally less expensive and can provide efficient control under islanding

and self-healing features in scenarios where there is limited communication between the

nodes [66].

Frequency control of power systems is a well-established field of research, which has

led to standard control designs. The decentralized PI controller is used along with the

proportional droop controller to achieve a zero steady-state error, which leads to global

stability results [67]. However, it suffers from performance degradation due to measurement

bias and clock drifts. Moreover, the steady-state control does not follow equal power-sharing,

and therefore, it is not economically efficient. The decentralized leaky-integrator approach,

introduced in [68], is robust to biases in the frequency measurements, and the steady-state

control leads to equal power-sharing. However, due to the leakage term, the steady-state

error is not zero in the case of constant power demand.

To achieve optimal load frequency control (OLFC), the economic dispatch is incorporated

with the frequency control problem [67]. Primal-dual gradient controllers achieve OLFC and

can handle constraints and convex cost functions; see the survey paper [69]. However, they

require knowledge about load and power flow, which are generally unknown. Distributed

consensus algorithms are used to achieve OLFC by assuming the existence of a commu-

nication layer. A distributed controller strategy called the distributed averaging integral

10

control (DAPI) is well-studied in the context of microgrids [70], [71] and synchronous gener-

ators [72]. The DAPI uses a communication network to exchange the integrator states among

the controllers, which leads to equal power-sharing, zero steady-state error, and robustness

to measurement bias [73]. The DAPI controller has been studied considering the linear [72]

and nonlinear [73] models of the power system.

The unknown power demand is assumed to be constant in the above-mentioned ap-

proaches. However, an increase in the use of renewable energy sources causes the power

demand to fluctuate at the same timescale as the power system dynamics, and therefore

approximating the power demand by a constant value becomes unrealistic [74]. This new

challenge necessitates the design of controllers that can handle time-varying power demand.

One of the strategies developed to deal with time-varying power demand is the distributed

internal model approach [74], [75], [76]. However, this approach requires the time-varying

power demand to be generated from a known exosystem model, which is difficult to know

in advance. More recently, an adaptive internal model-based approach has been developed

in [77], where the exosystem is unknown. Still, it uses a linear power system model and

assumes that the unknown power demand is a summation of sinusoids.

1.2 Contribution and Organization

1.2.1 Overview of Contribution

The main contribution of this dissertation is to use observers to increase the convergence

rate and reduce information exchange in multi-agent systems. The main contributions of

this thesis are discussed below.

We first show that observers can be used to construct missing information at each node

11

in a given network topology. We take advantage of the star topology network architecture

and design observer-based decentralized controllers that increase the convergence rate of

the consensus protocol. This is unlike high-gain consensus approaches where to reach fast

consensus, the control gain needs to be high. We also show that for sufficiently small observer

parameters, not only the convergence rate of the consensus protocol with the star topology

approaches the convergence rate of a complete graph but also the trajectories of the agents

with the star topology approach the trajectories of the agents with the complete graph.

However, this design procedure is only limited to the star topology.

Next, we study the effect of network size on the convergence rate. For non-expander

graphs, the convergence rate of the consensus algorithm becomes slow, with an increase

in network size. To solve this issue, we design a scalable consensus algorithm for first-

order agents using Proportional Derivative (PD) for general directed graphs in which we

guarantee that the convergence rate of the closed-loop system does not change with an

increase in network size. Therefore, using the proposed controller, the convergence rate to

achieve consensus does not slow down when the network size increases for general directed

non-expander graphs. The PD controller is realized using a high-gain observer. We show

that the trajectories of the closed-loop system when the high-gain observer is used can be

brought arbitrarily close to the trajectories under the PD controller.

The effect of an increase in network size on the system performance also carries over

to nonlinear systems since the nature of information exchange between the agents remains

the same, i.e., diffusive coupling. For a leader-follower network with a single leader in the

network, the smallest eigenvalue of the grounded Laplacian matrix approaches zero with an

increase in network size for undirected graphs. The controller gain of standard nonlinear

control approaches to achieve synchronization is inversely proportional to the smallest eigen-

12

value of the grounded Laplacian. Therefore, as the network size increases, these approaches

require a significantly high-gain to achieve synchronization. To alleviate this problem, we de-

sign a scalable consensus algorithm to achieve practical synchronization in a leader-follower

framework for second-order nonlinear heterogeneous systems. We assume that relative state

and velocity derivatives are available for feedback, and we realize the controller using a

reduced-order high-gain observer. We show that the synchronization error can be made

arbitrarily small by tuning a controller and observer parameter, respectively.

Next, we study the use of extended high-gain observers to achieve practical synchroniza-

tion in a leader-follower network of nonlinear multi-agent systems having the same relative

degree r under reduced information exchange. The class of systems considered in the previ-

ous cases is a special case of the one considered here. Moreover, the controller designed here

uses less information. The agents do not have access to their state or output and only have

relative output information from their neighbors. Extended high-gain observer, along with

feedback control, is the primary tool used to compensate for the heterogeneous dynamics

of each agent and then replace it with the dynamics of the leader. The proposed approach

has significant advantages over the other works in literature that solve the synchronization

problem for a general class of nonlinear heterogeneous agents:

i) It does not require the

explicit solution of the output regulator equation.

ii) Unlike the classical output synchro-

nization approach, which gives importance only to the steady-state synchronization error,

the proposed method also shapes the transient performance of the closed-loop system.

iii)

The convergence rate can be chosen by first designing the state feedback control. Then, the

convergence rate for output feedback can be made arbitrarily close to the convergence rate

under state feedback for a sufficiently small observer parameter. This is unlike [61], where the

convergence rate depends on the open-loop system. iv) Motivated by practical applications

13

like platooning of vehicles, and frequency synchronization of power systems, we specialize to

agents that have relative degree two and the leader having its first state unbounded. This

is unlike the scenario in the classical output synchronization approach, which requires the

leader or exosystem states to be bounded.

Finally, as an application, we use dynamic consensus and extended high-gain observers to

achieve practical frequency synchronization in power systems under unknown time-varying

power-demand. The controller design procedure is different from the previous chapter. We

do not use feedback linearization to cancel the power system dynamics; rather, we design

the controller to only compensate for the time-varying power demand. For stability analysis,

we follow a multiple time-scale approach. Unlike other algorithms in the literature, we do

not assume that a known model generates power-demand. We show that the steady-state

controller approaches the steady-state controller of [74], and the error between them can be

made arbitrarily small depending on the tuning of a controller and observer parameters. We

show that the synchronization error can be made arbitrarily small by tuning controller and

observer parameters. We also consider the communication topology for the exchange of in-

formation among the controllers to be directed, which relaxes the communication constraints

compared to [68], [72], [74], which assume the communication topology to be undirected.

1.2.2 Thesis Organization

The dissertation is organized as follows. Chapter 2 studies the use of high-gain observers in

a star topology to achieve fast consensus. In Chapter 3, we focus on the design of scalable

consensus algorithms in first-order agents. In Chapter 4, we design a scalable consensus con-

troller to achieve practical scalable synchronization in second-order nonlinear heterogeneous

agents. Chapter 5 studies the practical synchronization of a network of nonlinear heteroge-

14

neous agents under reduced information exchange. We study the power systems frequency

synchronization problem using dynamic consensus and high-gain observers in Chapter 6.

Finally, Chapter 7 discusses the conclusions and future work. Throughout the dissertation,
until and otherwise stated || · || would denote the Euclidean norm.

15

Chapter 2

Fast Consensus with Star Topology

Using High Gain Observers

2.1

Introduction

In this chapter, we show that the second smallest eigenvalue of the graph Laplacian can

be increased for a fixed star topology [78] by using extended high-gain observers. The star

topology is used in wireless sensor networks [79], VSAT communications [80], smart grids

in power networks [81] and the leader-follower configuration in multi-agent systems [82].

The star topology is robust against the failures of the individual nodes, i.e., except for

the root node, if one of the nodes get disconnected, the graph remains connected and the

remaining connected agents can still reach consensus. Furthermore, it has the least number

of connections required for a graph to be connected, which reduces the communication

cost. Such a configuration also arises in the consensus problem of mobile robots, where the

central robot uses an omnidirectional range sensor. In contrast, other robots use single ray

range sensors. The sensors are used to measure the relative distance among the robots. The

peripheral robots are within the field of view of the central robot, and therefore it can measure

its relative distance to the peripheral nodes. The peripheral robots are aligned towards the

central robot, and therefore they can measure the relative distance to the central robot. The

16

only information available to the robots is the relative distance among them based on the

range sensor measurements, and there is no other communication structure present in this

scenario. The consensus problem is solved based on the relative measurements available to

the robots by employing the consensus protocol [19].

The second smallest eigenvalue of the graph Laplacian increases with an increase in the

number of communication links, but it leads to high communication costs. For example,

the fastest convergence rate can be achieved when the communication topology is given by

a complete graph, which is costly in terms of the required number of communication links.

Therefore one of the trade-offs, as discussed in [19], is that undirected graphs with high

communication costs are expected to have a large second smallest eigenvalue of the graph

Laplacian and vice-versa. In this chapter, we show that we can increase the second smallest

eigenvalue of the graph Laplacian for a fixed star topology by adding observers to each agent

expect the root agent. The complete graph is chosen as a reference target system, and we

show that the convergence rate of the consensus protocol with the star topology approaches

the convergence rate of the consensus protocol with the complete graph for sufficiently small


, which is a high-gain observer parameter. Furthermore, we show that for sufficiently small


, the trajectories of the agents with the star topology approach the trajectories of the agents

with the complete virtual graph.

We make the following assumption about the communication topology.

Assumption 2.1: The communication topology is given by a time-invariant, undirected,

unweighted, star graph G.

The star topology is illustrated in 2.1

17

Figure 2.1: Star Topology

We consider agent one as the central or root agent and all the other agents as peripheral

agents.

2.2 Consensus on Different Network Topologies

In this section, the convergence rate of the consensus protocol is studied when the communi-

cation topology is in the form of complete and star graphs, respectively. The agent dynamics

are given by

˙xi = ui,

i = 1, . . . , N

The consensus protocol is given by

N(cid:88)

ui =

(xj − xi)

i = 1, . . . , N.

(2.1)

(2.2)

j=1

2.2.1 Consensus with Complete Graph

If the communication topology is given by a complete graph then the consensus protocol

takes the form

˙x = −Lcx,

18

(2.3)

where x = [x1, . . . , xN ]T , and Lc is the graph Laplacian for the complete graph. The

eigenvalues of Lc are given by [20]

λ(Lc) = {0, N, N, . . . , N, N}.

where λ(·) denotes the eigenvalue. Therefore, the convergence rate of the consensus protocol

is proportional to N .

2.2.2 Consensus with Star Graph

Now if the communication topology is given by a star graph we have

˙x = −Lsx,

(2.4)

where Ls is the graph Laplacian. The eigenvalues of Ls are given by λ(Ls) = {0, 1, 1, . . . , 1, N}.

Therefore, the convergence rate of the consensus protocol is proportional to 1. The conver-

gence rate of the consensus protocol for the complete graph is N times faster than the star

graph.

2.3 Design Preliminaries

We define our target system as the consensus protocol for the complete graph. The objective

of the observer-based controller designed in this chapter is to match the performance of the

target system.

19

2.3.1 Target System

The target system is defined as

˙x∗ = −Lcx∗,

x∗(0) = x(0),

where x∗(t) is the solution of the target system. Consider the change of variables

α∗ = x∗
1,
i+1 − α∗, 1 ≤ i ≤ N − 1.
i = x∗
δ∗

(2.5)

(2.6a)

(2.6b)

δ∗ = [δ∗

1, . . . , δ∗

N−1]T is called the disagreement vector and α∗ is the consensus manifold.

Differentiating (2.6) we have

N−1(cid:88)

i=1

δ∗
i ,

˙α∗ =

˙δ∗ = −N δ∗.

(2.7a)

(2.7b)

We choose a Lyapunov function candidate for the disagreement dynamics (2.7b) as ¯V =
2δ∗T δ∗. Differentiating ¯V along (2.7b) we have

1

˙¯V = −N δ∗T δ∗ = −N(cid:107)δ∗(cid:107)2.

From which we have

¯V (δ∗(t)) = e−2N t ¯V (δ∗(0)).

20

Therefore the disagreement vector converges exponentially to zero with a rate N . Hence we

can conclude that

t→∞ δ∗

lim

i (t) = 0 =⇒ lim

t→∞{x∗

i+1(t) − α∗(t)} = 0,

1 ≤ i ≤ N − 1.

2.3.2 Exploiting the Star Topology Structure

The root agent is numbered as 1 while the other agents are numbered from 2, . . . , N and the

root agent is connected to all the other agents in the star network.

In addition to the consensus controller a new decentralized controller will be designed

for each agent except the root agent. We do not modify the consensus controller of the root

agent as it is connected to all the other agents in the network and it is given by

N(cid:88)

˙x1 =

(xj − x1).

(2.8)

j=2

The controllers in all the other agents will be modified with an observer-based decentralized

controller.

2.4 Controller Design and Analysis

In this section we will discuss our strategy of designing the observer-based decentralized

controller.

21

2.4.1 Extended High Gain Observers

We add an extended high gain observer to each agent in the star network except the root

agent. The main motivation in adding the observer is that each agent estimates its missing

connections from the root agent. The information available to each agent except the root

agent is

ζi−1 = x1 − xi = −δi−1,

i = 2, . . . , N.

(2.9)

The control for the peripheral agents are given by

ui = ζi−1 + νi−1,

i = 2, . . . , N

where νi is the observer based controller to be defined later. Differentiating eq. (2.9) we

have

˙δi−1 = −( ˙x1 − ˙xi) = − N(cid:88)

(xj − x1) + (x1 − xi) + νi−1.

j=2

for i = 2, . . . , N . From which we have

˙δi−1 = −2δi−1 +

N(cid:88)

j=2,j(cid:54)=i

(x1 − xj) + νi−1.

If we knew the term

σi−1 =

N(cid:88)

j=2,j(cid:54)=i

(x1 − xj) = − N(cid:88)

j=2,j(cid:54)=i

δj−1,

i = 2, . . . , N

(2.10)

22

the missing connections then could be constructed by the following relation

(xj − xi) = −σi−1 − (N − 2)δi−1,

i = 2, . . . , N

N(cid:88)

j=2,j(cid:54)=i

This gives us the motivation to estimate σi−1 by using an extended high-gain observer. The

extended high-gain observer is constructed as [83]

˙ˆδi−1 = −2ˆδi−1 + ˆσi−1 + νi−1 +

(δi−1 − ˆδi−1),

α1



˙ˆσi−1 =

α2


2 (δi−1 − ˆδi−1),

(2.11a)

(2.11b)

for i = 2, . . . , N , where 
, α1 and α2 are positive constants with 
 << 1. The observer-based

controller is given by

νi−1 = −ˆσi−1 − (N − 2)δi−1.

Similar to (2.6), we define the change of variables

α = x1,
δi = xi+1 − α,

1 ≤ i ≤ N − 1.

(2.12)

(2.13a)

(2.13b)

and δ = [δ1, . . . , δN−1]T .

2.4.2 Peaking

When δi−1(0) − ˆδi−1(0) (cid:54)= 0 the transient response of the observer contains a term of the
form (1/
)e−at/
 for some a > 0. This is known as the peaking phenomenon [84]. Its impact

in feedback control is overcome by saturating the control outside a compact set of interest.

23

We define the Lyapunov function V (δ) = 1

2 δT δ. Let S be any compact set in RN−1.

Choose c > 0 such that

S ⊂ Ωc = {V (δ) ≤ c} ⊂ RN−1.

(2.14)

The observer-based controller is saturated outside the compact set Ωc. Let

Mi−1 > max
δ∈Ωc

|νi−1|, 2 ≤ i ≤ N,

The observer-based control is saturated as

¯νi−1 = Mi−1 sat

(cid:19)

(cid:18) νi−1

Mi−1

.

2.4.3 Observer Error Dynamics

The system (2.1) with the controller

ui = (x1 − xi) + ¯νi−1,

for i = 2, . . . , N

(2.15)

in the new coordinates become

(2.16a)

(2.16b)

N−1(cid:88)

i=1

δi,

˙α =

˙δ = − ˜Lsδ + ¯ν,

24

where ¯ν = [¯ν1, ¯ν2, . . . , ¯νN−1]T and ˜Ls is given by

˜Ls = [0 | IN−1]Ls

 0

IN−1

 + rrT ,

(2.17)

where IN−1 ∈ RN−1×N−1 is the identity matrix, r = 1 ∈ RN−1 is a column of all 1’s and
the eigenvalues of ˜Ls are the non-zero eigenvalues of Ls.

With the scaled estimation errors

ϕi−1 =

δi−1 − ˆδi−1




,

ηi−1 = σi−1 − ˆσi−1,

(2.18a)

(2.18b)

the observer error dynamics are given by


 ˙ϕi−1 = ˙δi−1 − ˙ˆδi−1

= −2δi−1 + σi−1 + νi−1 + 2ˆδi−1 − ˆσi−1 − νi−1 − α1


= −α1ϕi−1 + ηi−1 − 2
ϕi−1,

(δi−1 − ˆδi−1)


 ˙ηi−1 = −α2ϕi−1 + 
 ˙σi−1,

N(cid:80)

( ˙xj − ˙x1) = − N(cid:80)

˙δj−1.

j=2,j(cid:54)=i

where ˙σi−1 =

j=2,j(cid:54)=i

The foregoing equations can be rewritten as


 ˙ξi−1 = A0ξi−1 + 
∆i−1,

(2.19)

25

where ξi−1 = [ϕi−1, ηi−1]T , A0 =

−α1 1

−α2 0

 , ∆i−1 =

 −2ϕi−1
− N(cid:80)

j=2,j(cid:54)=i

˙δj−1

 and A0 is Hur-

witz. Let ˆΥ = [ ˆΥ1, . . . , ˆΥN−1]T where ˆΥi−1 = [ˆδi−1, ˆσi−1]T for i = 2, . . . , N , Υ =

[Υ1, . . . , ΥN−1]T where Υi−1 = [δi−1, σi−1]T for i = 2, . . . , N , and ξ = [ξ1, . . . , ξN−1]T
where ξ, ˆΥ, Υ ∈ R2(N−1). We have

ˆΥ = Υ − D
ξ,

where D
 = (IN−1 ⊗ D0) ∈ R2(N−1)×2(N−1) and D0 =

(2.20)


 0
 .

0 1

Theorem 2.1: Consider the disagreement dynamics (2.16b) obtained using the feedback

controller (2.15) and the extended high-gain observer (2.11), with 
, α1 and α2 chosen as
positive constants. Let S be defined as in (2.14) and Q be any compact set of R2(N−1). Then,
there exists 
∗ > 0, such that for every 0 < 
 ≤ 
∗, the solutions (δ(t), ˆΥ(t)) of (2.16b) and
(2.11), starting in S × Q, satisfy

lim
t→∞ δ(t) = 0

and

t→∞ ˆΥ(t) = 0
lim

(2.21)

Proof : We rewrite the closed loop system in the singularly perturbed form,

(2.22a)

(2.22b)

(2.22c)

˙α =

δi,

N−1(cid:88)

i=1

˙δ = F (δ, ξ),


 ˙ξ = Aξ + 
∆,

26

where F (δ, ξ) = − ˜Lsδ + ¯ν, ∆ = [∆1, ∆2, . . . , ∆N−1]T , A = (IN−1⊗A0) ∈ R2(N−1)×2(N−1).

The disagreement dynamics (2.22b) and the observer error dynamics (2.22c) are independent

of α.

The Lyapunov function for (2.22c) is defined as W = ξT ¯P ξ, where ¯P = (IN−1 ⊗ P )
∈ R2(N−1)×2(N−1) and P is the positive definite solution of the Lyapunov equation P A0 +
0 P = −I. We define the compact sets Σ = {W (ξ) ≤ β
2} and Λ = Ωc × Σ. Due to global
AT
boundedness of F and ∆ in ˆΥ, for all δ ∈ Ωc and ξ ∈ R2(N−1) we have

(cid:107)F (δ, ξ)(cid:107) ≤ k1, (cid:107)∆(δ, ξ)(cid:107) ≤ L1,

(2.23)

where L1, k1 are positive constants independent of 
. Because δ(0) is in the interior of Ωc
there exists T1 > 0 independent of 
 such that δ ∈ Ωc for t ∈ [0, T1] and during this time

interval (2.23) holds. Following the standard analysis for the high-gain observer theory [84,

Theorem 1] it can be argued that initially ξ(0) could be outside the set Σ but it quickly
reaches the set within a time interval [0, T (
)] where T (
) → 0 as 
 → 0 and there exists
positive constants β and 
1 such that the compact set Λ = Ωc × Σ is positively invariant
for every 0 < 
 ≤ 
1 and for all δ(0) ∈ S and ˆΥ(0) ∈ Q, the trajectory (δ(t), ξ(t)) enters Λ

within the interval [0, T (
)].

The saturation is no longer effective when the trajectory enters Λ. Therefore we have
¯νi−1 = νi−1, ∀ t ≥ T (
), and the closed-loop system is then defined by the linear singularly

27

perturbed form

N−1(cid:88)

i=1

δi,

˙α =

˙δ = −N δ − V ξ,


 ˙ξ = (A + 
R)ξ + 
M δ,

(2.24a)

(2.24b)

(2.24c)

for some matrices V , R and M . We define the following change of variables [85] to separate

the fast variables from the slow ones:

χ = ξ + Y (
)δ,

(2.25)

where Y (
) = 
A−1M + O(
2). The transformed system is

N−1(cid:88)

˙α =

δi,

i=1

˙δ = (−N IN−1 + O(
))δ − V χ,


 ˙χ = (A + 
R)χ.

(2.26a)

(2.26b)

(2.26c)

(2.27)

δ

 .

χ

−V



δ

 = Γ

χ

From (2.26b) and (2.26c) we have

 ˙δ

 =

−N IN−1 + O(
)

˙χ

0

(A + 
R)/


The matrix Γ is block triangular and hence its eigenvalues are the eigenvalues of (−N IN−1 +
O(
)) and (A+
R)/
. We can find 
∗ > 0 such that for all 
 ∈ (0, 
∗) the matrix Γ is Hurwitz.

28

Therefore,

lim
t→∞ δ(t) = 0 and lim

t→∞ χ(t) = 0.

From (2.25) we can conclude that lim

t→∞ ξ(t) = 0; hence (2.21) follows from (2.20).

(2.28)

(cid:4)

Remark 2.1: The convergence rate of the star topology will increase to b if the consensus

controller (2.2) is multiplied by a gain b. However there is a limit on how high the gain could

be and it depends on the physical constraints of the control. The observer based method

presented in this chapter speeds up the convergence rate not by increasing the gain in the

consensus equation but by providing more information to the nodes.

Theorem 2.2: Suppose Theorem 2.1 holds and let the target system be defined as in

(2.5). Then, given any µ > 0 there exists 
∗∗ > 0 such that for all 
 ∈ (0, 
∗∗)

(cid:107)x(t) − x∗(t)(cid:107) ≤ µ ∀ t ≥ 0,

(2.29)

Proof: As δ(0) is in the interior of Ωc we have (cid:107)δ(t) − δ(0)(cid:107) ≤ k1t during the interval
[0, T (
)], for some k1 > 0. Similarly it can be shown that (cid:107)δ∗(t) − δ∗(0)(cid:107) ≤ k1t during the
same time interval. As δ(0) = δ∗(0) we have

(cid:107)δ(t) − δ∗(t)(cid:107) ≤ 2k1T (
),

∀ t ∈ [0, T (
)].

Since T (
) → 0 as 
 → 0, there exists 0 < 
2 ≤ 
∗ such that, for every 0 < 
 ≤ 
2, we have

(cid:107)δ(t) − δ∗(t)(cid:107) = O(T (
)),

∀ t ∈ [0, T (
)].

(2.30)

29

Subtracting (2.7a) from (2.26a) we have

˙α(t) − ˙α∗(t) = rT [δ(t) − δ∗(t)].

(2.31)

Integrating (2.31) and using α(0) = α∗(0) results in

t(cid:90)

α(t) − α∗(t) = rT

[δ(τ ) − δ∗(τ )]dτ.

Taking the absolute value we have

0

|α(t) − α∗(t)| = O(T 2(
)), ∀ t ∈ [0, T (
)].

(2.32)

We know that

(cid:107)e−N IN−1t(cid:107) ≤ k2e−λ1t and (cid:107)e(−N IN−1+O(
))t(cid:107) ≤ k3e−λ2t,

(2.33)

where k2, k3, λ1 and λ2 are positive constants and for 
 small enough λ2 can be chosen such

that λ2 < λ1. From (2.26c) we have

(cid:107)χ(t)(cid:107) ≤ k4e−λ3(t−T (
))/
(cid:107)χ(T (
))(cid:107),

(2.34)

where k4 and λ3 are positive constants. Next we define the deviation of the system trajectory
from the target trajectory as ψ(t) = δ(t) − δ∗(t). Differentiating ψ(t) we have

˙ψ(t) = (−N IN−1 + O(
))ψ(t) + O(
)δ∗(t) − V χ(t).

30

Integrating ˙ψ(t) we have

ψ(t) = e(−N IN−1+O(
))(t−T (
))ψ(T (
)) +

t(cid:90)

T (
)

From (2.30) we have,

e(−N IN−1+O(
))(t−τ )[O(
)δ∗(τ ) − V χ(τ )]dτ.

(2.35)

(cid:107)ψ(T (
))(cid:107) = (cid:107)δ(T (
)) − δ∗(T (
))(cid:107) = O(T (
)).

From (2.35),

(cid:107)ψ(t)(cid:107) ≤ k3e−λ2(t−T (
))O(T (
)) +

[e−λ2(t−T (
)) − e−λ1(t−T (
))]


k3k5
(λ1 − λ2)

(cid:104)
e−λ2(t−T (
)) − e−λ3(t−T (
))/
(cid:105)

,

+


k3k6

(λ3 − 
λ2)

where k5, k6 and λ3 are positive constants. From the above inequality it can be shown that

(cid:107)ψ(t)(cid:107) ≤ (O(T (
)) + 
k7)e−λ2(t−T (
)) ∀ t ≥ T (
),

(2.36)

where k7 is a positive constant. Since T (
) → 0 as 
 → 0, there exists 0 < 
3 ≤ 
2 such that,
for every 0 < 
 ≤ 
3, we can conclude that

(cid:107)δ(t) − δ∗(t)(cid:107) ≤ µ,

∀ t ≥ 0.

(2.37)

31

Integrating (2.31) from the time T (
) we have

α(t) − α∗(t) = [α(T (
)) − α∗(T (
))] + rT

t(cid:90)

T (
)

[δ(τ ) − δ∗(τ )]dτ.

Using |α(T (
)) − α∗(T (
))| = O(T 2(
)) shows that

|α(t) − α∗(t)| ≤ O(T 2(
)) + (cid:107)rT(cid:107)(O(T (
)) + 
k7)

λ2

.

(2.38)

Since T (
) → 0 as 
 → 0, there exists 0 < 
4 ≤ 
3 such that, for every 0 < 
 ≤ 
4, we can

conclude that

|α(t) − α∗(t)| ≤ µ ∀ t ≥ 0.

Take 
∗∗ = min{
3, 
4}. Then (2.29) follows from (2.37) and (2.39).

(2.39)

(cid:4)

Remark 2.2: The consensus limit achieved by using a controller of the form (2.22) is the

average of the initial conditions of the system [19]. From Theorem 2.1 we can only conclude

that the agents achieve consensus. Theorem 2.2 shows that the consensus limit achieved

by using the observer-based controller can be made arbitrarily close to the one achieved by

using a standard consensus controller.

2.5 Simulations

The simulation results are provided for a network of 10 agents with agent 1 as the root agent.

The initial conditions for the agents are chosen as xi+1(0) = 5i, for i = 0, 1, . . . , 9.

32

Figure 2.2: Consensus on Complete Graph

Figure 2.3: Consensus on Star Graph with no Observers

Figure 2.4: Consensus on Star Graph with Observers and 
 = 0.001

Fig. 2.2 shows the trajectories of the agents under a complete graph. Fig. 2.3 shows the

trajectories of the agents under a star graph without observer where the convergence rate

33

00.10.20.30.40.50.6t01020304050x1,,x10x1x2x3x4x5x6x7x8x9x100123456t01020304050x1,,x10x1x2x3x4x5x6x7x8x9x1000.10.20.30.40.50.6t01020304050x1,,x10x1x2x3x4x5x6x7x8x9x10is 10 times slower than the complete graph. Fig. 2.4 shows the trajectories of the agents

under a star graph with observers. Fig. 2.2 and 2.4 have the same time scale as they achieve

consensus with a rate of 10.

Figure 2.5: Deviation of agent 6 trajectory from target trajectory

Fig. 2.5 illustrates the fact that as 
 decreases the deviation of the consensus limit of the

star topology with observers compared to the complete topology decreases and this trend is

representative for all other agent trajectories.

2.6 Conclusion

This chapter presented an algorithm which increases the convergence rate of the consensus

protocol on star topology using observers. The trajectories of the agents under the star

topology approach the trajectories of the agents under a complete graph for sufficiently

small 
. However adding observers increases the controller complexity as the closed loop

system also comprises the observer dynamics.

34

00.10.20.30.40.50.6t-1.4-1.2-1-0.8-0.6-0.4-0.2x6-x6*ǫ = 0.01ǫ = 0.005ǫ = 0.001Chapter 3

Scalable Consensus Using High Gain

Observers

3.1

Introduction

In this chapter, we design a PD consensus controller for general directed graphs in which

we guarantee that the convergence rate of the closed-loop system does not change with an

increase in network size. The effect of network size on the convergence rate is not known

for the case of general directed graphs. However, using the proposed controller, we can

guarantee that the convergence rate remains unchanged [86] with an increase in network size.

Moreover, the closed-loop system matrix has the properties of a Laplacian matrix. Therefore,

it is named as Virtual Laplacian matrix, which represents the virtual connections among the

agents for the closed-loop system. The real parts of the eigenvalues of the Virtual Laplacian

matrix approach one while the imaginary parts approach zero as a design parameter increases.

We realize the PD controller using a high-gain observer and show that the trajectories of

the closed-loop system when the high-gain observer is used can be brought arbitrarily close

to the trajectories under the PD controller.

35

3.2 Problem Definition

We consider the consensus problem in a network of N agents where a single integrator

represents each agent

˙xi = vi,

i = 1, . . . , N.

(3.1)

The communication topology is defined by G = (V,E,A). The agents can only measure

relative information, and we define a signal which represents the weighted sum of all the

measurements at agent i,

N(cid:88)

j=1

ζi =

aij(xj − xi)

(3.2)

where aij ≥ 0. Equation (3.2) in matrix form can be written as ζ = −Lx, where ζ =

col(ζ1, . . . , ζN ).

Assumption 3.1: The communication topology is defined by a weighted directed graph
G which contains a globally reachable node, i.e., G contains a node which can be reached

from any another node by traversing a directed path.

The above assumption implies that zero is a simple eigenvalue of the graph Laplacian L and

all other eigenvalues have positive real parts [20]. This is the most general assumption con-

sidered in the literature for time-invariant graphs. The objective of the consensus algorithm

is defined as

t→∞{xi(t) − xj(t)} = 0, for i (cid:54)= j and i, j = 1, . . . , N

lim

for which a standard consensus controller [19] is

vi = ks

aij(xj − xi)

(3.3)

N(cid:88)

j=1

36

where ks is a positive constant, which we introduce as a tunable controller gain. The closed-

loop system of (3.1) with the controller (3.3) is

˙x = −ksLx

(3.4)

where the convergence rate for undirected graphs [18, Chapter 7] is given by ksλ2(L). The

convergence rate decreases for certain classes of undirected graphs where increase in network
size results in λ2(L) → 0. Therefore, as the second smallest eigenvalue of the graph Laplacian

tends to zero the magnitude of the control signal needs to be very large to attain a specific

convergence rate.

3.3

State feedback Controller Design

In this section we design PD control for a network of N agents described in (3.1), assuming

that the state derivatives are available to the agents. Differentiating (3.2) in matrix form we

have

˙ζ = −Lv

where v = col(v1, . . . , vN ) and for each agent i we have

˙ζi =

aij(vj − vi)

N(cid:88)

j=1

37

Next we define a signal σi to be σi = ˙ζi, which can be written as σ = −Lv, where σ =

col(σ1, . . . , σN ). The control at each node is taken as

vi = kdσi + kpζi

(3.5)

where kp and kd are positive constants. In this section we assume that σi is available for

feedback. Substituting for σi we have,

vi = kd

N(cid:88)

j=1

aij(vj − vi) + kpζi

The above controller is distributed in nature as it relies only on local information. Next we

write the equations in matrix form,

(I + kdL)v = kpζ

The matrix (I + kdL) is nonsingular because λi(I + kdL) = 1 + kdλi(L), where Re (λi(L)) ≥

0, from Assumption 3.1. Choosing kd = k and kp = γk, we have the control as v =

γ

I + L

ζ, which results in the closed-loop system

(cid:19)−1

(cid:18) 1

k

˙x = −γ

I + L

Lx = −γ ˆLx

(3.6)

where ˆL represents the Virtual Laplacian matrix, defined as

(cid:19)−1

(cid:18) 1

k

(cid:18) 1

k

ˆL =

(cid:19)−1

I + L

L

(3.7)

38

Remark 3.1: The matrix L encodes the physical exchange of information among the agents

while the matrix ˆL encodes the virtual connections among the agents for the closed-loop

system.
Properties of a matrix of the form (I + eL)−1L for e ≥ 0, were identified in [87, Section II.B],

for the case of undirected graphs. In the next Lemma we show a property of ˆL for general

directed graphs.

Lemma 3.1: The matrix ˆL is a Laplacian matrix.

Proof: A matrix is a Laplacian matrix if [18, Chapter 6], i) its row-sums are zero, ii) its

diagonal entries are non-negative, and iii) its non-diagonal entries are non-positive. We now

show that ˆL satisfies the three properties. Using the identity

(I + A)−1 = I − (I + A)−1A

where A ∈ RN×N , equation (3.7) can be simplified to

ˆL = I − (I + kL)−1 .

The Laplacian matrix L is a singular Metzler or M matrix [6], which implies that (I + kL)
is a nonsingular M matrix. It follows from [88, Section III.G] that (I + kL)−1 is a positive

matrix. We first show that the row sums of ˆL are zero by

I − (I + kL)−1(cid:105)
(cid:104)

ˆL1 =

1 =

I + L

L1 = 0

(cid:19)−1

(cid:18) 1

k

where (1, 0) are columns of all 1’s and 0’s of appropriate dimension. From which we conclude
that (I + kL)−1 1 = 1.

39

it implies ˜wij ≥ 0. Moreover, from (I + kL)−1 1 = 1, we have(cid:80)n
the properties ˜wij ≥ 0 and(cid:80)n

Next, let ˜wij be an element of the matrix (I + kL)−1. Since (I + kL)−1 is a positive matrix
j=1 ˜wij = 1. The matrix ˆL
has the diagonal elements as 1 − ˜wii and the off-diagonal elements as − ˜wij for i (cid:54)= j. Using
j=1 ˜wij = 1 we can conclude that 1 − ˜wii is non-negative and
− ˜wij is non-positive, which satisfies property (ii) and (iii) and therefore we conclude that ˆL is
(cid:4)

a Laplacian matrix.

In the next Theorem we show properties of the eigenvalues of ˆL for general directed graphs.

Theorem 3.1: Consider the closed-loop system (3.6), with initial condition x(0) ∈ RN ,

obtained using the state feedback controller (3.5), with kp = γk, and kd = k where k and
γ are chosen as positive constants. Let the graph G be a weighted directed graph satisfying

Assumption 3.1, then for sufficiently large k,

(cid:19)

(cid:18) 1
(cid:18) 1

Re

λi(L)

λi(L)

(cid:19)

Re(λi( ˆL)) = 1 − 1
k

Im(λi( ˆL)) = − 1
k

Im

+ O

(cid:18) 1
(cid:19)
(cid:18) 1

k2

k2

+ O

(cid:19)

,

,

(3.8a)

(3.8b)

for i = 2, . . . , N, where λi( ˆL) and λi(L) are the eigenvalues of ˆL and L, respectively.
Moreover, the control signal v = −γ ˆLx is bounded uniformly in k.

Proof : First we decompose L in its Jordan form

L = ˜P J ˜P−1

where J = blkdiag(0, J2, . . . , Jm) and m ≤ N is the number of Jordan blocks. The first

Jordan block is the scalar zero and the other Jordan blocks J2, . . . , Jm contain the eigenvalues

40

with strict positive real parts. From the Jordan decomposition of L we have

(cid:20) 1

k

ˆL =

˜P ˜P−1 + ˜P J ˜P−1

˜P J ˜P−1 = ˜P

I + J

J ˜P−1

(cid:21)−1

(cid:21)−1

(cid:20) 1

k

I + J is a triangular matrix with diagonal elements

1
k
with diagonal elements λi(L). Therefore the eigenvalues of ˆL are given by

1
k

+ λi(L) and J is a triangular matrix

λi( ˆL) =

λi(L)

=

+ λi(L)

1 +

1
k

1

1

.

kλi(L)

For suffciently large k we have

1

1

kλi(L)

1 +

= 1 − 1

kλi(L)

+

(cid:18) 1

kλi(L)

(cid:19)2 −

(cid:18) 1

(cid:19)3

kλi(L)

+ . . .

for i = 2, . . . , N , from which

 λi(L)
 λi(L)

1
k

+ λi(L)

1
k

+ λi(L)

 = 1 − 1
 = − 1

k

k

Im

Re

Im

(cid:18) 1

(cid:19)

λi(L)

Re

(cid:18) 1

(cid:19)

k2

,

+ O

(cid:18) 1

(cid:19)

λi(L)

(cid:18) 1

(cid:19)

k2

+ O

Therefore, (3.8) follows from the above equations.
To show that the control signal is bounded we write it as v = − ˜Kx, where ˜K = γ ˆL. We show
that the norm of ˜K is bounded uniformly in k. First, we rewrite ˜K as ˜K = γI−γ(I +kL)−1.
The matrix (I + kL) is a nonsingular M matrix and it is strictly diagonally dominant,

41

therefore it follows from [89] that

From which we have

||(I + kL)−1||∞ = 1.

|| ˜K||∞ ≤ 2γ

(cid:21)−1

(cid:20) 1

k

Therefore, the infinity norm of ˜K is independent of k. For sufficiently large k, the matrix

˜Ji =

I + Ji

Ji is given by ˜Ji = I + O(1/k) and the solution of (3.6) is given by

x(t) = ˜P ¯L ˜P−1x(0) + O

(cid:18) 1

(cid:19)

k

where ¯L = diag(1, e−γt, . . . , e−γt). From which it can be shown that

||x(t)||∞ ≤ a1 + (a2 + a3e−γt)||x(0)||∞

where a1, a2 and a3 are positive constants independent of k. Therefore,

||v(t)||∞ ≤ || ˜K||∞ · ||x(t)||∞ ≤ 2γ[a1 + (a2 + a3e−γt)||x(0)||∞]

From which we can conclude that the control signal is bounded uniformly in k.

(cid:4)

Remark 3.2: For sufficiently large k, the eigenvalues of the Virtual Laplacian matrix ˆL

approach 1+0i, and therefore the directedness of the graph is irrelevant. The eigenvalues of
the closed-loop system (3.6) approach −γ for sufficiently large k. Therefore, the convergence

rate approaches γ and will not deteriorate as the network size grows, provided k is sufficiently

42

large. The convergence rate can further be increased by increasing γ, however with increase

in γ, the magnitude of the control signal increases.

Remark 3.3: A special case arises when the graph G is connected and undirected with

Laplacian L = U ΛLU T , where U is orthonormal and ΛL = diag(0, λ2, . . . , λN ). Then for

this case, L is irreducible [18], which implies that (I + kL) is also irreducible. Therefore,
it follows from [88, Section III.G] that (I + kL)−1 is a strictly positive matrix. Since,
(I + kL)−1 strictly positive, it implies that all the matrix elements are non-zero. Therefore,

we can conclude the virtual Laplacian matrix ˆL is symmetric with all the connections (virtual

complete graph). Moreover, with sufficiently large k

(cid:19)−1

(cid:18) 1

k

I + L

= L+ − 1
k

L+ + O

(cid:18) 1

(cid:19)

k2

where L+ is the pseudoinverse Laplacian matrix, see [18, Chapter 6], with L+ = U ΛP U T ,

where ΛP = diag(0, 1/λ2, . . . , 1/λN ). Therefore, we have

(cid:20)

(cid:18) 1

(cid:19)

k

=

I − 1
N

11T

(cid:21)

+ O

(cid:18) 1

(cid:19)

k

(3.9)

ˆL = L+L + O

3.4 Output-Feedback Controller

3.4.1 Preliminaries

For the design of the output feedback controller we first discuss the following controller

µ ˙vi = −vi + kσi + γkζi

(3.10)

43

which in vector form becomes

µ ˙v = −v + kσ + γkζ

The motivation for using a controller of the form (3.10) is because the analysis of the output

feedback requires the boundedness of the derivative of the control ( ˙v) which is achieved by

(3.10). First, we discuss the case when σ is available for feedback. The quasi-steady state

of (3.10) when µ = 0 is given by

v = kσ + γkζ = −kLv − γkLx =⇒ v = −γ ˆLx

which implies that for sufficiently small µ we recover the PD controller properties. From the

change of variable y = v + γ ˆLx, we have

µ ˙y = [A + µγ ˆL]y − µγ2 ˆL2x

(3.11)

where A = −(I + kL). We define a change of coordinates

1 0T

 z (cid:44) T ¯y

y =

1

I

From which we have

˜L = T−1 ˆLT =

0

˜L12
0 ˜L22

 , ¯L = T−1LT =

0

¯L12
0 ¯L22



44

where the eigenvalues of the matrix ˜L22, ¯L22 are the non-zero eigenvalues of ˆL and L,
respectively. Using the transformations x = ˜P z and y = T ¯y, the closed-loop system becomes

˙z1 = R1 ¯y
˙˜z = −γ ˜J ˜z + R2 ¯y

µ ˙¯y = ˜A¯y + kE ¯y + µγR3 ¯y + µγ2R4 ˜z

(3.12a)

(3.12b)

(3.12c)

where

˜A =

−1

0

0 −(I + k ¯L22)

 , E =

0

− ¯L12

0 0N−1×N−1

 ,

where ˜A is Hurwitz, 0N−1×N−1 is a matrix of all 0’s, z = col(z1, ˜z), for some matrices

R1, R2, R3, and R4 where the norm of these matrices are bounded uniformly in k and

˜J = blkdiag( ˜J2, . . . , ˜Jm). For sufficiently large k, we have

˜J = I + O(1/k)

Remark 3.4: The system (3.12b)-(3.12c) is a linear singularly perturbed system and it can

be easily shown following the standard singular perturbation results that for sufficiently small

µ, the origin of (3.12b)-(3.12c) is exponentially stable. We follow a Lyapunov analysis for

(3.12b)-(3.12c) for two reasons i) The analysis of the system (3.12) determines the saturation

levels that have to be used in output feedback to deal with the observer peaking. ii) To show

that an upper bound on parameter µ is independent of k.

45

Next we define a composite Lyapunov function for (3.12b)-(3.12c) as

V = ˜zT ˜z + ¯yT P ¯y

where P is a symmetric positive definite matrix given by

b/2 0T

0 W



P =

where b > 0 and W = W T > 0 is the solution to the Lyapunov equation W ¯L22 + ¯LT
22W = I.
From which we have P ˜A + ˜AT P = −Q, where Q is a symmetric positive definite matrix

given by

Q =

b

0T

0 (2W + kI)



We define the set of initial conditions as Γ = {V ≤ c}, where c is any positive constant.

Lemma 3.2: Consider the closed-loop system (3.12) obtained using the system (3.1) and
controller (3.10). Let (˜z(0), ¯y(0)) ∈ Γ and z1(0) lie in any compact set of R. Then, there
exists µ∗ > 0, such that for all µ ∈ (0, µ∗], the set Γ is positively invariant and the trajectories

of the system (3.12) are bounded. Moreover we have

lim
t→∞ ˜z(t) = 0,

lim
t→∞ ¯y(t) = 0, and lim

t→∞ z1 = constant

(3.13)

46

Proof : Taking the time-derivative of V along (3.12b)-(3.12c), we have

˙V = −γ ˜zT [ ˜J + ˜J T ]˜z + 2˜zT R2 ¯y − (1/µ)[˜yT (2W + kI)˜yT + b¯y2
1]

− (1/µ)bk¯y1 ¯L12 ˜y + 2γ ¯yT P R3 ¯y + 2γ2 ¯yT P R4 ˜z

where ¯y = col(y1, ˜y), ˜y = col(¯y2, . . . , ¯yN ). From which we have

˙V ≤ −2γ||˜z||2+γ(e1/k)||˜z||2+e2||˜z||·||¯y||+e3||¯y||2−(1/µ)[¯y1 ˜y]







¯y1

˜y

b

−bke4/2
−bke4/2 2λmin(W )

for some positive constants e1, e2, e3 and e4 independent of k. Choosing b ≤ 8λmin(W )/(k2

mine2
4),

where kmin is a lower bound on k, which without the loss of generality can be chosen as

kmin = 1, from which we have



−[¯y1 ˜y]

b

−bke4/2
−bke4/2 2λmin(W )

where b1 is a positive constant independent of k. For k ≥ e1, we have

 [¯y1 ˜y]T ≤ −b1||¯y||2


˜z


−e2/2
− e3
b1
µ

¯y

 γ

−e2/2

˙V ≤ −[˜z ¯y]

Therefore, there exists µ∗ > 0 such that for all µ ∈ (0, µ∗], the above matrix positive is

positive definite. From which it follows that

lim
t→∞ ˜z(t) = 0,

lim
t→∞ ¯y(t) = 0

47

Moreover, the set Γ is positively invariant as ˙V < 0 on the boundary {V = c}.

Because of the exponential stability of the system (3.12b)-(3.12c), z1(t) remains bounded

and all the agents converge to the consensus value lim

t→∞ z1(t) = constant. From which we can
(cid:4)

conclude that (3.13) follows.

3.4.2 Observer Design

The controller (3.10) assumes the availability of the state derivatives.

In this section we

estimate the extended state by adding a high-gain observer to each agent in the network to

estimate the signal σi. The driving signal to the observer is ζi. The high-gain observer [90]

is constructed as

˙ˆζi = ˆσi +

(ζi − ˆζi)

α1



˙ˆσi =

α2


2 (ζi − ˆζi)

(3.14a)

(3.14b)

where 
, α1 and α2 are positive constants with 
 << 1 for i = 1, . . . , N .

3.4.3 Peaking

When the initial condition of the estimate ˆζi(0) is not the same as ζi(0), then during the

transient period, the observer estimates will peak; see [90]. In order to mitigate this effect,

the control is saturated outside a positively invariant set under the state feedback controller

so that peaking does not affect the plant. We find the maximum value of the control by
finding the maximum value of v = y − γ ˆLx = Q1 ¯y + Q2 ˜z inside the positively invariant set

48

Γ, where Q1 and Q2 are some matrices independent of k. Let

Mi > max
(˜z,¯y)∈Γ

|(Q1)i ¯y| + |(Q3)i ˜z|

where (Q1)i, (Q2)i denotes the ith rows of Q1 and Q2. The constants Mi are independent
of k as the set {V ≤ c} is independent of k. The output feedback controller is given by

(cid:32)

(cid:33)

µ ˙vi = −vi + Misat

kˆσi + γk ˆζi

Mi

(3.15)

where sat(·) is the saturation function defined as sat(y) = sign(y) min{1,|y|}.

The change of variables y = v + γ ˆLx transforms (3.15) to

(cid:34)

µ ˙y = [A + µγ ˆL]y − µγ2R3 ˜z − ∆
(cid:32)

(cid:33)(cid:35)

kˆσ + γk ˆζ

, M = diag(M1, . . . , MN ) and

where ∆ =

(cid:32)

(cid:32)

(kσ + γkζ) − M sat

(cid:33)

(cid:32)

M

kˆσ1 + γk ˆζ1

kˆσN + γk ˆζN

sat = col

sat

, . . . , sat

MN
The observer scaled estimation error is defined as

M1

(cid:33)(cid:33)

.

ζi − ˆζi




δi =

and ηi = σi − ˆσi

49

and using the change of coordinates x = ˜P z and y = T ¯y the closed-loop system under the

output feedback controller (3.15) is given by

˙z1 = R1 ¯y
˙˜z = −γ ˜J ˜z + R2 ¯y

(3.16a)

(3.16b)

(3.16c)

(3.16d)

µ ˙¯y = ˜A¯y + kE ¯y + µγR3 ¯y + µγ2R4 ˜z + R5∆


 ˙ψ = A0ψ + (
/µ)

0

−LAT ¯y + L∆





−α1I

I

−α2I 0N×N

where A0 =

 is Hurwitz, ψ = col(δ, η), δ = col(δ1, . . . , δN ), η = col(η1, . . . , ηN )

and R5 is some matrix which is bounded uniformly in k.

Next we define the Lyapunov functions Vψ = ψT Hψ, where H = HT > 0 is the solution to

the Lyapunov equation HA0 + AT

0 H = −I.

We now state the main Theorems of the chapter.

Theorem 3.2: Consider the system (3.1) with the output feedback controller (3.15) and the

high-gain observer (3.14). Let S be a compact set in the interior of Γ and Q be any compact
set of R2N +1. Let (˜z(0), ¯y(0)) ∈ S, and (z1(0), ˆζ(0), ˆσ(0)) ∈ Q, and λ = max
, then
there exists λ∗ > 0, such that for every 0 < λ ≤ λ∗, the trajectories of the closed-loop system
(3.16) are bounded for all t ≥ 0, and

µ, 

µ

(cid:110)

(cid:111)

(cid:2)xi(t) − xj(t)(cid:3) = 0,

lim
t→∞

for i (cid:54)= j and i, j = 1, . . . , N

(3.17)

50

Proof : It proceeds in three steps:
Step 1. We show that there exist positive constants κ1 and ˜λ1 such that for ˜λ1 ≤ λ the set
Λ = {V ≤ c} × {Vψ ≤ κ1(
/µ)2} is positively invariant.
Step 2. We show that for any bounded (ˆζ(0), ˆσ(0)) and any (˜z(0), ¯y(0)) in the interior of
{V ≤ c} there exists ˜λ2 > 0, such that for λ ≤ ˜λ2, the trajectory (˜z(t), ¯y(t), ψ(t)) enters the
set {V ≤ c} × {Vψ ≤ κ2(
/µ)2} in finite time T1(
) with lim

→0
Step 3. We show that there exists ˜λ3 such that for λ ≤ ˜λ3, the system (3.16b)-(3.16d) is

T1(
) = 0.

exponentially stable from which (3.17) follows.
We show the first step by calculating the derivatives of V and Vψ on the boundaries {V = c}
and {Vψ = κ1(
/µ)2}, respectively. The saturation is no longer effective when the trajectory

is in Λ, from which we have ∆ = kη + 
γkδ. By taking the time derivative of Vψ along

(3.16d) we have


 ˙Vψ = −ψT ψ +

2

µ

ψT H

With λ ≤ 1, we have



0

−LAT ¯y + kLη + 
γkLδ




 ˙Vψ ≤ −||ψ||2 + (
/µ)q1||ψ||2 + (
/µ)q2||ψ||

where q1 and q2 are some positive constants. For 
/µ ≤ 1/(2q1) we have


 ˙Vψ ≤ −(1/2)||ψ||2 + (
/µ)q2||ψ||

51

Thus,


 ˙Vψ ≤ −(1/4)||ψ||2, ∀ ||ψ|| ≥ 4(
/µ)q2

Taking κ1 = 16λmax(H)q2

2 ensures that


 ˙Vψ ≤ −1
4

||ψ||2, ∀ Vψ ≥ κ1(
/µ)2

From which we conclude that for sufficiently small λ the set {Vψ ≤ κ1(
/µ)2} is positively
invariant as ˙Vψ ≤ 0 on the boundary Vψ = κ1(
/µ)2.
The derivative of V , for all ψ ∈ {Vψ ≤ κ1(
/µ)2}, satisfies

˙V = −γ ˜zT [ ˜J + ˜J T ]˜z + 2˜zT R2 ¯y − (1/µ)[˜yT (2W + kI)˜yT + b¯y2

1] − (1/µ)bk¯y1 ¯L12 ˜y

+ 2γ ¯yT P R3 ¯y + 2γ2 ¯yT P R4 ˜z + 2¯yT P R5∆

Similar to the proof of Lemma 3.2, it can be shown that for sufficiently small µ

˙V ≤ −b2||˜z||2 − b3||¯y||2 + (
/µ)q3||¯y||

for some positive constants b2, b3, and q3. From which we have

˙V ≤ −b4||Θ||2 + (
/µ)q3||Θ||

where Θ = col(||˜z||,||¯y||) and b4 = min{b2, b3}. Thus,

˙V ≤ −b4
2

||Θ||2, ∀ ||Θ|| ≥ 

µ

(2q3/b4)

52

˙V is negative on the boundary {V = c}, for sufficiently small λ. Therefore, there

Therefore,
exists ˜λ1 > 0, such that for all λ ≤ ˜λ1 the set Λ is positively invariant.
In the second step, because (˜z(0), ¯y(0)) lie in the interior of the set {V ≤ c}, and the right-

hand-side functions of (3.16b)-(3.16c) are bounded uniformly in 
, there exists a time ˜T1 > 0,
such that (˜z(t), ¯y(t)) ∈ {V ≤ c} for all t ∈ [0, ˜T1]. For ψ(0) /∈ {ψT Hψ ≤ κ1(
/µ)2}, it can

be seen

Vψ(t) ≤ q4e−q5t/
||ψ(0)||2

where q4 = λmax(H), q5 = 1/(4λmax(H)). Considering ψ(0) = O(1/
), we have

Vψ(t) ≤ (q6/
2)e−q5t/


for some q6 > 0. Since 
 ≤ 
/µ, for λ ≤ 1, the time taken by ψ(t) to reach the set
{ψT Hψ ≤ κ1(
/µ)2} can be estimated by the more conservative time T1(
) when Vψ = κ1
2,

which is given by T1 = (
/q5) ln(cid:0)q6/(κ1
4)(cid:1). By l’Hˆopital’s rule it can be shown that

T1(
) = 0. Therefore, there exists ˜λ2 > 0, such that for all λ ≤ ˜λ2, we can ensure
lim

→0
T1(
) < (1/2) ˜T1, which implies (˜z(t), ¯y(t), ψ(t)) ∈ {V ≤ c} × {Vψ ≤ κ1(
/µ)2} for all
t ≥ T1(
).

In the third step we show the exponential stability of the system (3.16b)-(3.16d), by forming

the composite Lyapunov function as Vc = V + Vψ and analyzing the system inside Λ. For

sufficiently small µ the time derivative of Vc along (3.16b)-(3.16d) is given by

 Ξ

0
− q8
2µ
− q1
µ

−q7
2
1





˙Vc ≤ −ΞT

b2

0

0
0 −q7
2

b3
− q8
2µ

53

where q7, q8 are positive constants, and Ξ = col(||˜z||,||¯y||,||ψ||).
The 2×2 principal minor is positive since b2 and b3 are positive. Choose 
/µ small enough to

make the determinant positive, which makes the above matrix positive definite. Therefore,
we can conclude there exists ˜λ3 > 0, such that for all λ ≤ ˜λ3,

lim
t→∞ ˜z(t) = 0,

lim
t→∞ ¯y(t) = 0 and lim

t→∞ ψ(t) = 0.

(3.18)

Because of the exponential stability of the system (3.16b)-(3.16d), z1(t) remains bounded

and all the agents converge to the consensus value

lim
t→∞ z1(t) = constant

and by choosing λ∗ = min{˜λ1, ˜λ2, ˜λ3, 1} we conclude that (3.17) follows.
(cid:4)
Theorem 3.3: Consider that Theorem 3.2 holds and let x∗(t) be the solution of the closed-
loop system under state feedback (3.6) with x∗(0) = x(0). Then, given any Υ > 0 there exists
λ∗∗ > 0 such that for all λ ∈ (0, λ∗∗)

(cid:107)x(t) − x∗(t)(cid:107) ≤ Υ ∀ t ≥ 0,

(3.19)

Proof : In this Theorem we show the solution x(t) under the output feedback control ap-
proaches the solution x∗(t) under the state feedback control which illustrates the performance

recovery property of the output feedback controller. We first show this in the transformed
coordinates z = ˜P−1x, where z(t) = col(z1(t), ˜z(t)), is the solution of the equations (3.16a)

and (3.16b). The system in the transformed coordinates under the state feedback control is

54

given by

˙z1 = 0

˙˜z = −γ ˜J ˜z

(3.20a)

(3.20b)

1(0) = z1(0).

z∗
1(t) and ˜z∗(t) are the solutions of the equations (3.20a) and (3.20b), with initial conditions
˜z∗(0) = ˜z(0) and z∗
The time interval is split into [0, T1(
)] and [T1(
),∞) and it is shown that ||z(t)−z∗(t)|| ≤ ˜Υ
√
2|| ˜P||). This is achieved by showing that
holds for each time interval, where ˜Υ = Υ/(
1(t)| ≤ ˜Υ and ||˜z(t) − z∗(t)|| ≤ ˜Υ hold for each time interval. Showing |z1(t) −
|z1(t) − z∗
z∗
1(t)| ≤ ˜Υ is more involved as z1(t) is in the direction of the zero eigenvalue of ˆL.
Since ˜z(0) is in the interior of {V ≤ c} we have (cid:107)˜z(t) − ˜z(0)(cid:107) ≤ ˜k1t during the interval
[0, T1(
)], where ˜k1 is a positive constant. Similarly, it can be shown that (cid:107)˜z∗(t)−˜z∗(0)(cid:107) ≤ ˜k1t
during the same time interval. By using ˜z(0) = ˜z∗(0), we have

(cid:107)˜z(t) − ˜z∗(t)(cid:107) ≤ 2˜k1T1(
), ∀ t ∈ [0, T1(
)].

Since T1(
) → 0 as 
 → 0, therefore we have

˜z(t) − ˜z∗(t) = O(T1(
)), ∀ t ∈ [0, T1(
)].

(3.21)

By using continuous dependence of the solutions of differential equations on initial conditions
and parameters and the exponential stability of ˙˜z = −γ ˜J ˜z we conclude that [91, Theorem

55

9.1],

˜z(t) − ˜z∗(t) = O(T1(
)), ∀ t ≥ 0

Therefore, for sufficiently small 
, we have

||˜z(t) − ˜z∗(t)|| ≤ ˜Υ, ∀ t ≥ 0

(3.22)

Next we integrate ˙z1(t) − ˙z∗

1(t) and by using z1(0) = z∗

1(0),

(cid:90) t

0

z1(t) − z∗

1(t) =

R1 ¯y(τ )dτ

During the period [0, T1(
)], |R1 ¯y(τ )| ≤ ˜k2, where ˜k2 is a positive constant independent of 

and µ. Since T1(
) → 0 as 
 → 0, therefore for sufficiently small 
 we have

z1(t) − z∗

1(t) = O(T1(
)),∀ t ∈ [0, T1(
)].

For t ≥ T1(
), the system (3.16b)-(3.16d) represents a three-time scale system where ψ is

the fast variable and (˜z, ¯y) are the slow variables. Following [85, Chapter 2], we perform a

change of coordinates χ = ψ + D1 ˜z + D2 ¯y, for some matrices D1, D2, which separates the

fast variable ψ from the slow ones (˜z, ¯y), and θ = ¯y + S ˜z, for some matrix S, to separate the

56

intermediate system ¯y from the slow one ˜z, from which we have

˙z1 = O(λ)˜z + R1θ

(cid:104)−γ ˜J + O(λ)
(cid:105)

˙˜z =

˜z + R2θ

µ ˙θ = [ ˜A + O(λ)]θ + R6χ


 ˙χ = (A0 + (
/µ)R7)χ

(3.23a)

(3.23b)

(3.23c)

(3.23d)

for some matrices R6, R7. For sufficiently small λ, the solution of (3.23c) and (3.23d) for
t ≥ T1(
) is bounded by

||χ(t)|| ≤ ˜k3e−(r1/
)(t−T1(
))

||θ(t)|| ≤ ˜k4e−(r2/µ)(t−T1(
))

for 
 << µ and some positive constants r1, r2, ˜k3 and ˜k4 independent of λ.
Defining φ(t) = ˜z(t) − ˜z∗(t) and by differentiating it we have

˙φ(t) = (−γ ˜J + O(λ))φ(t) + O(λ)˜z∗(t) + R2θ(t).

Integrating ˙φ(t) from time T1(
) we have

φ(t) = e(−γ ˜J+O(λ))(t−T1(
))φ(T1(
)) +

t(cid:90)

T1(
)

e(−γ ˜J+O(λ))(t−τ )[O(λ)˜z∗(τ ) + R2θ(τ )]dτ.

From (3.21), we have φ(T1(
)) = O(T1(
)), and therefore it can be shown that

(cid:107)φ(t)(cid:107) ≤ (O(T1(
)) + O(λ))e−r3(t−T1(
))

(3.24)

57

for all t ≥ T1(
), where r3 is a positive constant independant of λ. Next integrating ˙z1(t) −
˙z∗
1(t) from time T1(
) and by substituting ˜z(t) = φ(t) + ˜z∗(t), we have

t(cid:90)

[O(λ)φ(τ ) + O(λ)˜z∗(τ ) + R1θ(τ )] dτ.

z1(t) − z∗

1(t) = [z1(T1(
)) − z∗

1(T1(
))] +

T1(
)

1(T1(
)) = O(T1(
)), it can be shown that z1(t)− z∗
Using z1(T1(
))− z∗
for all t ≥ T1(
). Therefore, for all λ ∈ (0, λ∗∗), we can conclude that

1(t) = O(T1(
)) + O(λ)

|z1(t) − z∗

1(t)| ≤ ˜Υ, ∀ t ≥ 0.

(3.25)

From which we conclude that ||z(t)− z∗(t)|| ≤ √
we have x(t) − x∗(t) = ˜P [z(t) − z∗(t)]. By taking the norm we have

2 ˜Υ and by using the transformation x = ˜P z

||x(t) − x∗(t)|| ≤ || ˜P|| · ||z(t) − z∗(t)|| ≤

√

2|| ˜P|| ˜Υ

from which (3.19) follows.

(cid:4)

Theorem 3.2 shows that the agents achieve consensus under the output feedback controller

(3.15), while Theorem 3.3 shows x(t) under the output feedback controller approaches the
solution x∗(t) under the state feedback controller which illustrates the performance recovery

property of the output feedback controller.

3.5 Cyclic Pursuit on Plane

Motivated by robotic coordination problems we take the example of cyclic pursuit [92]. We

consider N robots on a plane with positions xi, for i = 1, . . . , N , and moving according to

58

˙xi = vi, where vi are the velocity commands. The objective of the robots is rendezvous at a

common point (while using only onboard sensors). A simple strategy to achieve rendezvous

is cyclic pursuit: where robot i pursues the next robot i + 1. In other words, the information
available to each robot is given by ζi = xi−1 − xi for i = 1, . . . , N .

(Here we consider

x0 = xN ). We can arrange the nodes such that the graph for this network of N robots is

represented by a circulant directed graph. Fig. 3.1 demonstrates the graph.

Figure 3.1: A directed circulant graph

We take the number of nodes to be 100. The initial conditions of the agents are chosen

to be in the range [1, 100]. The circular directed graph is balanced and strongly connected

and as a result the convergence rate is given by λs(Ls) = 0.0020, where λs is the smallest

non-zero eigenvalue of Ls = (L + LT )/2; see [19]. We simulate the system with the PD

controller with γ = 3 and k = 20 and the standard controller (3.3) with ks = 600 in order to

match the convergence rate of the system under the PD controller. The saturation levels are
chosen as Mi = ±170, for i = 1, . . . , 100. The saturation levels are chosen from simulations

to see the maximal values that the state trajectories (3.12) would take when using the state

PD controller.

In Fig. 3.2, xhgc denotes the state under high-gain consensus controller, xpd denotes the

state under PD controller (3.5), xpdo denotes the state under the output feedback controller

59

Figure 3.2: Agent 4 state

Figure 3.3: Agent 4 control signal

(3.15). We compare the state and control signals of agent 4 for the high-gain consensus

controller, the state feedback PD controller, and the output feedback PD controller. From

Fig. 3.2, the state under the high-gain consensus controller xhgc oscillates as it approaches

the consensus value. The oscillations are due to the presence of the imaginary component

in the eigenvalues of L. The state under the state feedback PD controller xpd does not

have any oscillations and achieves the same consensus value as in the case of a high-gain

consensus controller. The state under the output feedback PD controller xpdo is shown for

60

0123456789Time253035404550556065x4xhgcxpdxpdo, µ = 0.1, ǫ = 0.001xpdo, µ = 0.01, ǫ = 0.00010123456789Time-60-50-40-30-20-10010u4updupdo, µ = 0.1, ǫ = 0.001updo, µ = 0.01, ǫ = 0.00010246810-50000500010000uhgcFigure 3.4: Agent 4 control signal with µ = 0.1, 
 = 0.0001

two different values of 
 and µ, from Fig. 3.2 it can be seen that as 
 and µ decrease the

trajectory under output feedback approaches, the one under state feedback as predicted by

Theorem 3.3 and this trend is representative for all other agent trajectories.

In Fig. 3.3, uhgc denotes the control signal of the high-gain consensus controller, upd

denotes the control signal of the PD controller (3.5), updo denotes the control signal of the

output feedback controller (3.15). In Fig. 3.3, the control signal of agent 4 under control

uhgc is represented by a subfigure due to the large magnitude difference. As seen in Fig. 3.3

the control signal under the high-gain consensus controller is much higher than the proposed

controller for the same convergence rate. Finally, Fig. 3.4 shows the control signal of agent

4 with different values of k. We plot the control signals in the linear-log scale for different

values of k to improve the readability of the figure. It can be seen from the figure increasing

k does not increase the magnitude of the control signal of agent 4.

61

10-1010-810-610-410-2100102Time(log)-20-15-10-505u4k = 10k = 50k = 100k = 1503.6 Tree Graphs

In this section, we consider two kinds of self-similar graphs, namely, the Tree-Like Fractal

graph and the Vicsek Fractal graph. Self-similar graphs have a prescribed way of increasing

the graph size such that the structure of the graph is preserved. Other examples of self-

similar graphs include Torus and lattice graphs, where 1-D lattice graphs are line graphs

used in the platooning of vehicles [35].

Tree-Like Fractal graphs are generated in an iterative manner, starting from the first

generation, which consists of two nodes connected by an edge. Each family of these tree-like

fractal graphs is parameterized by a positive integer m. The procedure of creating the graph

of generation g+1 starts by replacing each edge with a path of length two in the graph of

generation g, thus creating a new node in the process. After that, m new nodes are added

to the newly created node; see [93]. Fig. 3.5 represents the first three generations of the

Tree-like fractal graphs with m = 2.

Figure 3.5: Tree-like Fractal graphs

Vicsek fractal graphs are parameterized by a positive integer d. The star graph with

d + 1 nodes is the first generation (g=1) graph. The graph for the next generation (g+1) is

generated from the graph of generation g. A detailed procedure for constructing the graph

of generation g+1 from the graph of generation g is given in [93]. Fig. 3.6 represents the

62

first three generations of the Vicsek fractal graphs for d = 2. Both graphs have an increasing

Figure 3.6: Vicsek Fractal graphs

tree diameter (greatest distance between any two nodes in a graph) as the generation of the

graph increases. From [94], for a tree with growing diameter the second smallest eigenvalue

of the Laplacian is bounded by

λ2(L) ≤ 2

(cid:18)

1 − cos

(cid:18) π

(cid:19)(cid:19)

dg + 1

where dg represents the diameter of the graph. Therefore λ2(L) decreases with increase in

the diameter of the graph.

The following tables illustrate the performance of the proposed controller on Tree-Like

Fractal graphs and Vicsek Fractal graphs with different network sizes.

63

Table 3.1: Laplacian Eigenvalues of Tree-like fractal graphs

N

2

5

17

65

g

1

2

3

4

λ2(L)

λ2( ˆL),

λ2( ˆL),

k = 20

k = 200

,

λ2( ˆL)
λ2(L)
k = 20

,

λ2( ˆL)
λ2(L)
k = 200

2

1

0.9756

0.9975

0.4878

0.4988

0.9524

0.9950

0.9524

0.9950

0.2087

0.8067

0.9766

3.8654

4.6794

0.0314

0.3859

0.8627

12.2898

27.4745

Table 3.1 shows the performance of the proposed controller with two different values of

k. As seen from the table as the size of the Tree-Like Fractal graph increases the second

smallest eigenvalue of the graph λ2(L) decreases. With increase in network size for higher

value of k the ratio between λ2( ˆL) and λ2(L) significantly increases.

Table 3.2: Laplacian Eigenvalues of Vicsek fractal graphs

N

5

25

121

g

1

2

3

λ2(L)

λ2( ˆL),

λ2( ˆL),

k = 60

k = 300

,

λ2( ˆL)
λ2(L)
k = 60

,

λ2( ˆL)
λ2(L)
k = 300

1

0.9836

0.9993

0.9836

0.9993

0.0692

0.8059

0.9540

11.646

13.78

0.0053

0.2396

0.6117

45.2075

115.4151

Table 3.2 shows the performance of the proposed controller with different size of Vicsek

Fractal graph with two different values of k. From the table for N = 121, the ratio between

λ2( ˆL) and λ2(L) is significantly higher for k = 300 than the other cases.

64

3.7 Conclusion

In this chapter we proposed a new consensus algorithm which is scalable as with increase in

the network size the eigenvalues of the closed-loop Laplacian matrix remain preserved. This

has benefits as in certain classes of graphs with increase in network size the convergence rate

becomes slow and to achieve fast consensus a large control effort is required.

We simulated the proposed controller on a circular, directed graph with 100 nodes. The

magnitude of the proposed controller compared to the high-gain consensus controller for

achieving the same convergence rate, was much lower. We also considered two classes of

graphs, which show the trend of decrease in the second smallest eigenvalue of the graph

Laplacian with an increase in network size. We showed the benefit of using the proposed

controller on these networks with different sizes.

65

Chapter 4

Practical Scalable Synchronization in

Leader-Follower Networks

4.1

Introduction

For a network of linear agents, if the performance measure degrades with an increase in

network size, then this is because the smallest eigenvalue of the grounded Laplacian matrix

approaches zero. We will discuss the effect of this behavior on some standard nonlinear

control techniques, which rely on local relative information. Therefore, we will exclude, [55]

from our discussion as it uses local output information.

Synchronization was achieved for homogeneous (identical dynamics) second-order non-

linear systems in [95] and [96], using only relative position and velocity feedback. However,

the gain of the controller is inversely proportional to the smallest eigenvalue of the grounded

Laplacian. As this quantity decreases with an increase in network size, the controller gain

becomes very large.

In [53] and [97], synchronization was achieved for homogeneous and

heterogeneous nonlinear agents, respectively. The feedback gain matrix in these methods

depends on the solution of a parametric algebraic Riccati equation (PARE), where the pa-

rameter is a lower bound on the real parts of the eigenvalues of the graph Laplacian. There-

fore, as the second smallest eigenvalue of the graph Laplacian approaches zero, it implies

66

that the parameter also approaches zero, which in turn makes the controller gain very small.

As a result, synchronization is achieved in a very long time.

In [61], practical synchronization is achieved in a network of nonlinear heterogeneous

agents by choosing the gain of the diffusive coupling sufficiently large. Other approaches

like [98], [99], [100] involve the use of adaptation where the nonlinearity is compensated using

neural networks. For undirected graphs, the controller gain in these algorithms is inversely

proportional to the smallest eigenvalue of the grounded Laplacian. A similar conclusion can

be drawn for the controller gain in [57], where the controller gain is required to be greater

than a constant that is inversely proportional to the smallest eigenvalue of the grounded

Laplacian.

As discussed above, decrease in the smallest eigenvalue of the grounded Laplacian, has an

adverse effect on the controllers of nonlinear systems. Therefore, in this chapter, we design

scalable consensus algorithms [101] for leader-follower networks of second-order nonlinear

systems. We consider second-order systems as they represent models of real-world systems

akin to platooning of vehicles [35], power systems with second-order swing dynamics [65],

Euler-Lagrange systems [99], and longitudinal vehicle dynamics [102].

67

4.2 Problem Setup

4.2.1 Class of Systems

We consider a network of N second-order nonlinear heterogeneous agents given by

˙xi = vi
˙vi = fi(xi, vi) + wi(t) + biui, ∀ i = 1, . . . , N.

(4.1a)

(4.1b)

where xi ∈ R is the position, vi ∈ R is the velocity, ui ∈ R is the control, fi(xi, vi) represents

the nonlinearity associated with each system, wi(t) represents a time-varying disturbance.

The system (4.1) represents models of real world systems like power systems [65], and lon-

gitudinal vehicle dynamics [102]. We make the following assumptions.

Assumption 4.1: The communication topology is given by a strongly connected weighted
directed graph G.

Assumption 4.2: The functions fi are unknown and continuously differentiable for all
(x, v) ∈ D where D is some domain containing the origins.

Assumption 4.3: The disturbances wi(t) and its derivative ˙wi(t) are unknown and bounded.
Assumption 4.4: The control coefficient bi is an unknown constant and satisfies bi ≥ ¯bi > 0.

4.2.2 Leader

In applications of multi-agent systems like platooning, power systems, etc., the agents are

required to follow or synchronize to a desired trajectory. The trajectory is generated by a

leader and, to reduce the information flow, it is assumed that only a subset of the agents

68

receives information from the leader. The leader dynamics is defined as

˙x0 = v0

˙v0 = f0(x0, v0) + u0(t, x0, v0)

(4.2a)

(4.2b)

The leader (4.2) with the control input u0(t, x0, v0) is used to generate trajectories depending

on the required objective. We do not require the control of the leader to be known to the

agents.

Assumption 4.5: The functions f0, u0 are continuously differentiable and the closed-loop
trajectories of the leader (x0, v0) belongs to a known compact invariant set W ⊂ R2.

Next we discuss the information exchange between the agents which is of the following form:

N(cid:88)
N(cid:88)

j=1

xri =

vri =

aij(xj − xi) + di(x0 − xi),

aij(vj − vi) + di(v0 − vi)

(4.3a)

(4.3b)

j=1

where aij ≥ 0, xri defines the relative position exchange, vri defines the relative velocity

exchange and di > 0 is satisfied at least for one agent. We define the synchronization error

as

exi = xi − x0, evi = vi − v0

69

for all i = 1, . . . , N . From which the information exchange can be redefined as

N(cid:88)
N(cid:88)

j=1

xri =

vri =

aij(exj − exi) − diexi,

aij(evj − evi) − dievi,

(4.4a)

(4.4b)

j=1

which in matrix form can be written as

xr = −LGex, vr = −LGev

where LG = L + D is the grounded Laplacian matrix [36], D = diag(d1, . . . , dN ), ex =

col(ex1, . . . , exN ) and ev = col(ev1, . . . , evN ).

The expanded graph is given by ¯G = ( ¯V, ¯E, ¯A).

Assumption 4.6: There is a directed path from the leader to all the agents in the expanded
graph ¯G.

Remark 4.1: The above assumption is standard to achieve synchronization in a network of

agents with a leader.

Under Assumption 4.1 and 4.6 we have the following properties

(i) L is an M-matrix [6] beacause it can be decomposed into L = ˜dI − ˜A, for some
nonnegative matrix ˜A and ˜d ≥ ρ( ˜A), where ρ(·) denotes the spectral radius of a

matrix.

(ii) LG is an M-matrix, since L is an M-matrix and D = diag(di) ≥ 0.

(iii) LG is nonsingular and all its eigenvalues have positive real parts [10]. Therefore, LG

is a nonsingular M-matrix.

70

The objective of this chapter is to design a controller using the information structure

(4.4) to achieve

for i = 1, . . . , N .

t→∞ exi(t) = 0, and lim
lim

t→∞ evi(t) = 0

(4.5)

Remark 4.2: In this chapter we will achieve practical synchronization where the synchro-

nization error can be made arbitrarily small by tuning a control and observer parameter,

respectively.

We transform system (4.1) into the error coordinates

˙exi = evi
˙evi = fi(exi + x0, evi + v0) + wi(t) − f0(x0, v0) − u0(t, x0, v0) + biui

(4.6a)

(4.6b)

for i = 1, . . . , N .

4.3 Controller Design With Full Information

4.3.1 Controller Design

In this section we assume that the relative velocity derivatives are available for feedback and

we design the following PD control for each agent, which is given by

us
i = kxxri + kvvri + kdσi

(4.7)

71

where us

i denotes the controller designed with full information, kx, kv, kd are positive con-

stants to be chosen later, σi is defined as

N(cid:88)

σi = ˙vri =

aij( ˙evj − ˙evi) − di ˙evi,

which in matrix form can be written as σ = −LG ˙ev, where σ = col(σ1, . . . , σN ). We assume

j=1

that σi is available for feedback. Substituting for σi in (4.7) we have,

N(cid:88)

us
i = kxxri + kvvri + kd

aij( ˙evj − ˙evi) − kddi ˙evi

j=1

The above controller is distributed in nature as it relies only on local information. Next we

write the equations in matrix form,

(I + kdLGB)us = kxxr + kvvr − kdLG(f + w) + kdLG1(f0 + u0)

where f = col(f1, . . . , fN ), B = diag(b1, . . . , bN ), w = col(w1, . . . , wN ) and us = col(us

1, . . . , us

N ).

The terms f , f0, u0 and w are written without their arguments for ease of notation.

Lemma 4.1: Suppose Assumption 4.6 holds, then the matrix (I + kdLGB) is nonsingular

with eigenvalues having positive real parts.

Proof: The matrix (I + kdLGB) is similar to the matrix (I + kdBLG), this follows by pre-
multiplying (I + kdLGB) by B and then post-multiplying it by B−1, where B is invertible

from Assumption 4.4. The matrix BLG can be thought of as a grounded Laplacian matrix

where each row is scaled by bi. Therefore, it follows that the eigenvalues of BLG remain on

72

the right half-plane for any positive diagonal matrix B and we have

λi(I + kdBLG) = 1 + kdλi(BLG),

where Re (λi(BLG)) > 0, from which we conclude that (I+kdLGB) is invertible and its eigen-

values have positive real parts.

(cid:4)

Choosing kd = k, kx = γ1k and kv = γ2k, where γ1 and γ2 are positive constants, we have

the control as

us = −γ1B−1 ˆLGex − γ2B−1 ˆLGev − B−1 ˆLG[f + w − 1(f0 + u0)]

(4.8)

where ˆLG represents the Virtual Grounded Laplacian matrix, defined as

(cid:19)−1

(cid:18) 1

k

ˆLG =

I + ¯LG

¯LG

(4.9)

where ¯LG = BLG is the scaled grounded Laplacian matrix from the proof of Lemma 4.1.
Lemma 4.2: Let the expanded graph ¯G satisfy Assumption 4.6, then for sufficiently large

k,

Re(λi( ˆLG)) = 1 − 1
k

Im(λi( ˆLG)) = − 1
k

Im

(cid:18) 1
(cid:18) 1

Re

λi( ¯LG)

λi( ¯LG)

(cid:19)

(cid:19)

+ O

(cid:18) 1
(cid:19)
(cid:18) 1

k2

k2

+ O

(cid:19)

,

,

(4.10a)

(4.10b)

for i = 1, . . . , N, where λi( ¯LG) and λi( ˆLG) are the eigenvalues of ¯LG and ˆLG, respectively.

The proof of this Lemma can be done by repeating the steps in Theorem 3.1 of chapter 3

73

and therefore it is omitted.

4.3.2 Closed-loop System Analysis

The system (4.6) with the controller (4.8) becomes

˙ex = ev

˙ev = −γ1 ˆLGex − γ2 ˆLGev + L1(f + w) − L11(f0 + u0)

(4.11a)

(4.11b)

where L1 = I − ˆLG = (I + k ¯LG)−1. Next we do a change of coordinates

ex = P ˜ex, and ev = P ˜ev

(4.12)

where ¯LG = P JP−1, J = blkdiag(J1, J2, . . . , Jm) is the Jordan block of ¯LG and m ≤ N is

the number of Jordan blocks. The system in the transformed coordinates becomes

˙˜ex = ˜ev

˙˜ev = −γ1 ˜J ˜ex − γ2 ˜J ˜ev + L2(f + w) − L21(f0 + u0)

(4.13a)

(4.13b)

where L2 = (I + kJ)−1P−1, ˜J =

I + J

J. The Jordan block ˜J contains eigenvalues

(cid:19)−1

(cid:18) 1

k

with strict positive real parts. For sufficiently large k, we have

˜J = I + O(1/k), and L2 = O(1/k).

74

Next we define the Lyapunov function Ve = eT W e, where W = W T > 0 is the solution to
the Lyapunov equation W A + AT W = −I, e = col(˜ex, ˜ev) and A is a Hurwitz matrix given

 0

I
−γ1I −γ2I

by

A =

 .

Theorem 4.1 Consider the system (4.6) with the controller (4.8). Let e(0) ∈ Ω. where
Ω = {Ve ≤ c}, for some c > 0, such that Ω ⊂ D. Then, there exists k∗ > 0, such that for all
k ≥ k∗, the trajectories of the closed-loop system are bounded for all t ≥ 0, and there exists

time T > 0, such that

(cid:19)
(cid:18) 1
(cid:18) 1
(cid:19)

k

k

, ∀ t ≥ T

, ∀ t ≥ T

x − 1x0 = O

v − 1v0 = O

(4.14a)

(4.14b)

Moreover, the control signal us(t) is bounded uniformly in k.

Proof: Taking the time-derivative of Ve along (4.13) we have

˙Ve ≤ −||e||2 +

||e||2 +

a1
k

a2
k

||e||(||f|| + ||w||) +

||e||(|f0| + |u0|)

a3
k

for some positive constants a1, a2, a3. For k ≥ k1 = 2a1, we have

˙Ve ≤ −1
2

||e||2 +

a2
k

||e||(||f|| + ||w||) +

||e||(|f0| + |u0|)

a3
k

75

Since fi(xi, vi) = fi(exi + x0, evi + v0) and the function fi is continuously differentiable, it
follows that for all e ∈ Ω and (x0, v0) ∈ W, we have

||f|| ≤ a4||e|| + a5|x0| + a6|v0| ≤ a4||e|| + a7

where a4, a5, a6 and a7 are positive constants. Moreover, since w(t) is bounded we have

˙Ve ≤ −(1/2)||e||2 + (a8/k)||e||2 + (a9/k)||e||

for some positive constants a8 and a9. For k ≥ k2 = 4a8, we have

˙Ve ≤ −(1/4)||e||2 + (a9/k)||e||

From which we have

˙Ve ≤ −(1/8)||e||2, ∀ ||e|| ≥ (8a9/k)

(4.15)

For all k ≥ k3 where k3 is a positive constant,

˙Ve < 0 on the boundary Ve = c and therefore

the set Ω is positively invariant, from which we can conclude that the trajectories of the
closed-loop system are bounded for all t ≥ 0. Finally, by choosing k∗ = max{k1, k2, k3}, we

can conclude that there exists time T > 0, such that

||e(t)|| = O(1/k), ∀ t ≥ T

Therefore, (4.14) follows from the above expression, which shows that the trajectories of the

agents become arbitrarily close to the trajectories of the leader (x0, v0).

Next we show that control signal us(t) is uniformly bounded in k. From (4.15), using [103,

76

Theorem 4.5] we have

||e(t)|| ≤

(cid:115)

(cid:26)

||e(0)|| · exp

(cid:18)

(cid:19)

(cid:27)

,

8a9
k

−t

16λmax(W )

λmax(W )
λmin(W )

max

(4.16)

The control signal is given by

us = B−1 ˆLG[− ˜Re − f − w + 1(f0 + u0)]

where ˜R = [γ1P γ2P ]. The norm of ||B−1|| is given by ||B−1|| = 1/˜b, where ˜b = min
1≤i≤N
and the norm of ˆLG is bounded uniformly in k as

bi

|| ˆLG|| ≤ ||P|| · || ˜J|| · ||P−1|| = a10 + O(1/k)

for some positive constant a10 independent of k. Since we have (x0(t), v0(t)) ∈ W and w(t)

is bounded, therefore the norm of the control signal satisfies

||us(t)|| ≤ a11||e|| + a12

for some positive constants a11 and a12, uniformly independent of k. Using (4.16) in the
(cid:4)

above inequality we can conclude that the control signal is bounded uniformly in k.

77

4.4 Output Feedback Controller

4.4.1 Observer Design

In the previous section we assumed σi is available for feedback, which is not true for practical

applications. Therefore, in this section we realize the proposed scalable controller (4.7) by

using a reduced-order high-gain observer. The stability analysis using a full-order high-gain

observer can be shown only with additional controller dynamics as shown in the previous

chapter. Since in this chapter we do not use additional controller dynamics, the stability of

the closed-loop system is shown using a reduced-order high-gain observer. However, using

reduced-order observers is at the expense of assuming that the initial state of the relative

sychronization errors is known. Each agent has measurement of relative state xri and relative

velocity vri locally. We estimate the relative velocity derivatives by adding a reduced-order

high gain observer to each agent in the network to estimate the signal σi = ˙vri. The driving

signal to the observer is vri. The reduced order high-gain observer [90] is constructed as

(cid:104)

˙φi = −p



(cid:105)

φi +

p



vri

, ˆσi = φi + (p/
)vri

(4.17)

where 
, and p are positive constants with 
 << 1 for i = 1, . . . , N . The transfer function of

the observer from vri to ˆσi is

s

(
/p)s + 1

. The output feedback controller is chosen as

ui = γ1kxri + γ2kvri + kˆσi

(4.18)

which in matrix form becomes

u = −γ1kLGex − γ2kLGev + kˆσ

78

where u = col(u1, . . . , uN ) and ˆσ = col(ˆσ1, . . . , ˆσN ).

4.4.2 Peaking

The estimation error is defined as ηi = σi − ˆσi, which satisfies


 ˙η = −pη + 
 ˙σ

where η = col(η1, . . . , ηN ) and ˙σ = −LG¨ev. After some simplifications the observer error

dynamics can be written as


 ˙η = −p[I + kLGB]η + 
∆

(4.19)

where ∆ = ∆( ˙w, ˙f0, ˙u0, ˙f , e). For all e ∈ Ω, (x0(t), v0(t)) ∈ W and bounded ˙w, the term ∆

is bounded by a constant independent of 
.
The output feedback control is given by u = −γ1kLGex − γ2kLGev + kσ − kη. Using
σ = −LG ˙ev, we arrive at

u = −γ1B−1 ˆLGex − γ2B−1 ˆLGev − B−1 ˆLG[f + w − 1(f0 + u0)] − k(I + kLGB)−1η (4.20)

Noting that η(0) = −LG[f (0) + w(0) − 1(f0(0) + u0(0))] − LGBu(0) − ˆσ(0), by substituting

u(0), we have

(I + kLGB)−1η(0) = −LG[f (0) + w(0)− 1(f0(0) + u0(0))] + γ1LG

+ LG

ˆLGev(0)
ˆLG[f (0) + w(0) − 1(f0(0) + u0(0))] − φ(0) + (p/
)LGev(0)

ˆLGex(0) + γ2LG

79

where we have used I−kLGB(I +kLGB)−1 = (I +kLGB)−1. The term LG
kLGB)−1LG = LG − (I + kLGB)−1LG, from which we have

ˆLG = kLGB(I +

(I + kLGB)−1η(0) = −(I + kLGB)−1LG[f (0) + w(0) − 1(f0(0) + u0(0))]
+ γ1[LG− (I + kLGB)−1LG]ex(0) + γ2[LG− (I + kLGB)−1LG]ev(0)− φ(0) + (p/
)LGev(0)

Multiplying the above equation by (I + kLGB), we have

η(0) = −LG[f (0) + w(0) − 1(f0(0) + u0(0))] + γ1kLGBLGex(0) + γ2kLGBLGev(0)

− (I + kLGB)φ(0) + (p/
)(I + kLGB)LGev(0)

Because η(0) = O(1/
), then during the transient period, the observer contains a term of
the form (1/
)e−¯a1t/
, for some ¯a1 > 0. This term is transmitted to the control as seen

in equation (4.20). By choosing φ(0) = γ1LGex(0) + γ2LGev(0) + (p/
)LGev(0), the initial

estimation error becomes

η(0) = −LG[f (0) + w(0) − 1(f0(0) + u0(0))] − γ1LGex(0) − γ2LGev(0),

which eliminates peaking and both the estimation error η(t) and the control u(t) become

bounded uniformly in 
 and k. Note that the parameters γ1, γ2, p, 
, and the initial conditions

LGex(0) and LGev(0) are locally available to each agent.

80

4.4.3 Analysis of the closed loop system

The system (4.6) with the controller (4.20) becomes

˙ex = ev

˙ev = −γ1 ˆLGex − γ2 ˆLGev + L1(f + w) − L11(f0 + u0) − L3η

(4.21a)

(4.21b)

where L3 = [(1/k)B−1 + LG]−1. Using the change of coordinates ex = P ˜ex, ev = P ˜ev, and
η = B−1 ˜η, we have

˙˜ex = ˜ev

˙˜ev = −γ1 ˜J ˜ex − γ2 ˜J ˜ev + L2(f + w) − L21(f0 + u0) − L4 ˜η

 ˙˜η = −p[I + k ¯LG]˜η + 
 ¯∆

(4.22a)

(4.22b)

(4.22c)

where L4 = [(1/k)I + J]−1P−1 and ¯∆ = B∆.

Next we recall the following Lemma.

Lemma 4.3, [104], [105]: A square matrix Am = [˜aij], where ˜aij ≤ 0 for i (cid:54)= j, is a
nonsingular M-matrix, if and only if one of the following equivalent conditions holds:

(i) There is a positive vector ˜x > 0 such that Am ˜x > 0

(ii) There is a positive vector ˜y > 0 such that AT

m ˜y > 0

(iii) All eigenvalues of Am have positive real parts

(iv) Am is nonsingular and A−1

m is nonnegative

81

Since ¯LG is a grounded Laplacian matrix with eigenvalues having positive real parts as
It follows that ¯LG is also a nonsingular M-matrix

shown in the proof of Lemma 4.1.

as premultiplying LG by a diagonal matrix does not change its structure of having outer

diagonal nonpositive elements and nonnegative diagonal elements.

The following Theorem constructs a Lyapunov function candidate for the system (4.22c).

Theorem 4.2, [104] : Suppose Assumption 4.6 holds. Let

˜x = col(˜x1, . . . , ˜xN ) = ¯L−1
G 1,
˜y = col(˜y1, . . . , ˜yN ) = ¯L−T
G 1,

˜P = diag(˜pi) = diag(˜yi/˜xi),

Q = ˜P ¯LG + ¯LT
G

˜P .

Then ˜P > 0 and Q > 0.

Proof : Since ¯LG is a nonsingular M-matrix, then by Lemma 4.3, ¯L−1
negative matrices with no zero rows. This implies ˜x, ˜y > 0 from which ˜P > 0. From which

G and ¯L−T

G are non-

we can conclude that Q has outer diagonal nonpositive elements and nonnegative diagnal

elements, since ˜P = diag(˜pi) > 0. Now

Q˜x = ˜P ¯LG ˜x + ¯LT
G

˜P ˜x

= ˜P 1 + ¯LT

G ˜y

= col(˜p1, . . . , ˜pN ) + 1 > 0

Therefore, from Lemma 4.3 we can conclude Q is a nonsingular M-matrix having eigenvalues

82

with positive real parts. Moreover, it is positive definite since Q = QT .

(cid:4)

We construct the Lyapunov function candidate for the system (4.22c) as V˜η = ˜ηT ˜P ˜η.

Theorem 4.3 Consider the system (4.6) with the output feedback controller (4.18) and the
reduced-order high-gain observer (4.17). Let S be a compact set such that S ⊂ Ω. Let
e(0) ∈ S, and φ(0) be chosen as φ(0) = γ1LGex(0) + (γ2 + (p/
))LGev(0)), then, there exists

∗ > 0, such that for every 0 < 
 ≤ 
∗, the trajectories of the closed-loop system are bounded
for all t ≥ 0, and there exists time ˜T > 0 such that

+ O (
) , ∀ t ≥ ˜T

+ O (
) , ∀ t ≥ ˜T

(4.23a)

(4.23b)

(cid:19)
(cid:18) 1
(cid:19)
(cid:18) 1

k

x − 1x0 = O

v − 1v0 = O

k

Moreover, the control signal u(t) is bounded uniformly in k and 
.

Proof : The proof proceeds in three steps:
Step 1. We show that there exists positive constants κ1 and 
1 such that for 
1 ≤ 
 the set
Υ = {eT W e ≤ c} × {˜ηT ˜P ˜η ≤ κ1
2} is positively invariant.

Step 2. We show that for ˆσ(0) = γ1LGex(0) + γ2LGev(0) and any e(0) in the interior of
{eT W e ≤ c}, there exists 
2 > 0, such that for 
 ≤ 
2, the trajectory (e(t), ˜η(t)) enters the
set {eT W e ≤ c} × {˜ηT ˜P ˜η ≤ κ1
2} in finite time T1(
) with lim
T1(
) = 0.

→0
Step 3. We show that there exists 
3 such that for 
 ≤ 
3, (4.23) holds.

We show the first step by calculating the derivatives of Ve = eT W e, and V˜η = ˜ηT ˜P ˜η on the
boundaries {Ve = c}, and {V˜η = κ1
2}, respectively. By taking the time derivative of V˜η

along (4.22c) we have


 ˙V˜η = −2p˜ηT ˜P ˜η − pk ˜ηT Q˜η + 2
˜ηT ˜P ¯∆

83

For all (x0(t), v0(t)) ∈ W, e(t) ∈ Ω and bounded ˙w, we have || ¯∆|| ≤ l1, where l1 is a positive

constant independent of 
. From which we have,


 ˙V˜η ≤ −p[2λmin( ˜P ) + kλmin(Q)]||˜η||2 + 
l2||˜η||

where l2 = 2l1|| ˜P||. Thus,


 ˙V˜η ≤ −α||˜η||2, ∀ ||˜η|| ≥ 
l3

where α = −(p/2)[2λmin( ˜P ) + kλmin(Q)] and l3 = l2/α. Taking κ1 = l2

3λmax( ˜P ) ensures

that


 ˙V˜η ≤ −α||˜η||2, ∀ V˜η ≥ κ1
2

From which we can conclude that ˙V˜η ≤ 0 on the boundary V˜η = κ1
2.
The derivative of Ve = eT W e satisfies

˙Ve ≤ −||e||2 +

||e||2 +

l4
k

l5
k

||e||(||f|| + ||w||) +

l6
k

||e||(|f0| + |u0|) + l7||e|| · ||˜η||

for some positive constants l4, l5, l6 and l7. For all (x0(t), v0(t)) ∈ W, e(t) ∈ Ω, bounded

w(t),

˙w(t) and ˜η ∈ {V˜η ≤ κ1
2}, we have

˙Ve ≤ −||e||2 +

||e||2 +

l8
k

l9
k

||e|| + 
l10||e||

where l8, l9 and l10 are positive constants. For k ≥ ˜k1 = 2l8, we have

˙Ve ≤ −(1/2)||e||2 +

||e|| + 
l10||e||

l9
k

84

From which we have,

˙Ve ≤ −(1/4)||e||2, ∀ ||e|| ≥ 4(l9/k + 
l10)

Therefore, for all k ≥ ˜k2 > 0 and 
 ≤ 
1 > 0,
the set {eT W e ≤ c} × {˜ηT ˜P ˜η ≤ κ1
2} is positively invariant.
In the second step, because e(0) lies in the interior of the set {eT W e ≤ c}, and the right-

˙Ve is negative on the boundary {Ve = c} and

hand-side function of (4.22a)-(4.22b) is bounded uniformly in 
, there exists a time ˜T1 > 0,
such that e(t) ∈ {eT W e ≤ c} for all t ∈ [0, ˜T1]. During this time period, we have ˜η(0) /∈
{˜ηT ˜P ˜η ≤ κ1
2}, it can be seen that

V˜η(t) ≤ l11e−l12t/
||˜η(0)||2

where l11 = λmax( ˜P ), l12 = α/(λmax( ˜P )). From the choice of initial condition of φ(0) =
−γ1xr(0) − γ2vr(0) − (p/
)vr(0), we have ˜η(0) = Bη(0) = B[σ(0) − ˆσ(0)], which is given by

˜η(0) = −BLG[f (0) + w(0) − 1(f0(0) + u0(0))] − γ1BLGex(0) − γ2BLGev(0)

For all (x0(t), v0(t)) ∈ W, e(t) ∈ Ω, bounded w(t), we can conclude that ˜η(0) = O(1), and

we have

V˜η(t) ≤ l13e−l12t/


(
/l12) ln(cid:0)l13/(κ1
2)(cid:1). By l’Hˆopital’s rule it can be shown that lim

for some l13 > 0. The time taken by ˜η(t) to reach the set {˜η ˜P ˜η ≤ κ1
2} is given by ˜T2 =
˜T2(
) = 0. Therefore,
there exists 
2 > 0, such that for all 
 ≤ 
2, we can ensure ˜T2(
) < (1/2) ˜T1, which implies


→0

85

(e(t), ˜η(t)) ∈ {Ve ≤ c} × {V˜η ≤ κ1
2} for all t ≥ ˜T2(
). Therefore, by choosing k∗∗ =
max{˜k1, ˜k2} and 
∗ = min{
1, 
2}, we conclude that all the trajectories of the closed-loop
system are bounded for all t ≥ 0 and there exists time ˜T > 0, such that

(cid:18) 1

(cid:19)

k

x − 1x0 = O

for all t ≥ ˜T .

+ O (
) , and v − 1v0 = O

+ O (
)

(cid:18) 1

(cid:19)

k

Next we show that the control signal u(t) is bounded uniformly in k. From (4.20), after

time t ≥ ˜T2(
), the control signal is given by

u = B−1 ˆLG[−γ1ex − γ2ev − f − w + 1(f0 + u0)] + O(
)

Therefore, following the same steps as in Theorem 4.1 it can be shown that ||u(t)|| for
all t ≥ ˜T2(
) is uniformly bounded in k. During the time period t ∈ [0, ˜T2(
)] since e(t) ∈
{eT W e ≤ c}, (x0(t), v0(t)) ∈ W, w(t) is bounded, and ˙V˜η ≤ −(α/(
λmax( ˜P )))V˜η. Therefore,

the norm of ˜η is given by

||˜η(t)|| ≤

(cid:115)

(cid:18) −αt

(cid:19)

2
λmax( ˜P )

λmax( ˜P )
λmin( ˜P )

||˜η(0)||exp

,∀ t ∈ [0, ˜T2(
)]

Since ˜η(0) = O(1), it implies ˜η(t) is bounded independent of 
 for all t ∈ [0, ˜T2(
)]. As shown
in Theorem 4.1 since || ˆLG|| is uniformly bounded in k and L3 = O(1), it follows that ||u(t)||
is uniformly bounded in k during the time period t ∈ [0, ˜T2(
)]. From which we conclude
that the control signal u(t) is uniformly bounded in k for all t ≥ 0.
(cid:4)

Theorem 4.4 Consider that Theorem 4.3 holds and let ¯e(t) and e(t) be the trajectories of

the closed-loop systems (4.13) and (4.21), respectively with ¯e(0) = e(0). Then, given any

86

Ξ > 0, there exists k∗∗∗ > 0, and for each k ∈ [k∗∗∗,∞), 
∗∗ = 
∗∗(k) exists such that for
each k ∈ [k∗∗∗,∞) and 
 ∈ (0, 
∗∗(k)],

||e(t) − ¯e(t)|| ≤ Ξ, ∀ t ≥ 0

(4.24)

Proof: The proof is done in three steps where we first show (4.24) during the time period
[ ˆT2,∞), where ˆT2 > 0 is some finite time followed by [0, ˆT1(
)] and then during [ ˆT1(
), ˆT2].
From Theorem 4.3, given any Ξ > 0, there exists k ≥ ˆk1 > 0 and ˆ
1 = 
1(k) > 0 such that
for each k ∈ [ˆk1,∞) and 
 ∈ (0, ˆ
1(k)]

||e(t)|| ≤ Ξ/2,

||¯e(t)|| ≤ Ξ/2, ∀ t ≥ ˆT2

Therefore, using ||e(t) − ¯e(t)|| ≤ ||e(t)|| + ||¯e(t)|| it follows that

||e(t) − ¯e(t)|| ≤ Ξ, ∀ t ≥ ˆT2

Next we show (4.24) during the time period [0, ˆT1(
)]. From Theorems 4.1 and 4.3, we know
that e(t) ∈ Ω and ¯e(t) ∈ Ω during the time period [0, ˆT1(
)]. Therefore, the right-hand-side

of (4.13a)-(4.13b) and (4.21a)-(4.21b) is bounded by a constant independent of 
. By using

e(0) = ¯e(0) we have

||e(t) − ¯e(t)|| ≤ 2b ˆT1(
),

for all t ∈ [0, ˆT1(
)], where lim

→0

ˆT1(
) = 0 and b is a positive constant. Therefore, there exists

87

k ≥ ˆk2 > 0 and ˆ
2 = ˆ
2(k) > 0 such that for each k ∈ [ˆk2,∞) and 
 ∈ (0, ˆ
2(k)], we have

||e(t) − ¯e(t)|| ≤ Ξ, ∀ t ∈ [0, ˆT1(
)]

Over the time interval [ ˆT2(
), ˆT2], equations (4.21a)-(4.21b) under output feedback is O(
)

perturbation from the corresponding model (4.13a)-(4.13b). Therefore, it follows from the

continuous dependence of solutions of differential equation on initial conditions and param-

eters [91, Theorem 3.5] that

||e(t) − ¯e(t)|| ≤ Ξ, ∀ t ∈ [ ˆT2(
), ˆT2]

Therefore, (4.24) follows by choosing k∗∗∗ = max{ˆk1, ˆk2} and 
∗∗ = min{ˆ
1, ˆ
2}.

(cid:4)

4.5 Examples

4.5.1

IEEE 300-Bus System

In this example we consider the synchronization problem on a network of oscillators on the

IEEE-300 Bus system. Fig. 4.1 which is modified from [106] represents the network. The

edge weights are taken to be 1 or 0 depending on whether an edge exists or not between two

nodes. The leader node 0 is connected to node with highest degree which is node 268. The

smallest eigenvalue of the grounded Laplacian is given by λmin(LG) = 0.0010. Therefore,

standard nonlinear control approaches will require a very high controller gain to achieve

synchronization.

88

Figure 4.1: Diagram of node connections for the IEEE 300-Bus System

The oscillator dynamics is given by

˙xi = vi
˙vi = −xi + gi(1 − x2

i )vi + ˜giui

(4.25a)

(4.25b)

where gi and ˜gi are constants. The constants are chosen to lie in the range gi ∈ [−5.5, 16.5],
˜gi ∈ [1, 3]. The leader dynamics is given by

˙x0 = v0,

˙v0 = −x0 − x2
0v0

The initial conditions of the leader are chosen as x0(0) = 2, v0(0) = −2 and the initial
conditions of the other oscillators are chosen to lie in the range xi ∈ [−1, 1], vi ∈ [−1, 1].

89

The controller parameters are chosen as γ1 = 20, γ2 = 20, k = 300, p = 1, 
 = 0.01.

Figure 4.2: State one synchronization error ex = x − x0

Figure 4.3: State two synchronization error ev = v − v0

Fig. 4.2 and Fig. 4.3 illustrate the synchronization error of the first state and second

state for the first 50 oscillators. From the figures it can be seen that the steady-state

synchronization error is very small.

90

0510152025Time-3-2.5-2-1.5-1-0.500.510152025-4-2024610-30510152025Time-0.500.511.522.5310152025-0.01-0.00500.0050.01Figure 4.4: Control signal of the oscillators

Figure 4.5: Log-Linear plot of the contol signal for different value of k

Fig. 4.4 represents the control signal of the first 50 oscillators. It can be seen from figure

that the control signal is bounded independent of the value of k. In order to better illustrate

the uniform boundedness of the control signal with respect to k, we use log-linear plot for

the control signal of the 200th oscillator for different values of k. From Fig. 4.5 it can be

seen that the control signal does not increase as the value of k increases.

91

0510152025Time-15-10-5051015202510-610-410-2100102Time (log)-5-4-3-2-1012k = 50k = 100k = 300k = 600Figure 4.6: Performance recovery with k = 300

Finally, Fig. 4.6 shows the results of Theorem 4.4, where the trajectory ex under the

output feedback controller approaches the trajectory ¯ex under the controller with full infor-

mation (4.7) as the value of 
 decreases. The trajectories ex and ¯ex are shown for the 100th

oscillator.

4.5.2 Platooning of Vehicles

We consider a platoon of vehicles which are described by their longitudinal dynamics given

by [102]:

˙˜xi = vi
mi ˙vi = −˜aiv2

i − ˜bimig cos θ − mig sin θ + di(t) + ui

(4.26a)

(4.26b)

where ˜xi = xi − x0d
denotes the position and velocity of the ith vehicle, mi denotes the mass of vehicle i, ˜ai

i denotes the desired distance of vehicle i from the leader, (xi, vi)

, x0d

i

is the drag factor, ˜bi is the rolling resistance coefficient, θ is the slope of the road, and

di(t) = ai

m sin(2πfit) denotes a time-varying disturbance acting on the vehicles, ui is the

control input. All the units are in SI system. We consider that the desired inter-vehicle

92

0246810Time-0.200.20.40.60.81 = 0.1 = 0.05 = 0.01distance between two adjacent vehicles is 50m and therefore x0d
i = 50i. The
i
system parameters are chosen as mi ∈ [1545, 1550], ˜ai ∈ [1, 2], ˜b = [0.01, 0.04], θ = 0.049,
fi ∈ [0.1, 0.9], g = 9.8 and ai

is chosen as x0d

m ∈ [1, 6].

The leader decides the synchronizing velocity of the platoon of vehicles and its dynamics is

given by

˙x0 = v0,

˙v0 = 0

Note that although x0 is unbounded, the Theorems of this chapter still apply because in

system (4.26), xi does not appear in the second equation. A more general class of second-

order system will be treated in the next chapter. Moreove, we will provide a rigorous analysis

of the closed-loop system.

The leader sends its information to the first vehicle. The initial condition of the leader is

chosen as x0(0) = 0, v0(0) = 25. We assume that the followers start at the desired inter-

vehicle distance and speed, which implies ˜xi(0) = 0 and vi(0) = 25 for i = 1, . . . , 100. The

minimum eigenvalue of the grounded Laplacian is given by λmin(LG) = 0.002. The controller

parameters chosen are γ1 = 40, γ2 = 40, k = 600, p = 1, 
 = 0.001.

Figure 4.7: Line graph representing platoon of vehicles

Fig. 4.7 illustrates the network topology. In the figure node 0 is the leader, while nodes

1, . . . , 100 are the followers.

Fig. 4.8 shows the position error of vehicle ten (˜x10), which is the difference between

the position of vehicle ten and the desired distance of vehicle ten from the leader. As seen

from the figure, the position error is small, and it approaches steady-state. Fig. 4.9 shows

93

Figure 4.8: Position error of vehicle ten, ˜x10 = x10 − x0d

10

Figure 4.9: Velocity error of vehicle ten, ˜ev10 = v10 − v0

the velocity synchronization error of vehicle ten. It can be seen from the figure that the

steady-state velocity synchronization error is small.

Figure 4.10: Control signal of vehicle ten

94

02004006008001000Time (s)-0.1-0.08-0.06-0.04-0.020˜x10(m)02004006008001000Time (s)-3-2-10123ev(m/s)×10-302004006008001000Time (s)-0.200.20.40.60.811.21.41.6u (m/s2)Fig. 4.10 shows the control signal of vehicle ten. The fluctuation in the control signal is

because it compensates for the time-varying disturbance di(t).

Remark 4.3: The string stability or network coherence [35] in a platoon of vehicles is also

an important performance measure. In this chapter we do not study coherence in a platoon

of vehicles. However, scalable coherence in a platoon of vehicles represented by second-order
linear systems has been studied in [107]. It is shown the H2-norm from the disturbance input

to the difference between the output of a vehicle to the average output of all the vehicles in

the network is scalable with respect to the network size.

4.6 Conclusion

In this chapter, we proposed a distributed scalable controller for the practical synchro-

nization of second-order nonlinear heterogeneous systems using high-gain observers. The

synchronization error can be made arbitrarily small by tuning a controller and observer pa-

rameter, respectively. The tuning of these parameters does not affect the magnitude of the

control signal as we show that it is uniformly bounded in these parameters. The system

performance remains almost invariant as the smallest eigenvalue of the grounded Laplacian

approaches zero, as the controller gain can be chosen sufficiently large without affecting the

magnitude of the control signal. This is unlike the case for standard nonlinear controller ap-

proaches where the magnitude of the controller signal is inversely proportional to the smallest

eigenvalue of the grounded Laplacian. We also showed the performance recovery property

of the output feedback controller. In other words, the trajectories under output feedback

can be made arbitrarily close to the trajectory under the controller using relative velocity

derivative information by tuning the controller and observer parameters, respectively. The

95

performance of the controller was shown for two large-scale nonlinear systems motivated by

practical applications.

96

Chapter 5

Practical Synchronization in

Networks of Nonlinear Heterogenous

Agents under Reduced Information

5.1

Introduction

More recently, the research in multi-agent systems is geared towards the synchronization

of agents with complex individual dynamics and reduced information exchange.

In this

chapter, we consider the practical state synchronization problem in a network of nonlinear

heterogeneous agents exchanging only relative output information. The class of systems

studied in the previous chapter is a special case of the one considered in this chapter. But

unlike the previous chapter, we do not focus on scalable synchronization but instead on

synchronization under reduced information exchange. The controller design approach used in

this chapter is different from the previous one, and it also uses less information. An extended

high-gain observer is a primary tool used in this chapter to achieve synchronization. The use

of extended high-gain observers for attaining synchronization is challenging as it is not clear

how to use it for the original problem formulation described in Isidori et al. [55]. Therefore,

the first step before applying the tool requires the original problem to be transformed into the

97

relative coordinates. In the relative coordinates, the synchronization problem is converted

into a stabilization problem of N nonlinear systems. It is then that we use the extended high-

gain observer based on the relative output exchange to estimate the uncertain terms, and then

using feedback control, we cancel them. In other words, we compensate for the heterogeneous

dynamics of each agent using extended high-gain observer and feedback control.

The proposed approach shapes the transient performance of the closed-loop system and

guarantees a steady-state error that can be made arbitrarily small [108] by tuning the ob-

server parameter. This is achieved by first designing the state feedback controller with full

information to shape the transient response of the closed-loop system. The transient response

under output feedback is shaped by the performance recovery property which is achieved

by bringing the trajectories under output feedback arbitrarily close to the trajectories under

state feedback for sufficiently small observer parameter.

5.2 Problem Formulation

We consider a network of N non-identical uncertain heterogeneous nonlinear agents having
the same relative degree r, which exchange information through a communication graph G.

We assume that there is a leader which can communicate with a subset of the agents in the

network. The objective is to synchronize the trajectories of the agents to the trajectory of the

leader. The controller structure is decentralized in nature and therefore the local controller

of each agent depends only on the relative information exchanged with its neighbors.

98

The dynamics of the agents are defined as

˙ηi = fi(ηi, ξi)

˙ξi = Aξi + B [bi(ηi, ξi)ui + φi(t, ηi, ξi)]

yi = Cξi

(5.1a)

(5.1b)

(5.1c)

for i = 1, . . . , N , where ηi ∈ Rni−r is the internal state, ni denotes the dimension of each
agent, r is the relative degree of the agent, ξi ∈ Rr is the external state and the matrices
A ∈ Rr×r, B ∈ Rr×1 and C ∈ R1×r are defined as



(cid:20)

0 1

0 0
...
...

0

1
. . .

. . . 0

. . . 0
...
. . .

A =

, B =

,

0 0 . . .

0

1

0 0 . . .

. . . 0





0

0
...

0

1



(cid:21)

C =

1 0 . . . 0 0

.

The agents are minimum-phase which implies that the origin of ˙ηi = fi(ηi, 0) is asymptoti-

cally stable.
Assumption 5.1: There exists continuously differentiable functions Vi(ηi), class K func-

tions Ψi, Γi and Xi such that

Ψi(||ηi||) ≤ Vi(ηi) ≤ Γi(||ηi||)

99

fi(ηi, ξi) ≤ 0,

||ηi|| ≥ Xi(||ξi||)

∂Vi
∂ηi

for all (ηi, ξi) ∈ Si ⊂ Rni where Si is a domain that contains the origin.

Assumption 5.1 implies that the internal dynamics of the agents are regionally input-to-state

stable.

Assumption 5.2: The functions bi and φi are continuously differentiable with locally Lip-

schitz derivatives and the function fi is locally Lipschitz.

Assumption 5.3: There exists ˜bi such that

bi(ηi, ξi) > ˜bi > 0, ∀ (ηi, ξi) ∈ Si

The agents class defined in (5.1) covers a wide variety of practical systems used for engineer-

ing applications.

5.2.1 Leader

The agents are required ro synchronize to a desired trajectory. The desired trajectory can

be generated by a leader or exosystem and, to reduce the information flow, it is assumed

that only a subset of the agents receives information from the leader. The leader dynamics

is defined as

˙ξ0 = A0ξ0 + Bν(t, ξ0)

y0 = Cξ0

(5.2a)

(5.2b)

100

where ξ0 ∈ Rr and y0 ∈ R. In this problem we are interested in the synchronization between

the external state of the agents ξ and the state of the leader ξ0 and therefore the leader is not

required to have internal dynamics. The leader (5.2) with the control input ν(t, ξ0) covers

both the class of linear and nonlinear dynamics. The control ν(t, ξ0) is not required to be

known to the agents in the network and is chosen such that the following assumption holds.

Assumption 5.4: The closed-loop trajectories of the leader ξ0(t) belongs to a known com-
pact invariant set W ⊂ Rr.
Remark 5.1: If the control ν(t, ξ0) = 0 or ν(t, ξ0) = − ˜Kξ0, then (5.2) corresponds to a

leader with linear dynamics and then for Assumption 5.4 to hold the eigenvalues of A0 or
(A0 − B ˜K) should lie in the closed left-half complex plane with distinct eigenvalues on the

imaginary axis.

We require the matrix A0 defined in (5.2) to have the following structure

A0 = A + ˜A

(5.3)

where

˜A =



0

0
...

0

0

0

. . .

. . .

. . .

. . .

. . .

. . .

0

0

0
...

0

a0

. . .

. . . ar2 ar−1



r×r

We define the expanded graph as ¯G = ( ¯V, ¯E, ¯A), where ¯V = {v0, v1, v2, . . . , vN}, ¯E ⊆ ¯V× ¯V

and ¯A contains the weights of the edges from the leader to the other agents.

Assumption 5.5: There is a directed path from the leader to all the other nodes in the

101

expanded graph ¯G.

The above assumption is standard for achieving synchronization in a network of agents with

a leader. We assume that the agents do not have access to their own state or output. The

information available to the agents is a linear combination of its own output relative to its

neighbors. The information structure is written as

N(cid:88)

χi =

aij(yj − yi) + di(y0 − yi)

(5.4)

j=1

where aij ≥ 0 are the edge coefficients of the communication graph G, di > 0 if agent i

receives information from the leader otherwise di = 0. The leader does not receive any infor-

mation from the agents in the network. In applications like deep space [43] and underwater,

it is difficult for the agent to measure its own output or state as GPS is not available, and

it is more likely that the agent can measure relative information between itself and those of

its neighbors. The physical meaning of the coefficients in the weighted sum could represent

the scaling in measuring the relative outputs of the agents. For engineering applications like

swarm robotics, installing sensors for each robot so that it can measure its own state or out-

put may become costly. On the other hand, it is cheaper to install sensors in agents that can

measure relative states or outputs for example in low-cost swarm robots like Kilobots [109].

The main goal is to find a distributed control law using only the relative information struc-

ture (5.4) such that synchronization between the external state of the agents ξ and the state

of the leader ξ0 is achieved which implies that

t→∞[ξi(t) − ξ0(t)] = 0

lim

102

for i = 1, 2, . . . , N.

5.3 Relative Dynamics

In this section we define the relative dynamics among the agents in the network and hence

transform system (5.1). We first define the relative states as

N(cid:88)

ρi =

aij(ξj − ξi) + di(ξ0 − ξi)

(5.5)

j=1

i = 1, 2, . . . , N. Defining ρ = col(ρ1, . . . , ρN ), ξ = col(ξ1, . . . , ξN ) we have

ρ = −(LG ⊗ Ir)ξ + (D ⊗ Ir)(1 ⊗ ξ0)

where LG = L+D, D = diag(d1, . . . , dN ), Ir represents an identity matrix of dimension r×r

and 1 represents column of all ones. From Assumption 5.5 the matrix LG is nonsingular.

Using the Kronecker product properties we have

ρ = −(LG ⊗ Ir)ξ + (d ⊗ ξ0)

(5.6)

where d = D1 = col(d1, . . . , dN ). The ξi dynamics of all the agents in the network can be

written in the following compact form

˙ξ = (IN ⊗ A)ξ + (IN ⊗ B) ˜B(η, ξ)u + (IN ⊗ B)φ(t, η, ξ)

(5.7)

103

where η = col(η1, . . . , ηN ), ˜B(η, ξ) = diag(b1(.), . . . , bN (.)), u = col(u1, . . . , uN ) and φ =

col(φ1, . . . , φN ) . Taking the time derivative of (5.6) yields

˙ρ = −(LG ⊗ Ir) ˙ξ + (d ⊗ ˙ξ0)
= {LG ⊗ Ir}{(IN ⊗ A)ξ + (IN ⊗ B) ˜B(η, ξ)u + φ(t, η, ξ)} + (d ⊗ ˙ξ0)

From (5.6) we have

ξ = (LG ⊗ Ir)−1{−ρ + (d ⊗ ξ0)}

= (L−1

G ⊗ Ir){−ρ + (d ⊗ ξ0)}

˙ρ = (IN ⊗ A)ρ − (d ⊗ Aξ0) − (LG ⊗ B)φ(t, η, ρ, ξ0) − (LG ⊗ B) ˜B(η, ρ, ξ0)u

+ (d ⊗ (A0ξ0 + Bν(t, ξ0)))

Using (5.3) we have

˙ρ = (IN ⊗ A)ρ + (IN ⊗ B)

(cid:104)

d1T(cid:16) ˜Aξ0 + Bν(t, ξ0)
(cid:17)(cid:105)

(cid:105)

(5.8)

(cid:104)

− (IN ⊗ B) LG

φ(t, η, ρ, ξ0) + ˜B(η, ρ, ξ0)u

104

Therefore, the system dynamics in the new coordinates becomes

˙ηi = fi(ηi, δi(ρ, ξ0))

−

 N(cid:88)

 biui + ∆i

u + ∆i

φ + ∆i
0

aij + di



(5.9a)

(5.9b)

(5.9c)

˙ρi = Aρi + B

χi = Cρi

where δ = col(δ1, . . . , δN ) =

j=1

(cid:110)
L−1
G ⊗ Ir

(cid:111)

N(cid:80)
(cid:32) N(cid:80)
aijφj −

u =

j=1

[(d ⊗ ξ0) − ρ], ∆i
N(cid:80)

φ =

j=1

j=1

aijbjuj represents the

(cid:33)

control inputs of the neighbors of the agent, ∆i

aij + di

φi repre-

sents the uncertain heterogeneous nonlinearities of the agent and its neighbors and ∆i

0 =

di1T(cid:16) ˜Aξ0 + Bν(t, ξ0)

(cid:17)

represents the leader states and control. The terms φi, and bi

are written without their arguments, their complete forms are φi = φi(t, ηi, δi(ρ, ξ0)), and

bi = bi(ηi, δi(ρ, ξ0)).

From the definition of δ we have

(cid:16)
(cid:16)
(cid:16)

ρ

(cid:17) −(cid:16)
(cid:17) −(cid:16)
(cid:17) −(cid:16)

(cid:17)
L−1
G ⊗ Ir
(cid:17)
L−1
G ⊗ Ir
(cid:17)
L−1
G ⊗ Ir

L−1
G d ⊗ ξ0
L−1
G D1 ⊗ ξ0
L−1
G LG1 ⊗ ξ0

ρ

ρ

δ =

=

=

where the term D1 is written as LG1 since the Laplacian L satisfies the property L1 = 0.

Therefore

δ = 1 ⊗ ξ0 −(cid:16)

(cid:17)

L−1
G ⊗ Ir

ρ

105

Taking the norm we have

where κ = ||L−1

G ⊗ Ir||.

√

||δ|| ≤

N||ξ0|| + κ||ρ||

The internal dynamics (5.9a) in the new coordinates now satisfies

fi(ηi, δi(ρ, ξ0)) ≤ 0,

||ηi|| ≥ Xi(κ||ρ|| +

∂Vi
∂ηi

√

N||ξ0||)

for all (η, ρ, ξ0) ∈ S × W where S = S1 × S2 × . . . SN . Therefore, the internal dynamics in

the new coordinates still has the regionally input-to-state stable property.

5.4 Synchronization under full information

In this section we assume that each agent has full knowledge of the system states (η, ρ, ξ0),

leader control ν, functions ∆i

0. The system (5.8) can be treated as a multi-input

u, ∆i

φ and ∆i

multi-output system where the coefficient of the control is given by

G(η, ρ, ξ0) = LG

˜B(η, ρ, ξ0).

In order to make the representation compact we drop the argument of G. A centralized

feedback linearizing controller is given by

(cid:17)(cid:111)
u = G−1(cid:110)−LGφ(t, η, ρ, ξ0) + d1T(cid:16) ˜Aξ0 + Bν(t, ξ0)

+ G−1(IN ⊗ K)ρ

(5.10)

where K is designed such that the matrix (A − BK) is Hurwitz.

Theorem 5.1: Consider the closed-loop system formed of the system (5.9) with the controller

106

(5.10). Suppose Assumptions 5.1-5.5 are satisfied. Then

t→∞[ξi(t) − ξ0(t)] = 0,

lim

for i = 1, 2, . . . , N.

(5.11)

Proof: In the new coordinates we need to solve the stabilization problem of N nonlin-

ear systems represented by (5.9) in order to achieve state synchronization in the original

coordinates. The closed loop system of the plant (5.9) with the controller (5.10) is given by

˙ρ = {IN ⊗ (A − BK)} ρ

(5.12)

From (5.6) we have

(cid:16)

(cid:17)

(cid:16)
L−1
G ⊗ Ir

(cid:17)

(D1 ⊗ ξ0)

which can be simpfied into

ρ = −ξ +

L−1
G ⊗ Ir
(cid:16)
L−1
G ⊗ Ir

(cid:17)

ρ = −ξ + 1 ⊗ ξ0

From (5.12) we have exponential stability of ρ which implies (5.11) using the above equation.

Therefore, the stabilization of N nonlinear systems (5.9) results in

t→∞ [ξ(t) − 1 ⊗ ξ0(t)] = 0

lim

(5.13)

which implies that all the states of the agents synchronize to the state of the leader.
We define a Lyapunov function as Vρ(ρ) = ρT (IN ⊗ P ) ρ where P = P T > 0 is the solution
of the Lyapunov equation P (A − BK) + (A − BK)T P = −Q for some Q = QT > 0. Let

(cid:4)

107

c > 0 be chosen such that {Vρ(ρ) ≤ c} ⊂ S × W . Next we define the compact set

Ω = {V1(η1) ≤ c1} × . . . × {VN (ηN ) ≤ cN} × {Vρ ≤ c}

where ci ≥ Γi(Xi(γ(c))) such that the set {Vi(ηi) ≤ ci} is compact and contained in Si and

γ(c) is defined as

γ(c) =

max

ρ∈{Vρ≤c},ξ0∈W

(κ||ρ|| +

√

N||ξ0||)

Since ci should be chosen such that the set {Vi(ηi) ≤ ci} is contained in Si, this may put a

restriction on the choice of c.

The compact set Ω is positively invariant with respect to the system

˙ηi = fi(ηi, δi(ρ, ξ0)),

˙ρ = {IN ⊗ (A − BK)}ρ

(5.14)

for i = 1, 2, . . . , N because on the boundary {Vi = ci}

Γi(||ηi||) ≥ ci ≥ Γi(Xi(γ(c))) =⇒ ||ηi|| ≥ Xi(γ(c))
=⇒ ||ηi|| ≥ Xi(κ||ρ|| +

√
N||ξ0||) =⇒ ˙Vi ≤ 0

and on the boundary {Vρ = c},

˙Vρ < 0.

Remark 5.2: Under Assumptions 5.1-5.5, the synchronization achieved will be regional.
However, if all the assumptions hold globally, i.e. Si = Rni and ˙ηi = fi(ηi, ξi) is input-to-

state stable, then the constants c1, . . . , cN and c, can be chosen arbitrarily large, and any
compact set of Rn1−r × . . . × RnN−r × RN r can be put in the interior of Ω = {V1(η1) ≤
c1} × . . . × {VN (ηN ) ≤ cN} × {Vρ ≤ c}. Then, the synchronization achieved for this case

108

will be semi-global.

5.5 Synchronization under output feedback

The control design of the previous section is not realizable as the agents only have the

knowledge of the relative outputs (5.4). The terms ∆i

u, ∆i

φ, ∆i

0 in (5.9b) are unknown

and therefore are treated as disturbance inputs. In this section we use extended high-gain

observers to estimate the unknown terms and cancel them using feedback control. The

extended high-gain observer is built for each agent locally and then the local controllers are

designed to stabilize the network of N nonlinear systems (5.9b) in order to achieve state

synchronization.

We assume that we have no knowledge of the functions φi(t, ηi, δi(ρ, ξ0)). Let gi be

positive constants, then the ˙ρi

r equation can be written as

r = σi − giui
˙ρi

where σi = ∆i

G = −[Gi − ˆGi], where Gi and ˆGi are the ith rows of G
and ˆG respectively where ˆG is defined as ˆG = diag(g1, g2, . . . , gN ). Since we assume that an

φ + ∆i

0 + ∆i

Gu and ∆i

agent has no information about the control input of its neighbors, this constrains ˆG to be

diagonal matrix.

Defining σ = col(σ1, . . . , σN ), we have

σ = −LGφ(t, η, ρ, ξ0) + d1T(cid:16) ˜Aξ0 + Bν(t, ξ0)

(cid:17) −(cid:104)

(cid:105)

G − ˆG

u

We augment σi as an additional state to the chain of integrators (5.9b) for each agent,

109

therefore a high-gain observer for the extended system of the agent is taken as

for 1 ≤ k ≤ r − 1,

1 − ˆρi
(ρi
1),
1 − ˆρi
αr
εr (ρi
1),

k+1 +

αk
k = ˆρi
˙ˆρi
εk
r = ˆσi − giui +
˙ˆρi
1 − ˆρi
1),

αr+1
εr+1 (ρi

˙ˆσi =

(5.15a)

(5.15b)

(5.15c)

for i = 1, 2, . . . , N , where α1 to αr+1 are chosen such that the roots of the polynomial

sr+1 + α1sr + . . . + αr+1

(5.16)

are real and negative and ε > 0 is a small parameter. The output feedback control is chosen

as

ui =

(cid:41)

(cid:40)

ˆσi + K ˆρi

gi

= ψi(ˆρi, ˆσi).

(5.17)

A characteristic of using high-gain observers is the peaking phenomenon, which occurs when
ˆρi(0) (cid:54)= ρi(0) and the estimates become O(1/εr−1) and when these estimates are used in

feedback control they may destabilize the closed-loop system [90]. Denote the ith component

of

G−1(cid:110)−LGφ + d1T(cid:16) ˜Aξ0 + Bν

(cid:17)

(cid:111)

+ (IN ⊗ K)ρ

by Ti(η, ρ, ξ0) and let

Mi >

max

(η,ρ)∈Ω,ξ0∈W

|Ti(η, ρ, ξ0)|

(5.18)

The control is saturated outside the compact set Ω× W in order to protect the system from
the peaking effect of the observer. We saturate the expression of ui (5.18) at ±Mi using the

110

saturation function sat(.) which gives the output feedback controller

(cid:32)

(cid:33)

ui = Mi sat

ψi(ˆρi, ˆσi)

Mi

(5.19)

Theorem 5.2: Consider the closed-loop system formed of the plant (5.9), the observer (5.15)

and the controller (5.19) for i=1,2,. . . ,N. Suppose that Assumptions 5.1-5.5 are satisfied,

(cid:40)

max
|ei|≤1

max

(η,ρ)∈Ω,ξ0∈W

(cid:13)(cid:13)(cid:13)[G(η, ρ, ξ0) − ˆG]E ˆG−1(cid:13)(cid:13)(cid:13)(cid:41)

< 1,

(5.20)

where E = diag(e1, . . . , eN ), the initial states of the observers (ˆρi(0), ˆσi(0)) belong to a
compact subset of Rr+1, and the initial states of the plant (ηi(0), ρi(0)) belong to a compact

set in the interior of Ω for i = 1, 2, . . . , N. Then

• there exists ε∗

1 > 0 such that for every 0 < ε ≤ ε∗

1, the trajectories of the closed-loop

system are bounded for all t ≥ 0;

• given any µ > 0, there exists ε∗

2 > 0, dependent on µ, such that for every 0 < ε ≤ ε∗

2

||ρ(t) − ρtr(t)|| ≤ µ ∀ t ≥ 0,

(5.21)

where ρtr is the solution of the system (5.12) with ρtr(0) = ρ(0);

• given any µ > 0, there exist ε∗

3 > 0 and T1 > 0, both dependent on µ, such that for

every 0 < ε ≤ ε∗
3,

||ρ(t)|| ≤ µ, ∀ t ≥ T1

(5.22)

Remark 5.3: The first bullet shows that for sufficiently small 
, the trajectories of the

111

closed-loop system are bounded. The second bullet shows the performance recovery property

of the output feedback controller, as the trajectories under output feedback ρ(t) can be

made arbitrarily close to the trajectories of the target system ρtr(t), for sufficiently small 
.

Finally, the third bullet shows that the output feedback controller achieves practical state
synchronization, as the ultimate bound on ||ρ(t)||, can be made arbitrarily small, by choosing


 small.

Remark 5.4: From the change of coordinates (5.6), we have

ξ − 1 ⊗ ξ0 = −(cid:16)

(cid:17)

L−1
G ⊗ Ir

ρ

Taking the norm on both sides we have

||ξ − 1 ⊗ ξ0|| ≤ κ||ρ||

where κ is defined earlier in section 5.3. Therefore, if the desired synchronization error is
required to be ˜µ, then by choosing µ ≤ ˜µ/κ, from the third bullet we have

||ξ − 1 ⊗ ξ0|| ≤ ˜µ, ∀ t ≥ T1

which implies we achieve practical state synchronization in the original coordinates.

Proof: We define the change of variables

ζi
k = (ρi

k − ˆρi

k)/εr+1−k, for 1 ≤ k ≤ r,

ζi
r+1 = ∆i

φ + ∆i

0 + ∆i

Gψs(ρ, ˆσ) − ˆσi

(5.23a)

(5.23b)

112

where

(cid:32)

ψs = col

M1gε

(cid:18) ψ1(ρ1, ˆσ1)

(cid:19)

M1

, . . . , MN gε

(cid:33)(cid:33)

(cid:32)

ψN (ρN , ˆσN )

MN

and gε is on odd function defined by



gε(y) =

y

y +

for 0 ≤ y < 1,

y − 1

ε

− 0.5(y2 − 1)

ε

for 1 ≤ y ≤ 1 + ε,

1 + 0.5ε

for y ≥ 1 + ε.

The function gε is nondecreasing, continuously differentiable with a locally Lipschitz deriva-
(cid:48)
tive, bounded uniformly in ε for any bounded interval of ε and satisfies |g
ε(y)| ≤ 1 and
|gε(y) − sat(y)| ≤ ε/2 for all y ∈ R.

The observer error dynamics are

(5.24a)

(5.24b)

(5.24c)

k = −αkζi
ε ˙ζi
r = −αrζi
ε ˙ζi

1 + ζi

k+1 for 1 ≤ k ≤ r − 1

1 + ζi

r+1 + Υi(η, ρ, ˆρ, ˆσ, ε)

r+1 = −αr+1ζi
ε ˙ζi

1 + αr+1∆i

GF ˆG−1(IN ⊗ C)ζ + εΠi

113

where

Υi = −∆i

Πi = ˙∆i

G [ψs(ρ, ˆσ) − ψsat(ˆρ, ˆσ)] ,
(cid:32)
(cid:19)

(cid:18)ψ1(ˆρ1, ˆσ1)

(cid:48)
ρ + ˙∆i

0 + ∆i

GF ψ

φ + ˙∆i

ψsat = col

M1sat

M1
(cid:48)
(cid:48)
(cid:19)

(ψN (·))),

(ψ1(·)), . . . , g
∂ψN
∂ρN

∂ρ1 ˙ρ1, . . . ,

(cid:18)∂ψ1

˙ρN

,

F = diag(g
(cid:48)
ρ = col

ψ

Gψs(ρ, ˆσ),

, . . . , MN sat

(cid:32)

ψN (ˆρN , ˆσN )

MN

(cid:33)(cid:33)

,

and ζi = col(ζi

1, . . . , ζi

r+1) where ζ = col(ζ1, . . . , ζN ).

From (5.24c), the term Πi on the right hand side of ε ˙ζi

r+1 is a continuous function of

(η, ρ, ζ, ξ0, ˙ξ0, ν, ˙ν, ε). Since gε is continuously differentiable with locally Lipschitz derivatives

and globally bounded using (5.23a) and from the definition of gε it can be shown that Υi/ε

is a locally Lipschitz function [83].

The local observer error dynamics is given by

ε ˙ζi = Λζi + αr+1 ¯B1∆i

GF ˆG−1(IN ⊗ C)ζ + ε

(cid:104) ¯B1Πi + ¯B2Υi/ε
(cid:105)

(5.25)

where



Λ =

1

0

0
...

1
. . .

−α1
−α2
...
−αr
0 . . .
−αr+1 0 . . .



. . . 0

. . . 0
...
. . .

0

1

. . . 0

 , ¯B2 =

 0

B

 .

B

0

, ¯B1 =

114

The observer error dynamics of the network is given by

ε ˙ζ = (IN ⊗ Λ)ζ − αr+1(IN ⊗ ¯B1)∆ζ (IN ⊗ C)ζ + ε(cid:2)(IN ⊗ ¯B2)Υ + (IN ⊗ ¯B1)Π(cid:3)

(5.26)

where Π = col(Π1, . . . , ΠN ), Υ = col(Υ1, . . . , ΥN ) and can be expressed as

(cid:104)
(cid:105)
(G − ˆG) (ψs(ρ, ˆσ) − ψsat(ˆρ, ˆσ))

,

Υ =

1
ε

Π = −(G − ˆG)F ψ

(cid:48)
ρ + ˙∆0 + ˙∆φ − ( ˙G − ˙ˆG)ψs(ρ, ˆσ),

˙∆0 = col( ˙∆1
0, . . . , ˙∆N
0 ),
∆ζ = (G − ˆG)F ˆG−1.

˙∆φ = col( ˙∆1

φ, . . . , ˙∆N

φ ) and

The functions Υ, Π and ∆ζ are locally Lipschitz in their arguments and bounded from above
by ka + kb||ζ||, where ka and kb are positive constants independent of ε. The matrix Λ is

Hurwitz by design, the fast dynamics of the observer (5.26) is the same as seen in high-gain
observer theory [84] except for the term αr+1(IN ⊗ ¯B1)∆ζ (IN ⊗ C)ζ. The system (5.26)

without O(ε) terms on the right hand side is given by

ε ˙ζ = (IN ⊗ Λ)ζ − αr+1(IN ⊗ ¯B1)∆ζ (IN ⊗ C)ζ.

(5.27)

Equation (5.27) can be represented by a negative feedback connection of the transfer function

(cid:18)

(cid:19)

Θ(εs) =

αr+1

(εs)r+1 + α1(εs)r + . . . + αr+1

IN

and the time-varying gain ∆ζ . Since we consider that the observer poles are real, we have

115

||Θ||∞ = 1. It can be shown that the origin of the system (5.27) is globally exponentially

stable using (5.20) by applying the circle criterion [83].

The change of variables (5.23b) is well defined if

∂ζr+1

dˆσ

= −IN + ∆ζ

is nonsingular for (η, ρ) ∈ Ω and ξ0 ∈ W , where ζr+1 = col(ζ1
ˆσ = col(ˆσ1, . . . , ˆσN ), which is the case because (cid:107)∆ζ(cid:107) < 1.

r+1, . . . , ζN

r+1), and

Applying a loop transformation to (5.27) and using the Kalman-Yakubovich-Poppov
lemma [103, Lemma 5.3] we can obtain a quadratic Lyapunov function W (ζ) = ζT (IN ⊗ Y )ζ
whose derivative with respect to (5.27) is bounded from above by −(λ/ε)W (ζ) for some

positive constant λ, independent of ε. Using W (ζ) as a Lyapunov function candidate for

(5.27), we can show that all the state variables are bounded. We show that for any ˜c1 > 0,
the set Σ = Ω × {W (ζ) ≤ ˜c1ε2} is positively invariant.

Since the initial conditions (η(0), ρ(0)) are defined in the interior of the set Ω and due to the

global boundedness of the control ui there exists a finite time T∗ > 0 independent of ε such
that the trajectories of the system (5.9) do not leave the compact set Ω for all t ∈ [0, T∗].
Initially the scaled estimation error ζ(0) could be outside the set {W (ζ) ≤ ˜c1ε2}. But from
the time-derivative of W (ζ) it can be shown that ζ(t) enters the set {W ≤ 
2˜c1} in finite
T (
) = 0. By choosing 
 small enough we can ensure that T (
) < T∗.
time T (
), where lim

→0
While the system trajectory (η, ρ, ζ) ∈ Σ, we have ζr+1(t) = O(ε) and ρ(t) − ˆρ(t) = O(ε).

Therefore,

ψ(ˆρ, ˆσ) = ψ(ρ, ˆσ) + O(ε)

ˆσ = −LGφ(t, η, ρ, ξ0) + d1T ( ˜Aξ0 + Bν(t, ξ0)) − (G − ˆG)ψs(ρ, ˆσ) + O(ε)

116

Therefore, up to an O(ε) error, the control ψ(ρ, ˆσ) satisfies

ψ(ρ, ˆσ) = ˆG−1{ˆσ + (IN ⊗ K)ρ}

= ˆG−1{−LGφ(t, η, ρ, ξ0) + d1T ( ˜Aξ0 + Bν(t, ξ0)) − (G − ˆG)ψs(ρ, ˆσ) + (IN ⊗ K)ρ}

ψ + ˆG−1(G − ˆG)ψs(ρ, ˆσ) = ˆG−1{−LGφ(t, η, ρ, ξ0) + d1T ( ˜Aξ0 + Bν(t, ξ0)) + (IN ⊗ K)ρ}

Next we define the map F : RN → RN as

F(s) = s + ˆG−1(G − ˆG)ψs(s)

The map F is proper which implies |F(s)| → ∞ as |s| → ∞. This follows from the definition

of ψs and from the fact that G is bounded. Using Hadamard’s Theorem [110], if the Jacobian
of F(s) is nonsingular in RN , then the map F is one-to-one and onto which implies that
the map has globally defined inverse. The Jacobian of F(s) is defined as

∂F
∂s

= IN + ˆG−1(G − ˆG−1)F

(5.28)

Rearranging the terms we have

∂F
∂s

= ˆG−1(cid:16)

IN + (G − ˆG−1)F ˆG−1(cid:17) ˆG = ˆG−1(cid:0)IN + ∆ζ

(cid:1) ˆG

The Jacobian (5.28) is nonsingular because (cid:107)∆ζ(cid:107) < 1. Therefore, the mapping F is globally

117

invertible and by direct substitution it can be seen that the unique solution is

ψ = G−1(cid:110)−LGφ(t, η, ρ, ξ0) + d1T(cid:16) ˜Aξ0 + Bν(t, ξ0)
(cid:17)(cid:111)

+ G−1(IN ⊗ K)ρ

Hence,

ψ(ˆρ, ˆσ) = G−1(cid:110)−LGφ(t, η, ρ, ξ0) + d1T(cid:16) ˜Aξ0 + Bν(t, ξ0)
(cid:17)(cid:111)

+ G−1(IN ⊗ K)ρ + O(ε).

The saturation levels Mi are chosen greater than Ti, and therefore for sufficiently small ε,

ψ(ˆρ, ˆσ) will be in the linear region of the saturation function and therefore the closed-loop

system is represented by

˙ηi = fi(ηi, δi(ρ, ξ0)) for 1 ≤ i ≤ N
˙ρ = {IN ⊗ (A − BK)}ρ + O(ε)
ε ˙ζ = (IN ⊗ Λ)ζ − αr+1(IN ⊗ ¯B1)∆ζ (IN ⊗ C)ζ + O(ε)

(5.29a)

(5.29b)

(5.29c)

From the equations (5.29), it can be easily argued that the set Ω×{W (ζ) ≤ 
2˜c1} is positively

invariant for sufficiently small 
, which establishes boundedness of all state variables.
Since ρ(t) satisfies (5.29b) for t ≥ T (ε), ˙ρ, ˙ρtr are bounded uniformly in ε and ρ(0) = ρtr(0),

therefore it follows that

ρ(t) − ρtr(t) = O(T (ε)),

for 0 ≤ t ≤ T (ε)

(5.30)

Hence, ρ(T (ε))− ρtr(T (ε)) = O(T (ε)). Using (5.30), (5.12), (5.29b), continuous dependence

of the solutions of differential equations on initial conditions and parameters [91, Theorem

118

9.1], and exponential stability of ˙ρ = {IN ⊗ (A − BK)}ρ, we conclude that

ρ(t) − ρtr(t) = O(ε) + O(T (ε)), ∀ t ≥ T (ε)

(5.31)

Therefore from (5.30) and (5.31) we conclude that

ρ(t) − ρtr(t) = O(ε) + O(T (ε)), ∀ t ≥ 0

(5.32)

and therefore given any µ > 0 there exists 
∗

2 > 0 such that for all ε ∈ (0, 
∗

2] (5.21) is

satisfied.

Because lim

t→∞ ρtr(t) = 0, given any µ > 0 there is a finite time T1 > 0 such that ||ρtr(t)|| ≤
3], ||ρ(t) −

3 > 0 such that for all ε ∈ (0, 
∗

µ/2 for all t ≥ T1. From (5.30) there exists 
∗
ρtr(t)|| ≤ µ/2 for all t ≥ T1. Hence

||ρ(t)|| = ||ρ(t) − ρtr(t) + ρtr(t)|| ≤ ||ρ(t) − ρtr(t)|| + ||ρtr(t)||

||ρ(t)|| ≤ µ ∀ t ≥ T1,

which proves (5.22).

(cid:4)

We consider three special cases for which condition (5.20) holds.

In all cases, we take

ˆG = gmI with gm > 0. Therefore,

(cid:107)(G − ˆG)E ˆG−1(cid:107) = (cid:107)(G − ˆG) ˆG−1E(cid:107) ≤

(cid:13)(cid:13)(cid:13)(cid:13)I − 1

gm

(cid:13)(cid:13)(cid:13)(cid:13)

G

as ||E|| ≤ 1.

119

Case I: The graph G is undirected with control coefficients bi = 1 for i = 1, . . . , N . In this

case G = LG is symmetric and positive definite. The condition (5.20) is satisfied by taking

gm >

1
2

λmax(LG)

Proof of Case I: In this case,(cid:13)(cid:13)(cid:13)(cid:13)I − 1

gm

Choosing gm >

λmax(LG)

2

(cid:13)(cid:13)(cid:13)(cid:13) ≤ max

1≤i≤N

G

(cid:12)(cid:12)(cid:12)(cid:12)1 − λi(LG)

gm

(cid:12)(cid:12)(cid:12)(cid:12)

=⇒ 0 <

λi(LG)

gm

< 2,

=⇒

(cid:13)(cid:13)(cid:13)(cid:13)I − 1

gm

Therefore, we can conclude

(cid:12)(cid:12)(cid:12)(cid:12)1 − λi(LG)
(cid:13)(cid:13)(cid:13)(cid:13) < 1.

gm

G

=⇒ −1 < 1 − λi(LG)

< 1,

gm

(cid:12)(cid:12)(cid:12)(cid:12) < 1.

Case II: G = LG, where LG is a normal matrix and the control coefficients bi = 1 for

i = 1, . . . , N . We take

gm > max
1≤i≤N

i + β2
α2
i
2αi

where αi + jβi, for i = 1, . . . , N, are the eigenvalues of LG. The set of graphs where the

matrix LG is normal includes undirected graphs but is more general as the eigenvalues can

be complex [111], [112].

120

Proof of Case II: Since the matrix LG is a normal matrix, it is unitarily diagonalizable

LG = Z ˜ΛZ∗

where Z is a unitary matrix and ˜Λ = diag(˜λ1, . . . , ˜λN ) is a diagonal matrix where the

diagonal elements are the eigenvalues of LG. Therefore

where ||Z|| = ||Z∗|| = 1 and

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

gm

gm

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)IN − Z ˜ΛZ∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) =
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)ZZ∗ − Z ˜ΛZ∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)||Z∗||
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)IN − ˜Λ
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ||Z||
(cid:33)
(cid:113)
(1 − αi/gm)2 + (βi/gm)2

Z∗

gm

gm

(cid:13)(cid:13)(cid:13)(cid:13) =

(cid:13)(cid:13)(cid:13)(cid:13)IN − LG
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z
(cid:32)
IN − ˜Λ
gm
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)IN − ˜Λ
(cid:13)(cid:13)(cid:13)(cid:13) < 1 to hold

gm

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ max

1≤i≤N

where αi and βi are the real and imaginary parts of the eigenvalues of LG. The eigenvalues

of the matrix LG are in the open right half plane which implies that αi > 0. Therefore for

(cid:13)(cid:13)(cid:13)(cid:13)I − 1

gm

G

(1 − αi/gm)2 + (βi/gm)2 < 1

max
1≤i≤N

and therefore gm should satisfy

gm > max
1≤i≤N

i + β2
α2
i
2αi

121

Case III: G(η, ξ) satisfies

G(η, ξ) + GT (η, ξ) ≥ β1I and ||G(η, ξ)|| ≤ β2

for some positive constants β1 and β2, for all (ηi, ξi) ∈ Si, for i = 1, . . . , N. The condition
(5.20) is satisfied by taking gm > β2

2/β1. A special case arises when G(η, ξ) is symmetric

and satisfies

0 < ˜g1I ≤ G(η, ξ) ≤ ˜g2I

G

gm

Proof of Case III: (cid:13)(cid:13)(cid:13)(cid:13)IN − 1
(cid:115)
(cid:115)
(cid:115)
(cid:118)(cid:117)(cid:117)(cid:116)(cid:32)

≤ max
||y||=1

= max
||y||=1

= max
||y||=1

yT

yT

= max
||y||=1

(cid:13)(cid:13)(cid:13)(cid:13)(cid:20)

(cid:13)(cid:13)(cid:13)(cid:13) = max
(cid:20)
||y||=1
IN − 1
(cid:26)
gm
IN − 1
gm

(cid:21)

(cid:13)(cid:13)(cid:13)(cid:13)

y

G

IN − 1
(cid:21)T(cid:20)
gm
IN − 1
gm

G

(cid:21)

G

y

(cid:27)

[G + GT ] +

1
g2
m

GT G

y

yT y − β1
gm

yT y +

yT y

β2
2
g2
m

(cid:33)

yT y

1 − β1
gm

+

β2
2
g2
m

Choosing gm > β2

2/β1, there is 0 < ˜k < 1 such that

(cid:33)

≤ ˜k

(cid:32)

1 − β1
gm

+

β2
2
g2
m

122

(cid:13)(cid:13)(cid:13)(cid:13)IN − 1

gm

(cid:13)(cid:13)(cid:13)(cid:13) ≤(cid:112)˜k < 1

G

Hence

5.6 Example

We illustrate our method by applying to an example that was treated in [55]. In [55] the

network consisted of three subsystems where each subsystem comprised of an agent and an

exosystem. Each exosystem exchanged information with neighboring exosystems and with

the agent in its subsystem. We consider that there is only one exosystem in the network,

which is connected to agent 1 and, instead of the exosystems, the agents exchange information

with their neighbors. The agents are labeled as 1, 2, 3, while the exosystem is labeled as 0.

The communication topology is taken to be the same as in [55]. In this case the L and D

matrix become

L =



0 −1

1
−1
1
0 −1

0

1

 , D =





1 0 0

0 0 0

0 0 0

Figure 5.1: Network interconnection between the agents

Fig. 5.1 illustrates the connections between the leader and the agents. The exosystem is

123

taken as the leader and its dynamics are defined as

˙ξ01 = ξ02
˙ξ02 = 2(1 − ξ2

01)ξ02 − ξ01

y0 = ξ01

(5.33a)

(5.33b)

(5.33c)

It can be seen that the leader is a Van der Pol oscillator and the compact set W consists

of the limit cycle of the oscillator and its interior [103]. The set W is globally uniformly

attractive with respect to the dynamics (5.33). The agent dynamics are defined as

˙ξi1 = ξi2
˙ξi2 = φi(µi, ξi) + ui, 1 ≤ i ≤ 3

yi = ξi1

(5.34a)

(5.34b)

(5.34c)

where φ1 = −µ1ξ11, φ2 = µ2(1 − ξ2

21)ξ22 − ξ21 and φ3 = −µ3ξ31 + ξ3

31. The relative states

are defined as

ρ1 = (ξ3 − ξ1) + (ξ0 − ξ1), ρ2 = (ξ1 − ξ2), ρ3 = (ξ2 − ξ3)

(5.35)

Defining ρ = col(ρ1, ρ2, ρ3) we have

ρ = −(LG ⊗ I2)ξ + (d ⊗ ξ0)

(5.36)

124

where ξ = col(ξ1, ξ2, ξ3) and d = col(d1, d2, d3) = col(1, 0, 0). Differentiating (5.36) we have

˙ρ = (I3 ⊗ A)ρ + (I3 ⊗ B)

(cid:104)
(cid:105)
d1T Bν(ξ0) − Gφ(ξ) + Gu

(5.37)

where G = LG, φ = col(φ1, φ2, φ3) and ν = 2(1 − ξ2

01)ξ02 − ξ01. The local dynamics are

defined as

˙ρi = Aρi − B



 3(cid:88)

j=1

aij + di

 ui − 3(cid:88)

j=1

aijuj −

 3(cid:88)

j=1

+ B(di1T Bν(ξ0))

 φi − 3(cid:88)

j=1



aijφj

aij + di

χi = Cρi

The extended high-gain observer is constructed as

(ρi1 − ˆρi1),

α1
ε

˙ˆρi1 = ˆρi2 +
˙ˆρi2 = ˆσi − gmui +
ε3 (ρi1 − ˆρi1).

˙ˆσi =

α3

α2

ε2 (ρi1 − ˆρi1),

(5.38a)

(5.38b)

(5.38c)

with α1 = α2 = 3 and α3 = 1, which assigns all three roots of the polynomial

s3 + α1s2 + α2s + α3

(5.39)

at −1, and ε > 0 is a small parameter. The output feedback control is chosen as

ui = Misat

,

i = 1, 2, 3.

(5.40)

(cid:18) ˆσi + K ˆρi

(cid:19)

gmMi

125

where K = [18 9] and the estimates (ˆρi, ˆσi) are provided by extended high-gain observer

and Mi is the saturation level of control. For simulation, the initial condition of the agents

were chosen as ξ0 = (1, 1), ξ1 = (2.4, 2.7), ξ2 = (2.6, 3.2), ξ3 = (3.2, 3.8) and saturation
levels were chosen as M1 = ±35, M2 = ±50 and M3 = ±100. The saturation levels were

chosen from simulation of the closed-loop system under the state feedback controller to see

the maximum value of the control signals. For this example the matrix G = L + D satisfies

case III with β1 = 0.43 and β2 = 2.46 so that β2

2/β1 = 14.07 and therefore gm is chosen as

gm = 15.

(a) ξi,1 for i=1,2,3.

Figure 5.2: States of the agents under the proposed controller with 
 = 0.001

(b) ξi,2 for i=1,2,3.

126

024681012141618Time-3-2-101234i1LeaderAgent 1Agent 2Agent 366.577.5800.511.52024681012141618Time-8-6-4-20246i2LeaderAgent 1Agent 2Agent 32.533.544.55-4-3-2-10Figure 5.3: Control of the agents with 
 = 0.001

(a) ξi,1 for i=1,2,3.

Figure 5.4: States of the agents under the proposed controller with 
 = 0.0001

(b) ξi,2 for i=1,2,3.

127

024681012141618Time-100-50050uAgent 1Agent 2Agent 3024681012141618Time-3-2-101234i1LeaderAgent 1Agent 2Agent 366.577.5800.511.52024681012141618Time-10-8-6-4-2024i2LeaderAgent 1Agent 2Agent 322.22.42.62.8-4-3.5-3-2.5-2-1.5Figure 5.5: Control of the agents with 
 = 0.0001

Fig 5.2. and Fig. 5.4 show that after the transient period the states of the agents track

the states of the exosystem and the synchronization error is small for sufficiently small value

of 
. Fig 5.3. and Fig 5.5. show the control of the agents and it can be seen that the control

saturates during the peaking period of the observer. As seen from Fig 5.2. and Fig 5.4. that

as 
 decreases the synchronization error also decreases.

5.7 Case with Some Unbounded Leader States

The designed state and output feedback controllers developed in section 5.4 and 5.5 deal

with the case when the leader state is bounded. In applications like formation control of

mobile robots [66] and frequency control of power systems [113] some of the leaders state

need be unbounded. Therefore in this section we deal with a special case where the systems

have relative degree two (r = 2) and the second-order equation of the leader has unbounded

128

024681012141618Time-100-50050100uAgent 1Agent 2Agent 3first component of the state. The systems take the form

˙ξi1 = ξi2

˙ξi2 = ¯bi(ξi2)ui + ¯φi(ξi2) + hi



mij(ξi − ξj)

 N(cid:88)

j=1

yi = Cξi

(5.41a)

(5.41b)

(5.41c)

where hi : R2 → R is continuously differentiable and mij = 0 if there no coupling between

agent i and j. The system (5.41) is a special case of (5.1) where there is no zero dynamics,

r = 2, and the functions ¯bi and ¯φi depend only on the second component of ξi. On the

other hand, (5.41) allows for a coupling term between the agents that depends only on the
differences ξi − ξj. Examples of systems that appear in the form (5.41) are phase oscillators

[114], power systems [72], [113] and double integrator models of mobile robots [66], [115].

The leader dynamics are defined as

˙ξ01 = ξ02

˙ξ02 = 0

(5.42a)

(5.42b)

which implies that ξ02 is bounded and ξ01 is unbounded. The information structure (5.4)

can be rewritten as

χi =

N(cid:88)

j=1

aij{(ξj1 − ξ01) − (ξi1 − ξ01)} + di(ξ01 − ξi1)

129

Next we define the relative states as

ϑi =

N(cid:88)

j=1

aij( ˜ξj − ˜ξi) − di

˜ξi

(5.43)

where ˜ξi = col(ξi1 − ξ01, ξi2 − ξ02) defines the synchronization error. Equation (5.43) in

matrix form can be written as

ϑ = −{LG ⊗ I2} ˜ξ

(5.44)

where ˜ξ = col( ˜ξ1, . . . , ˜ξN ), ϑ = col(ϑ1, . . . , ϑN ), I2 represents an identity matrix of dimension

2. The synchronization error dynamics of the network is given by

˙˜ξ = (IN ⊗ A) ˜ξ + (IN ⊗ B)

(cid:16)(cid:80)N

and H = col(h1

j=1 m1j( ˜ξ1 − ˜ξj)

, . . . , hN

The relative dynamics is given by

(cid:105)

(5.45)

(cid:104) ¯B( ˜ξ, ξ02)u + ¯φ( ˜ξ, ξ02) + H( ˜ξ)
(cid:17)
(cid:16)(cid:80)N
j=1 mN j( ˜ξN − ˜ξj)

).

(cid:17)

where ¯B = diag(¯b1( ˜ξ12 + ξ02), . . . , ¯bN ( ˜ξN 2 + ξ02)), ¯φ = col( ¯φ1( ˜ξ12 + ξ02), . . . , ¯φN ( ˜ξN 2 + ξ02))

˙ϑ = (IN ⊗ A)ϑ − (IN ⊗ B)[LG
χ = (IN ⊗ C)ϑ

where χ = col(χ1, . . . , χN ) and ˜ξ = −(cid:104)

(cid:105)

L−1
G ⊗ I2

¯B(ϑ, ξ02)u + LG

¯φ(ϑ, ξ02) + LGH(ϑ)]

(5.46a)

(5.46b)

ϑ. It can be seen that equation (5.46)

does not depend on the unbounded state of the leader ξ01. The control coefficient for system

(5.46) is given by

¯G = LG

¯B(ϑ, ξ02).

130

5.7.1 State feedback Controller

The state feedback controller that achieves synchronization will be of the form

u = − ¯G−1LG{ ¯φ(ϑ, ξ02) + H(ϑ)} + ¯G−1(IN ⊗ K)ϑ

(5.47)

Theorem 5.3: Consider the closed-loop system formed of the system (5.46) with the con-

troller (5.47). Then

t→∞ [ξi1(t) − ξ01(t)] = 0,

lim

t→∞ [ξi2(t) − ξ02(t)] = 0 for i = 1, 2, . . . , N.

lim

(5.48)

Proof: The closed loop system is given by

˙ϑ = {IN ⊗ (A − BK)} ϑ

(5.49)

t→∞ ˜ξ(t) = 0 from which (5.48) follows.
lim

which implies that ϑ exponentially converges to zero. And from (5.44) we can conclude
(cid:4)
that
We define a Lyapunov function as Vϑ(ϑ) = ϑT (IN ⊗ P ) ϑ. Let a > 0 be chosen such that
Ωa = {Vϑ(ϑ) ≤ a} is a compact positively invariant set with respect to the dynamical system

(5.49).

5.7.2 Output feedback Controller

The ˙ϑ2 equation for (5.46) can be written as

˙ϑ2 = Ξ − ˆGu

131

where ϑ2 = col(ϑ1

2, . . . , ϑN

2 ), and Ξ = col(Ξ1, . . . , ΞN ) is defined as

¯φ(ϑ, ξ02) − LGH(ϑ) −(cid:104) ¯G − ˆG
(cid:105)

Ξ = −LG

The extended high-gain observer is given by

1 − ˆϑi
(ϑi
1)
1 − ˆϑi
α2
ε2 (ϑi
1)

α1
ε

2 +

˙ˆϑi
1 = ˆϑi
2 = ˆΞi − giui +
˙ˆϑi
1 − ˆϑi
1)

˙ˆΞi =

α3
ε3 (ϑi

u

(5.50a)

(5.50b)

(5.50c)

for i = 1, 2, . . . , N , where α1 to α3 are chosen such that the roots of the polynomial

s3 + α1s2 + α2s + α3

(5.51)

are real and negative and ε > 0 is a small parameter. The output feedback control is chosen

as

ui =

(cid:40) ˆΞi + K ˆϑi

(cid:41)

gi

(cid:44) ¯ψi( ˆϑi, ˆΞi).

(5.52)

The control is saturated outside the compact set Ωa × Ωs where Ωs is the set of the synchro-

nization trajectories ξ02. Denote the ith component of

− ¯G−1LG{ ¯φ(ϑ, ξ02) + H(ϑ)} + ¯G−1(IN ⊗ K)ϑ

by ¯Ti(ϑ, ξ02) and let

(cid:12)(cid:12) ¯Ti(ϑ, ξ02)(cid:12)(cid:12)

¯Mi >

max

ϑ∈Ωa,ξ02∈Ωs

132

(5.53)

We saturate the expression of ui (5.53) at ± ¯Mi using the saturation function sat(.), which

gives the output feedback controller

ui = ¯Mi sat

(cid:33)

(cid:32) ¯ψi( ˆϑi, ˆΞi)

¯Mi

(5.54)

where the estimates ( ˆϑi, ˆΞi) are provided by extended high-gain observers and ¯Mi is the

saturation level of control.

Theorem 5.4: Consider the closed-loop system formed of the system (5.46), observer (5.50)

and the controller (5.54). Suppose

(cid:40)

max
|ri|≤1

max

ϑ∈Ωa,ξ02∈Ωs

(cid:13)(cid:13)(cid:13)[G(ϑ, ξ02) − ˆG]R ˆG−1(cid:13)(cid:13)(cid:13)(cid:41)

< 1,

(5.55)

where R = diag(r1, . . . , rN ), the initial states of the observers ( ˆϑi
2(0), ˆΞi(0)) for
i=1,2,. . . ,N belong to a compact subset of R3, and the initial states of the plant ϑ(0) be-
long to a compact set in the interior of Ωa. Then there exists ε∗ > 0 such that for every
0 < ε ≤ ε∗, the trajectories of the closed-loop system are bounded for all t ≥ 0 and there is

1(0), ˆϑi

¯T > 0 such that

for i=1,2,. . . ,N.

|ξi1(t) − ξ01(t)| = O(
) ∀ t ≥ ¯T
|ξi2(t) − ξ02(t)| = O(
) ∀ t ≥ ¯T

(5.56a)

(5.56b)

133

Proof: We define the change of variables

1 = (ϑi
ϕi

2 = (ϑi
ϕi

3 = ∆i
ϕi

1)/ε2,

1 − ˆϑi
2 − ˆϑi
d − ( ¯Gi − ˆGi) ¯ψs(ϑ, ˆΞ) − ˆΞi

2)/ε,

(5.57a)

(5.57b)

(5.57c)

where ¯Gi and ˆGi are the ith rows of ˆG and ˆG respectively, ∆d represents the nonlinear

couplings of the agents neighbors,

(cid:32)

(cid:32) ¯ψ1(ϑ1, ˆΞ1)

¯M1

(cid:33)

, . . . , ¯MN gε

(cid:33)(cid:33)

(cid:32) ¯ψN (ϑN , ˆΞN )

¯MN

¯ψs = col

¯M1gε

Following similar steps as in the section 5.5 the observer error dynamics of the network are

given by

ε ˙ϕ = (IN ⊗ Λ)ϕ − α3(IN ⊗ ¯B1)∆ϕ(IN ⊗ C)ϕ + ε(cid:2)(IN ⊗ ¯B2) ¯Υ + (IN ⊗ ¯B1) ¯Π(cid:3)

(5.58)

¯M1sat

¯ψsat = col
¯Π = col( ¯Π1, . . . , ¯ΠN ), ¯Πi = ¯Πi( ˙∆d, ϑ, Ξ), ∆ϕ = ( ¯G − ˆG) ¯F ˆG−1,

, . . . , ¯MN sat

¯MN

¯M1

,

¯F = diag(g

(cid:48)

( ¯ψ1(·)), . . . , g

(cid:48)

( ¯ψN (·))), ϕi = col(ϕi

1, ϕi

2, ϕi

3), ϕ = col(ϕ1, . . . , ϕN ).

134

 , ¯Υ =


(cid:32) ¯ψ1( ˆϑ1, ˆΞ1)

−α1 1 0
−α2 0 1
−α3 0 0
(cid:33)

Λ =

(cid:32)

(cid:104)

( ¯G − ˆG)

(cid:17)(cid:105)
(cid:16) ¯ψs(ϑ, ˆΞ) − ¯ψsat( ˆϑ, ˆΞ)

1
ε

(cid:32) ¯ψN ( ˆϑN , ˆΞN )

(cid:33)(cid:33)

Similarly, the system (5.58) without O(ε) terms on the right hand side is given by

ε ˙ϕ = (IN ⊗ Λ)ϕ − α3(IN ⊗ ¯B1)∆ϕ(IN ⊗ C)ϕ.

(5.59)

Equation (5.59) can be represented by a negative feedback connection of the transfer function

(cid:18)

Θ(εs) =

(cid:19)

α3

(εs)3 + α1(εs)2 + α2(εs) + α3

IN

and the time-varying gain ∆ϕ. Similar to the proof of Theorem 5.2, the origin of (5.59) is

globally exponentially stable using (5.55) by applying the circle criterion [83].

Using the same steps as in the proof of Theorem 5.2 it can be shown that after a finite

time ¯T > 0, the closed-loop system is given by

˙ϑ = {IN ⊗ (A − BK)}ϑ + O(ε)
ε ˙ϕ = (IN ⊗ Λ)ϕ − α3(IN ⊗ ¯B1)∆ϕ(IN ⊗ C)ϕ + O(ε)

from which (5.56) follows.

(5.60a)

(5.60b)

(cid:4)

Remark 5.5: Similar to Theorem 5.2, we can consider three special cases where inequality

(5.55) is satisfied. The first two cases are exactly the same as the general case, since they

depend on the communication topology which is considered the same for both the cases.

The third case will be different as it depends on ¯G(ν, ξ02) and therefore the inequalities will

be satisfied by constants different than the general case.

Remark 5.6: The performance recovery property can also be shown by following the same

steps as in the last part of the proof of Theorem 5.2.

135

5.8 Frequency Control of Power Systems

We consider the frequency control of power systems when only relative phase measurement is

available. Each synchronous generator is equipped with an encoder to measure the angular

position of the rotor, which is the phase of the generator [116]. The generators can exchange

their phase measurements with their neighbors. The frequency measurement is not available,

which might be due to the lack of speed sensor. In [117], it is illustrated that the lack of

a speed sensor reduces the system cost and can enhance the reliability of the generator.

We consider a network of generators interconnected by transmission lines. The objective is

to achieve frequency control where the frequencies of the buses synchronize to a reference

frequency. For this system the per unit nonlinear swing equation is given by [65]

˜mi ˙ωi + ˜diωi = − N(cid:88)

kij sin(θi − θj) + pm

i + ui

(5.61)

j=1

where θi is the phase angle of bus i, ˜mi = 2H/ωs, H is the inertia constant defined as

H = miω2

s /(2S), S is the MVA rating of the generator, mi and ˜di are the inertia and

damping coefficients respectively, ωs is the synchronizing frequency, ω is the frequency, pm
i

is the electrical power load at bus i and ui is the mechanical input power. The coefficient
kij is nonnegative and is given by kij = |Vi||Vj|bij where |Vi| is the absolute value of the

voltage of bus i and bij is the susceptance of the line (i, j). Next we define a change in the

time scale by τ = t/ ˜mi from which we get the modified equation as

kij sin(θi − θj) + pm

i + ui

(5.62)

= − ˜diωi − N(cid:88)

dωi
dτ

j=1

136

In state space form the swing equation is defined as

˙θi = ωi

˙ωi = − ˜diωi − N(cid:88)

j=1

kij sin(θi − θj) + pm

i + ui

where ˙(·) denotes the derivative w.r.t τ . The leader dynamics is defined as

˙θ0 = ω0

˙ω0 = 0

(5.63a)

(5.63b)

(5.64a)

(5.64b)

This definition implies that ω0 is bounded and θ0 is unbounded and therefore the problem

formulation is the same as discussed in section 5.7. The information structure can be written

as

N(cid:88)

j=1

χi =

aij(θj − θi) + di(θ0 − θi)

(5.65)

where aij = βkij, β is a positive constant, di > 0 if agent i receives information from the

leader otherwise di = 0. The terms kij might be large and therefore they are scaled by β.

The synchronization error dynamics of the network is given by

˙ξ = (In ⊗ A)ξ + (IN ⊗ B)

(cid:104)− ˜D(IN ⊗ ¯C)ξ − ˜D(1 ⊗ ¯Cξ0)
(cid:105)

+ (IN ⊗ B)

(cid:104)
Pm + u − BΓL sin(BT (IN ⊗ C)ξ

(cid:105)

(5.66)

where ξ = col(ξ1, . . . , ξN ), ξi = col(θi − θ0, ωi − ω0), ¯C = [0 1], ˜D = diag( ˜d1, . . . , ˜dN ),
Pm = col(p1

m), u = col(u1, . . . , uN ), ΓL is a diagonal matrix whose diagonal ele-

m, . . . , pN

137

ments represents the edges weights of the graph G and B is the graph incidence matrix [17]
which is related to the Laplacian by L = BΓLBT . The design of the output feedback

controller can be done by following the same steps as in section 5.7 first by designing the

extended high-gain observer (5.50) and then designing the control (5.54). The extended

high-gain observer estimates the nonlinear couplings and disturbances and then by using

feedback control compensates for them in the closed-loop system.

The exosystem (5.64) with the initial condition ω0(0) = 100π will guarantee that the fre-

quencies of the buses track the reference frequency of 50 Hz. The constants were cho-
sen as β = 1/(|V |2) where Vi = V for i = 1, 2, . . . , N and D = diag(0, 0, 3, . . . , 0),

d = col(0, 0, 3, . . . , 0). The fact that d3 = 3 implies that bus 3 receives information about the

reference phase and frequency from the exosystem (5.64). The proposed controller is tested

on the IEEE 30 bus test system [118]. The line admittances were extracted from [118] and

the line voltages were assumed to be 11 kV for all buses. The mass and damping coefficients

are given by mi = 105 kgm2, ˜di = 1s−1, ωs = 100π rad/s, S = 100 MVA. For simulation the

initial condition of the exosystem (5.64) was chosen as (θ0, ω0) = (0, 100π) and the initial
condition of the system (5.63) was chosen to be in the range θi ∈ [0, π], ωi ∈ (290, 314.15)

for i = 1, 2, . . . , N and the controller parameters were chosen as K = [9 6]. The observer

parameters were chosen as α1 = 3, α2 = 3, α3 = 1, 
 = 0.0001 and the saturation levels

were taken as Mi = 436 for i = 1, . . . , 30. The saturation levels were chosen from simu-

lation of the closed-loop system under the state feedback controller to see the maximum

value of the control signal. In power systems the graph is undirected because the generators

are connected by physical links, which makes the coupling bidirectional. Since the graph

is undirected it implies that the matrix G is symmetric and positive definite and therefore

138

satisfies case I with

1
2

λmax(LG) = 52.4. Therefore, gm was chosen as gm = 54. The power

Figure 5.6: Phase of the buses

Figure 5.7: Frequency of the buses

system was initially not in the operational equilibrium and therefore as seen from Fig. 5.6

and Fig. 5.7 the phase and frequency synchronize to the trajectory of the exosystem (5.64).

The power load at buses 2, 3 and 7 was increased by 30 per unit at τ = 6, which results in

the immediate desynchronization of the frequencies. After the desynchronization the phases

and frequencies under the distributed controller again synchronize towards the phase and

frequency of the exosystem (5.64). Fig. 5.8 shows the per unit control signal at each of the

buses.

139

Figure 5.8: Control

5.9 Conclusion

In this chapter, we presented a new distributed control design for practical synchronization in

a network of nonlinear heterogeneous agents. A state transformation was done to transform

the system into relative dynamics where the agents are coupled. The problem then changes

into a stabilization problem for the network of agents. Extended high-gain observers were

used to estimate the nonlinear coupling terms and then cancel them by feedback control.

The proposed distributed controller requires only relative output measurement from its

neighbors, which is a less restrictive information requirement from a practical standpoint.

The efficacy of the controller was tested on the example from [55] and on the IEEE 30 bus

system [118] for the frequency control of power systems.

140

Chapter 6

Power Systems Frequency

Synchronization

6.1

Introduction

In this chapter, we study the frequency control of power systems when the frequency mea-

surement is available. The main reason behind assuming the availability of the frequency

measurement is that the controller designed in this chapter does not use feedback lineariza-

tion to cancel the power system dynamics compared to the previous chapter. The designed

controller compensates for the time-varying power demand and does not cancel the rich

power system dynamics.

The existing approaches to achieve frequency synchronization assume the unknown power

demand to be constant. However, increasing use of renewable energy sources causes the power

demand to fluctuate at the same timescale as the power system dynamics and therefore

approximating the power demand by a constant value becomes unrealistic [74].

In this

chapter we design a dynamic consensus based extended high-gain observer algorithm [119]

that achieves practical frequency synchronization in the presence of unknown time-varying

power demand.

141

6.1.1 System Model

We consider a lossless, connected and network-reduced power system with N generators

modeled by the following swing equation [65]

˙θ = ˜ω − ωref

M ˙˜ω = −D(˜ω − ωref ) − P (t) − ∇U (θ) + u

(6.1a)

(6.1b)

where θ = col(θ1, . . . , θN ) are the generator rotor angles, ˜ω = col(˜ω1, . . . , ˜ωN ) are the

velocities of the generator rotors with respect to a fixed reference frame [65], ωref is the

synchronous frequency which is typically 120π for a 60 Hz system, M = diag(m1, . . . , mN )

represents the inertia matrix with mi > 0, D = diag(d1, . . . , dN ) represents the droop coef-

ficient matrix with di > 0, P (t) = col(P1(t), . . . , Pn(t)) represents the time-varying vector of
unknown power demand, ∇U (θ) ∈ RN represents the power flow and u = col(u1, . . . , uN ) is

the control input. All quantities are normalized following the per unit representation of the

power system models [65]. The time-varying power demand can be decomposed into

P (t) = ¯P + ˜P (t)

(6.2)

where ¯P and ˜P (t) represent the unknown constant and time-varying components of the

power-demand.

The power system network can be viewed as an undirected graph G = (V,E), where
the nodes, V, represent the number of generators and the edges, E, represent the power

142

transmission lines connecting the generators. We introduce the potential function given by

N(cid:88)

i,j=1

U (θ) = −1
2

BijViVj cos(θi − θj)

where Bij = Bji ≥ 0 is the susceptance of the line connecting generators i and j with
constant terminal voltage magnitudes Vi, Vj > 0. The ith component of the gradient of the

potential function is

(∇U (θ))i =

N(cid:88)

j=1

BijViVj sin(θi − θj)

(6.3)

∇U (θ) satisfies a zero net power flow balance given by 1T∇U (θ) = 0. The terms U (θ) and
∇U (θ) can be compactly written as

U (θ) = −1T Γcos(BT θ), ∇U (θ) = BΓsin(BT θ)

where B ∈ RN×m is the incidence matrix of the power system network, with N generators
and m transmission lines, cos(·) = col(cos(·), . . . , cos(·)), sin(·) = col(sin(·), . . . , sin(·)),
Γ ∈ Rm×m is a diagonal matrix which encodes the susceptance Bk of the power lines and
the voltage amplitudes Vi, Vj at each edge as [Γ]kk = BkViVj, for each edge k = (i, j) ∈ E

and Bk = Bij.

For ease of analysis we apply a change of coordinates as ω = ˜ω−ωref , which represents the

frequency deviation and δ = Πθ =
θ, which is inspired by the center-of-inertia
BT 11T θ = BT θ,
coordinates, see [68], [65]. From which we have BT δ = BT Πθ = BT θ − 1
N
since BT 1 = 0. By a slight abuse of notation we refer to the potential function of δ by the

(cid:18)

I − 1
N

11T

(cid:19)

143

same symbol U . Therefore, the system in the new coordinate becomes

˙δ = Πω

M ˙ω = −Dω − P (t) − ∇U (δ) + u

(6.4a)

(6.4b)

where ∇U (δ) = BΓsin(BT δ).

6.2 Control Objective

The control objective is to achieve frequency synchronization, i.e., to regulate the frequency

deviation ω to zero while achieving proportional power sharing among the generators. We
characterize the steady-state solution (δ∗, ω∗) of (6.4), with steady-state control u(t) = u∗(t),
where ω∗ is a constant that belongs to ker(Π), which implies that ω∗ is a constant vector

with all the elements being the same. The constant steady-state solution satisfies

0 = Πω∗

0 = −Dω∗ − P (t) − ∇U (δ∗) + u∗(t)

(6.5a)

(6.5b)

Lemma 6.1, [74] : If there exists (δ∗, ω∗) ∈ Im Π × RN , such that (6.5) holds, then
ω∗ = ωs1, with

1T (u∗(t) − P (t))

1T D1

ωs =

(6.6)

Remark 6.1: From, equation (6.5), we can observe that the left-hand-side is constant

while the right-hand-side contains a time-varying quantity P (t). This equation can only be

satisfied when P (t) satisfies a certain condition which will be defined later in this section.

144

From equation (6.6), it is clear that in order to regulate the frequency deviation ω to zero,
there are infinitely many choices for u∗(t) to satisfy 1T (u∗(t) − P (t)) = 0. This also implies
that the controller u∗ is time-varying. Therefore, we split the control into constant and

time-varying parts as

u∗(t) = ¯u∗ + ˜u∗(t)

where ¯u∗ and ˜u∗(t) should satisfy 1T (¯u∗− ¯P ) = 0 and 1T (˜u∗(t)− ˜P (t)) = 0. This freedom in
the choice of ¯u∗ leads to the design of controllers that are optimal with respect to a certain

cost function. The controllers are designed based on the concept of optimal power dispatch

given a constant power demand, where the cost is dependent only on the amount of power

produced at each generator. A solution to the optimal dispatch problem is given by [74]

opt = Q−1 11T ¯P
¯u∗
1T Q−11

(6.7)

where Q = diag(Q1, . . . , Qn) is the marginal cost matrix. From (6.7), it is clear that the

power generated at each node i, is inversely proportional to its marginal cost Qi and optimal
steady-state controller is chosen as ¯u∗ = ¯u∗
synchronization, we need to satisfy 1T (u∗(t) − P (t)) = 0, and therefore u∗(t) is

opt. However, in order to achieve frequency

u∗(t) = Q−1 11T P (t)
1T Q−11

= ¯u∗ + ˜u∗(t)

(6.8)

where ˜u∗(t) = Q−1 11T ˜P (t)
1T Q−11
Equation (6.5b) can now be satisfied with u∗(t) = ¯u∗ + ˜u∗(t) and ω∗ = 0, from which

, represents the time-varying part of the steady-state controller.

BΓsin(BT δ∗) =

P (t)

(6.9)

(cid:33)

− I

(cid:32)
Q−1 11T

1T Q−11

145

From, equation (6.9), we can observe that left-hand-side is constant while the right-hand-side

contains a time-varying quantity ˜P (t); therefore we require ˜P (t) to belong to

(cid:32)

ker

Q−1 11T

1T Q−11

− I

(cid:33)

.

It is shown in [74], that R(Q−11) is included in ker

make the following assumption.

(cid:32)
Q−1 11T

1T Q−11

(cid:33)

− I

. Therefore, we

Assumption 6.1, [74], [76]: The time-varying part of the power demand satisfies

˜P (t) = Q−11˜p(t)

(6.10)

where ˜p(t) is some bounded unknown time-varying function with bounded derivative.

Assumption 6.1 guarantees the existence of the steady-state solution (6.5), despite the pres-

ence of the time-varying quantity P (t). From Assumption 6.1, equation (6.9) is simplified

as

BΓsin(BT δ∗) =

(cid:32)
Q−1 11T

1T Q−11

(cid:33)

− I

¯P

(6.11)

Next we make a feasibility assumption.
Assumption 6.2, [74]: For a given constant ¯P , there exists δ∗ ∈ Im Π, such that equation

(6.11) has a solution.

Assumptions 6.1 and 6.2 guarantees the existence of a constant steady-state solution (6.5),

despite the presence of the time-varying quantity P (t).
Assumption 6.3, [68] : The synchronous solution (6.5) is such that BT δ∗ ∈ Θ :=

(cid:16)−π

2

(cid:17)m

, for a constant ρ ∈(cid:16)

(cid:17)

.

0,

π
2

+ ρ,

π
2

− ρ

The assumption on δ∗, is standard in stability analysis of power systems [68], and is usually

called the security constraint [71].

146

Remark 6.2: Assumptions 6.1, 6.2 and 6.3 are made in order to guarantee the existence

of the synchronous solution (6.5). These assumptions are standard in the power systems

literature [68], [73], [74], [76].

6.3 Controller Design Under known Time-Varying Power

Demand

6.3.1 Communication Topology

We will proceed with a two-step approach for the design of the controller, where in the first

step, the controller will be designed assuming that the power demand is known.

In the

second step, we will use observers to remove the requirement of knowing the power demand.

In this section we discuss the first step.

Assumption 6.4: There exists a balanced, directed, strongly connected communication topol-

ogy to allow the exchange of information among the controllers.
The communication graph is represented by ˜G = (V, ˜E, ˜A) and the graph Laplacian is de-

noted by L. In general, the communication graph can be different from the graph represented
by the power transmission lines i.e., ( ˜E, ˜A) can be different from (E,A). The generators are
coupled physically through the graph G, which evolve in a slow time-scale, while the dynamic
controllers exchange information through the graph ˜G and evolve in a fast time-scale.

6.3.2 Dynamic Average Consensus

To achieve a steady-state controller of the form (6.8), a natural intuitive idea is that if each

controller knows its local power demand Pi, then it can communicate its local Pi to its

147

neighbors. Such a strategy would require an all-to-all communication graph so that each

controller can calculate the sum of the total power demand. However, such design would be

quite restrictive because an all-to-all communication topology (complete graph ) is a costly

communication requirement. In this section, for the design of the controller, we will assume

that each generator has knowledge about its own power demand. Next, to alleviate the

requirement of a complete graph, we will apply ideas from dynamic average consensus [120].

The dynamic average consensus algorithm states that if each node i has access to a time-

varying reference signal ri(t), then using a distributed consensus algorithm it is possible that

each node is able to track the average of the reference signals given by(cid:80)N

j=1 ri(t)/N . The

dynamic average consensus algorithm is given by [120]:

˙zi = − N(cid:88)

aij(zi(t) − zj(t)) + ˙ri(t), ∀ i ∈ {1, . . . , N}

(6.12)

j=1

with the initial condition zi(0) = ri(0). In order to obviate the requirement of the derivative
of the reference signal, we do a change of coordinates [120], zi = ri(t) − pi, from which we

get

N(cid:88)

j=1

˙pi =

aij(zi(t) − zj(t)), pi(0) = 0

(6.13)

The applications and a tutorial of the dynamic average consensus algorithm can be found

in [120]. We choose the reference signal ri(t) at each generator to be σi, which is given by

σi = −(∇U (δ))i − Pi(t), ∀ i = 1, . . . , N.

or in vector form σ = −∇U (δ)) − P (t). For the controller design we consider that each

generator has access to the signal σi. The signal σi is treated as the known time-varying

148

reference signal as in the case of the dynamic consensus algorithm.

The distributed dynamic average consensus algorithm is given by

µ ˙p = −Lp + Lσ = Ly

(6.14)

where µ is a small positive consant, p ∈ RN is the state of the dynamic consensus algorithm
and y = σ − p.

Remark 6.3 : In the literature the dynamic consensus algorithm is not coupled with any

other dynamics. In the current problem it will be coupled with the power system dynamics

as the controller will depend on the state p. Therefore, we require p to be faster than the

power system dynamics and we achieve it by introducing the parameter µ.

6.3.3 Analysis of the Closed-loop System

We define the dynamic consensus tracking error as

e = y − 1T σ1
N

= σ − p − 1T σ1
N

=

(cid:34)

(cid:35)

I − 11T
N

σ − p = Πσ − p

Differentiating the above equation and using L1 = 0 we have,

µ ˙e = −Le + µΠ ˙σ

We define the change of coordinates as ˜e = T T e where T =

(cid:21)

1 R

(cid:20) 1√

N

(6.15)

in which RT R =

I, RT 1 = 0. Hence, T T T = T T T = I. By the change of coordinates we have

µ ˙˜e = −(T T LT )˜e + µT T Π ˙σ

149

0

0

0 RT LR

 , T T Π =

 1√

1T

N
RT

(cid:20)

I − 11T
N

(cid:21)

=



 0

RT

T T LT =

The system (6.15) is transformed to

˙˜e1 = 0

µ ˙s = − ˜Ls + µRT ˙σ

(6.16a)

(6.16b)

where s = col(˜e2, . . . , ˜eN ), ˜L = RT LR and the eigenvalues of ˜L are the nonzero eigenvalues

of L.

The initial condition of the dynamic consensus state p(t) is taken as p(0) = 0, from which

the initial condition of ˜e1(0) is given by

˜e1(0) =

1T e(0)√
N

=

1T [Πσ(0)]

√

N

=

1T Πσ(0)√

N

= 0

where we have used 1T Π = 0. Therefore, we conclude that

˜e1(t) ≡ 0, ∀ t ≥ 0

Assumption 6.5: The controller at each generator has access to the quantity Q−1

i /1T Q−11.

Remark 6.4: Assumption 6.5 requires a central authority which will communicate the
quantity Q−1

i /1T Q−11 to the generators. This communication occurs only when the system

is initialized or when there is a change in the cost matrix Q. This assumption is not restrictive

because the optimal dispatch problem is solved centrally offline in order to obtain the optimal

controller (6.8) and then the cost Qi is made available to the generators. Therefore, the

150

information Q−1

i /1T Q−11 can be made available to the controllers after the optimal dispatch

problem is solved centrally.

The controller is chosen as

u = −F1(σ − p)

(6.17)

where F1 = (N/(1T Q−11))Q−1. Substituting y = e + (1T σ1)/N and after some simplifica-

tions we have

u(t) = u∗(t) − F1e

The error e can be written as e =

1√
N

becomes

˜e1 + Rs = Rs, since ˜e1 = 0, from which the control

u(t) = u∗(t) − F1Rs

and the closed-loop system is given by

˙δ = Πω
M ˙ω = −Dω − ∇U (δ) − P (t) + u∗(t) − F1Rs
µ ˙s = − ˜Ls − µRT∇2U (δ)ω − µRT Q−11 ˙˜p(t)

(6.18)

(6.19a)

(6.19b)

(6.19c)

The closed-loop system (6.19) does not have an equilibrium point due to the presence of

the time-varying component ˙˜p(t) in equation (6.19c), therefore we proceed with a singular

perturbation framework. The reduced-order model of (6.19), obtained by setting µ = 0, is

˙δ = Πω

M ˙ω = −Dω − ∇U (δ) − P (t) + u∗(t)

(6.20a)

(6.20b)

151

and the boundary-layer-model, obtained by choosing t = µτ and setting µ = 0, is

= − ˜Ls

ds
dτ

(6.21)

Theorem 6.1: Consider the closed-loop system (6.19) obtained from the power system model

(6.4), the dynamic consensus algorithm (6.14) and the controller (6.17). Let (¯δ(t), ¯ω(t)) and

(δ(t), ω(t)) be the trajectories of the reduced-order model (6.20) and the closed-loop system
(6.19), respectively, with (δ(0), ω(0)) = (¯δ(0), ¯ω(0)). Then, there exists µ∗ > 0 such that for
all µ ∈ (0, µ∗),

δ(t) − ¯δ(t) = O(µ), ω(t) − ¯ω(t) = O(µ), ∀ t ≥ 0

(6.22)

Proof : The reduced-order model (6.20) has a synchronous solution (δ∗, 0), which follows

from Assumptions 6.1 and 6.2. We first show the exponential stability of the reduced-order

system by using the following Lyapunov function from [73],

V (δ, δ∗, ω) = U (δ) − U (δ∗) − ∇U (δ∗)T (δ − δ∗) +

ωT M ω + h(∇U (δ) − ∇U (δ∗))T M ω

1
2

where h is a small positive constant. It can be shown from [73, Lemma 2] that

γ1||δ − δ∗||2 ≤ ||∇U (δ) − ∇U (δ∗)||2 ≤ γ2||δ − δ∗||2

(6.23)

where γ1 and γ2 are positive constants and

cmin||x||2 ≤ V (δ, δ∗, ω) ≤ cmax||x||2

(6.24)

152

where x = col(δ − δ∗, ω), cmax is a positive constant and cmin is positive for sufficiently

small h and therefore V is a Lyapunov function candidate. Taking the time derivative of V ,
along the reduced-order model (6.20), and substituting P (t) = −∇U (δ∗) + u∗(t) from the

steady-state of (6.20b), we have

˙V = −ωT Dω−h(∇U (δ)−∇U (δ∗))T (∇U (δ)−∇U (δ∗))−h(∇U (δ)−∇U (δ∗))T Dω+hωT H(δ)ω

where H(δ) = (M∇2U (δ) + ∇2U (δ)M )/2 which is obtained by using ∇2U (δ)Π = ∇2U (δ)
since ∇2U (δ)1 = 0. Since ∇2U (δ) is bounded for any δ, we have ||H(δ)|| ≤ k1 where k1 is

a positive constant. From which we have

||∇U (δ) − ∇U (δ∗)||
T

||ω||

||∇U (δ) − ∇U (δ∗)||


||ω||

Υ

˙V ≤ −

where

Υ =



h

2

−hλmax(D)



−hλmax(D)
λmin(D) − hk1

2

By choosing h sufficiently small we can make the determinant of Υ positive thereby making Υ
a positive definite matrix. Let χ = col(||∇U (δ)−∇U (δ∗)||,||ω||). Since Υ is positive definite,
we have ˙V ≤ −k2||χ||2, for some positive constant k2. From (6.23), ||χ||2 ≥ k3||x||2, where

k3 = min(1, γ1). Hence

˙V ≤ −k2k3||x||2 ≤ −k4||x||2

where k4 = k2k3. Therefore, we can conclude that (6.20) is exponentially stable.

The boundary-layer system is exponentially stable and a Lyapunov function candidate for

(6.21) is given by Vs = sT Gs, where G = GT > 0, is the solution to the Lyapunov equation

153

G ˜L + ˜LT G = I. The time-derivative of Vs along (6.21) is given by ˙Vs = −||s||2.

The time-derivative of V and Vs by taking into account the coupling between the systems

(6.19b) and (6.19c) is given by,

(cid:21)

(cid:112)

˙V ≤ −k5
2
˙Vs ≤ − k7
2µ

(cid:20)√
(cid:20)√
γ2)/(cid:112)cmin · λmin(G), k7 = 1/λmax(G), k8 =
2l||GRT||/(cid:112)cmin · λmin(G), k9 = 2pm||GRT Q−11||/(cid:112)λmin(G), || ˙˜p(t)|| ≤ pm, and ||∇2U (δ)|| ≤

√
where k5 = k4/cmax, k6 = ||F1R||(1 + h

V − k5
2
Vs − k7
2µ

V s − 2µk8
k7

V − 2µk9
k7

V − 2k6
k5

Vs
√

√

V
√
V s

(6.25a)

(6.25b)

(cid:21)

l, for some positive constants pm and l. From (6.25a) and (6.25b), in order to have ˙V < 0

and ˙Vs < 0, we need

(cid:112)

√
Vs ≤ (k5/2k6)

V , and

(cid:112)
√
Vs ≥ (2µk8/k7)

V + 2µk9/k7

√
Considering equality, we have two lines in (
√
V = µk10 and

and the intersection points are given by

V ,

√

√
Vs), which intersect for sufficiently small µ

Vs = µk11, for some positive con-

√

√

V = µk10,

stants k10 and k11. Moreover, for sufficiently small µ, on the boundaries
V s =
µk11, we have ˙V < 0 and ˙Vs < 0. From which we can conclude that ˙V < 0 for V ≥ µ2k2
and ˙Vs < 0 for Vs ≥ µ2k2
11.
Next we find a compact set Ωx = {V ≤ d}, which is contained in Θ. The constant d can be
chosen [68] as d = cmin˜c2/λmax(BBT ), where ˜c > 0 is such that for ||BT δ − BT δ∗|| ≤ ˜c, also
satisfies BT δ ∈ Θ. Therefore, for sufficiently small µ, we have ˙V < 0 on V = d and ˙Vs < 0
on Vs = a, where a > 0. Therefore, the set Ω = Ωx × {Vs ≤ a} is positively invariant.

10

Since the reduced-order model and boundary layer are exponentially stable, it follows from

154

[91, Theorem 11.2] that (6.19) has a unique solution x(t, µ) on [0,∞) and

x(t, µ) − ¯x(t) = O(µ)

(6.26)

holds uniformly for t ∈ [0,∞), where ¯x(t) = (¯δ(t) − δ∗, ¯ω(t)) is the solution of the reduced-
(cid:4)

order model (6.20). Therefore, (6.22) follows from (6.26).

6.4 Controller Design Under Unknown Time Varying

Power Demand

In this section we proceed with the second step and design the controller when the power

demand is unknown.

6.4.1 Extended High-Gain Observer

In the previous section we assumed that each local controller had access to the signal σi.

However, in practice this signal is typically not available because the power demand is not

known beforehand. We use a high-gain observer to estimate the signal σi, which can then

be used by the dynamic consensus algorithm. The signal σi, composed of a time-varying

power demand and the power-flow, is estimated using an extended high-gain observer. The

extended high-gain observer is constructed as

˙ˆω = −M−1D ˆω + M−1ˆσ + M−1u +

(ω − ˆω)

α1



˙ˆσ =

α2


2 (ω − ˆω)

(6.27a)

(6.27b)

155

where α1, α2 are any positive constants, 
 is a small positive constant, ˆω ∈ RN , ˆσ ∈ RN

are the estimates of ω and σ. Next we define the scaled estimation errors as

ω − ˆω




ζ1 =

ζ2 = σ − ˆσ

,

where ζ1 ∈ RN , ζ2 ∈ RN . Using (6.4) and (6.27) we write the observer error dynamics as


 ˙ζ = A0ζ + 
B1ζ + 
B2∆(t, δ, ω)

(6.28)

where ζ = col(ζ1, ζ2), ∆(t, δ, ω) = −Q−11 ˙˜p(t) − ∇2U (δ)ω,

−α1I M−1

 , B1 =

−M−1D

 , B2 =

A0 =

 ,

 0

−I

−α2I

0

0

The eigenvalues of A0 are given by the roots of the polynomials

s2 + α1s + α2m−1

i

, for i = 1, . . . , N.

Since the coefficients of the polynomials are positive it follows that the roots of the poly-

nomials are negative, from which we conclude that A0 is Hurwitz. To avoid the peaking

phenomenon [90], we saturate the observer estimates outside the positively invariant set Ω

before using them in feedback control. The signal ˆσ is saturated as

(cid:18) ˆσi

(cid:19)

Mis

ˆσis = Missat

156

where sat is the saturation function and

Mis >

max

δ∈Ω, ||P (t)||≤Pm

σi

where Pm represents a measure of the maximum power demand. The distributed dynamic

average consensus algorithm using the saturated estimate ˆσs = col(ˆσ1s, . . . , ˆσns) is given by

and the controller is chosen as

µ ˙p = −Lp + Lˆσs

u = −F1(ˆσs − p)

(6.29)

(6.30)

6.4.2 Analysis of the Closed-loop System

By defining the dynamic consensus tracking error as e = σ− p− 1T σ1
N

and using the change

of coordinates ˜e = T T e, the error dynamics is given by

µ ˙˜e = −(T T LT )˜e + T T L(ˆσs − σ) + µT T Π ˙σ

where T T L = col(0, RT L), from which we have

˙˜e1 = 0

µ ˙s = − ˜Ls + RT L(ˆσs − σ) + µRT ˙σ

(6.31a)

(6.31b)

157

where s = col(˜e2, . . . , ˜en). Since, we choose p(0) = 0, it follows that

˜e1(t) = 0,

∀ t ≥ 0

Next we define the Lyapunov function Vζ = ζT Sζ, where S is the solution to the Lyapunov

0 S = −I.

equation SA0 + AT
Lemma 6.2: Let (δ(0) − δ∗, ω(0), s(0)) lie in the interior of the set Ω, and let the initial
observer states lie in (ˆω(0), ˆσ(0)) ∈ Y , where Y is a compact subset of R2n. Then, there
exists a positive constant κ1 such that the set Λ = Ω × {Vζ ≤ κ1
2} is positively invariant
and the trajectory (δ(t) − δ∗, ω(t), s(t), ζ(t)) enters the set Λ in finite time T (
) > 0, where

T (
) = 0.

lim

→0

Proof : We first show that Λ is positively invariant by calculating the derivatives of V , Vs

and Vζ = ζT Sζ on the boundaries V = d, Vs = a and Vζ = κ1
2, respectively. The saturation

is no longer active when the system trajectory is in Λ, which implies ˆσis(t) = ˆσi(t). Taking

the time derivative of Vζ we have


 ˙Vζ ≤ −||ζ||2 + 
q1||ζ||2 + 
q2||ζ||

where q1 = 2||SB1||, q2 = 2du||SB2||, and ||∆|| ≤ du. After some simplifications we have


 ˙Vζ ≤ −(1/4)||ζ||2, ∀ ||ζ|| ≥ 4
q2

By taking κ1 = 16λmax(S)q2

2, we have


 ˙Vζ ≤ −(1/4)||ζ||2, ∀ Vζ ≥ κ1
2

158

Next by taking the time derivative of V and Vs, we have

˙V ≤ −(k5/4)V − k5
2

√
V

V − 2k6
k5

˙Vs ≤ − k7
4µ

Vs − k7
2µ

√
V s

√

V s − 2µk8
k7

(cid:20)√
(cid:20)√

(cid:21)

(cid:112)

Vs

,∀ V ≥ 
2q3
(cid:21)

V − 2µk9
k7

for all Vs ≥ 
2q4, where q3, q4 are positive constants. It can be shown that for sufficiently
small µ, ˙V < 0 for V ≥ µ2q5 and ˙Vs < 0 for Vs ≥ µ2q5, for some positive constants q5 and q6.
Furthermore, we can choose 
 sufficiently small, such that the set {V ≤ 
2q3} × {Vs ≤ 
2q4}
is in the interior of the set {V ≤ µ2q5} × {Vs ≤ µ2q6}. Therefore, we have ˙V < 0 on the
boundary V = d and ˙Vs < 0 on the boundary Vs = a, which implies Λ is positively invariant.

By following the standard work in high-gain observer theory [90], it can be shown that
(δ(t) − δ∗, ω(t), s(t)) ∈ Ω for all t ∈ [0, ¯T ], where ¯T > 0 is some finite time independent of
µ. The state ζ(t) enters the set {Vζ ≤ κ1
2} within time T (
) > 0, where lim

→0
therefore by choosing 
 sufficiently small we have T (
) < ¯T , from which we can conclude
that (δ(t) − δ∗, ω(t), s(t), ζ(t)) ∈ Λ for all t ≥ T (
).
(cid:4)
From Lemma 6.2, we have ζ(t) = O(
),∀ t ≥ T (
), which implies ˆσi(t) = σi(t) + O(
),∀ t ≥
T (
) and the closed-loop system for t ≥ T (
) is given by

T (
) = 0 and

˙δ = Πω
M ˙ω = −Dω − ∇U (δ) − P (t) + u∗(t) − F1(Rs − ζ2)
µ ˙s = − ˜Ls − RT Lζ2 + µRT ˙σ


 ˙ζ = A0ζ + 
B1ζ + 
B2∆

(6.32a)

(6.32b)

(6.32c)

(6.32d)

159

Theorem 6.2: Consider the closed-loop system (6.32) obtained from the power system model

(6.4), the dynamic consensus algorithm (6.29) and the controller (6.30). Let (¯δ(t), ¯ω(t)) and

(δ(t), ω(t)) be the trajectories of the reduced-order model (6.20) and the closed-loop system

(6.32), respectively with (δ(0), ω(0)) = (¯δ(0), ¯ω(0)). Then given any Ξ > 0, there exists
µ∗ > 0, and for each µ ∈ (0, µ∗], 
∗ = 
∗(µ) exists such that for each µ ∈ (0, µ∗] and

 ∈ (0, 
∗(µ)], all system trajectories are bounded, and

||δ(t) − ¯δ(t)|| ≤ Ξ,

||ω(t) − ¯ω(t)|| ≤ Ξ, ∀ t ≥ 0

(6.33)

Proof : The proof is done in three steps where we first show (6.33) during the time period
[ ˜T ,∞), where ˜T > 0 is some finite time followed by [0, T (
)] and then during [T (
), ˜T ].
Let ˜x(t) = col(δ(t)−δ∗, ω(t), s(t)) be the solution of (6.32) and ˜xr(t) = col(δr(t)−δ∗, ωr(t), sr(t))

be the solution of the reduced-order model of system (6.32) obtained by setting 
 = 0, which

is represented by Eq. (6.19), with ˜xr(0) = ˜x(0). From Theorem 6.1 and Lemma 6.2, given
any Ξ > 0, there exists µ1 > 0 and 
1 = 
1(µ) > 0 such that for each µ ∈ (0, µ1] and

 ∈ (0, 
1(µ)]

||˜x(t)|| ≤ Ξ/4,

||˜xr(t)|| ≤ Ξ/4, ∀ t ≥ ˜T

Therefore it follows that

||˜x(t) − ˜xr(t)|| ≤ Ξ/2, ∀ t ≥ ˜T

Using δ(t) − ¯δ(t) = δ(t) − δr + δr(t) − ¯δ(t) and ω(t) − ¯ω(t) = ω(t) − ωr + ωr(t) − ¯ω(t), there
exists µ2 > 0 and 
2 = 
2(µ) > 0, such that for all µ ∈ (0, µ2] and 
 ∈ (0, 
2(µ)], we have

||δ(t) − ¯δ(t)|| ≤ Ξ, ||ω(t) − ¯ω(t)|| ≤ Ξ, ∀ t ≥ ˜T

160

Next we show (6.33) during the time period [0, T (
)]. From Lemma 6.2, we know that
(δ(t) − δ∗, ω(t), s(t)) ∈ Ω during the time period [0, T (
)]. Therefore the right-hand-side of

(6.32a)-(6.32b) and (6.19a)-(6.19c) is bounded by a constant independent of 
, from which

we have

||δ(t) − δr(t)|| ≤ 2bT (
),||ω(t) − ωr(t)|| ≤ 2bT (
),

for all t ∈ [0, T (
)], where lim
T (
) = 0 and b is a positive constant. Similar to the first step

→0
using δ(t) − ¯δ(t) = δ(t) − δr + δr(t) − ¯δ(t) and ω(t) − ¯ω(t) = ω(t) − ωr + ωr(t) − ¯ω(t), there
exists µ3 > 0 and 
3 = 
3(µ) > 0 such that for each µ ∈ (0, µ3] and 
 ∈ (0, 
3(µ)], we have

||δ(t) − ¯δ(t)|| ≤ Ξ, ||ω(t) − ¯ω(t)|| ≤ Ξ,∀ t ∈ [0, T (
)]

Over the time interval [T (
), ˜T ], equations (6.32a)-(6.32c) under output feedback is O(
)

perturbation from the corresponding model (6.19a)-(6.19c). Therefore, it follows from the

continuous dependence of solutions of differential equation on initial conditions and parame-

ters [91, Theorem 3.5] and by repeating the arguments in the previous steps that there exists
µ4 > 0 and 
4 = 
4(µ) > 0 such that for each µ ∈ (0, µ4] and 
 ∈ (0, 
4(µ)], we have

||δ(t) − ¯δ(t)|| ≤ Ξ, ||ω(t) − ¯ω(t)|| ≤ Ξ, ∀ t ∈ [T (
), ˜T ]

Therefore, (6.33) follows by choosing µ∗ = min{µ1, µ2, µ3, µ4} and 
∗ = min{
1, 
2, 
3, 
4}.
(cid:4)

161

6.5 Simulation

We illustrate the performance of the proposed controller on a connected four area network

taken from [74]. The area connections taken from [74] is illustrated in Fig. 6.1.

Figure 6.1: Four area network

The four area equivalent network can be obtained for the IEEE New England 39-bus

system; see [121]. The parameters of the system in per unit taken from [74], are given

by M = diag(5.22, 3.98, 4.49, 4.22), D = diag(1.60, 1.22, 1.38, 1.42). All the line voltages

are chosen to 1 per unit and line coefficients of the power-flow are taken as B12 = 25.6,

B23 = 33.1, B34 = 16.6, B14 = 21. The generator cost coefficient matrix is given by

Q = diag(1, 0.75, 1.5, 0.5) and the nominal frequency is chosen as ωref = 120π rad/s.

6.5.1 Case I : Unknown Constant Power Demand

For the first scenario the power demand is constant which implies P = ¯P (per unit). The
system is initially at steady-state with P = [2.00, 1.00, 1.50, 1.00]T , t ∈ [0, 10). At timestep 10
the load is increased for t ≥ 10 to P = [2.20, 1.05, 1.55, 1.10]T . For the design of the controller

(6.30) we choose the parameters as α1 = 2, α2 = 1, 
 = 0.001, µ = 0.01 and the saturation
level as ˆMis = ±3 for i = 1, 2, 3, 4. The saturation levels are chosen from simulations to

162

see the maximal values that the trajectories would take when using the controller with the

worst case power demand.

Figure 6.2: Frequency deviation (ω) under unknown constant power demand

Figure 6.3: Control (u) under unknown constant power demand

Fig. 6.2 and 6.3 illustrates the frequency deviation under the controller (32). Compared

to the results in [74], the frequency deviation is roughly ten times smaller. Morever the

control signal quickly reaches steady-state as seen from Fig. 6.3.

163

020406080100120Time(s)-10-8-6-4-202468Frequency deviation, ω (pu)×10-4Area 1Area 2Area 3Area 4708090100110120-4-2024×10-6020406080100120Time(s)0.60.811.21.41.61.822.22.4Generation, u (pu)Area 1Area 2Area 3Area 46.5.2 Case II : Unknown Time-Varying Power Demand with vari-

able Frequency

For the second scenario we consider the unknown power-demand to be time-varying with

variable frequency. The power demand is taken as:



t ∈ [0, 10)
[2.00, 1.00, 1.50, 1.00]T ,
[2.20, 1.05, 1.55, 1.10]T + 0.05Q−11˜p1(t), t ∈ [10, 50)
[2.20, 1.05, 1.55, 1.10]T + 0.07Q−11˜p2(t), t ∈ [50, 80)
[2.20, 1.05, 1.55, 1.10]T + 0.09Q−11˜p3(t), t ≥ 80
(cid:19)

(cid:19)

(cid:19)

t

, ˜p2(t) = sin

t

, ˜p3(t) = sin

P (t) =

(cid:18)2π

30

where ˜p1(t) = sin

(cid:18)2π

20

(cid:18)2π

10

t

.

Figure 6.4: Frequency deviation (ω) under unknown time-varying power demand

164

020406080100120Time(s)-10-8-6-4-202468Frequency deviation, ω (pu)×10-460708090100110120-505×10-5Area 1Area 2Area 3Area 4Figure 6.5: Control (u) under unknown time-varying power demand

Fig. 6.4 shows the frequency deviation under the proposed controller (6.30) with µ = 0.01,
and 
 = 0.001. The steady-state error is of the order of 10−5 and has oscillations of small

amplitude due to the time-varying power demand. Finally, Fig. 6.5 shows the control,

which is time-varying, due to the time-varying component of P (t), and it quickly reaches

the time-varying steady-state control.

6.6 Conclusion

In this chapter, we presented an observer-based dynamic consensus algorithm to achieve

practical frequency synchronization and power-sharing in the presence of unknown time-

varying power demand. The proposed approach does not assume that the power demand is

generated from a known exosystem compared to the internal model approach [74]. Moreover,

the trajectories of the closed-loop system under the controller (6.30) is arbitrarily close to

the system trajectories (6.20) under the steady-state controller.

165

020406080100120Time(s)0.511.522.5Generation, u (pu)Area 1Area 2Area 3Area 4Chapter 7

Conclusion and Future Work

7.1 Overview of Conclusion

This thesis investigated the use of observers in multi-agent systems to reduce information

exchange and increase the convergence rate. The conclusion is summarized below.

First, we showed that by adding extended high-gain observers to each agent except the

root agent, the convergence rate of the star topology could be increased to match the conver-

gence rate of a complete graph. Furthermore, we showed that the trajectories of the agents

under the star topology approach the trajectories of the agents under a complete graph for

sufficiently small 
.

Second, we design a scalable consensus algorithm for first-order agents using PD control,

where the eigenvalues of the closed-loop Laplacian matrix are invariant with respect to the

size of the network for general directed graphs. As a result, the convergence of the consensus

protocol does not slow down when the network size increases for non-expander graphs. The

PD controller is realized using a high-gain observer, and we show that the trajectories of

the closed-loop system when the high-gain observer can be brought arbitrarily close to the

trajectories under the PD controller. Simulation results were presented to demonstrate the

efficacy of the proposed algorithm on a circular directed graph with 100 nodes and two classes

of graphs, which show the trend of decrease in the second smallest eigenvalue of the graph

166

Laplacian with an increase in network size.

Third, motivated by real-world systems akin to power systems with second-order swing

dynamics [65], and longitudinal vehicle dynamics [102], and effect of an increase in network

size on the performance of nonlinear systems we design a scalable consensus algorithm to

achieve practical synchronization in a leader-follower framework for second-order nonlinear

heterogeneous systems. The synchronization error can be made arbitrarily small by increas-

ing a controller parameter. We show that the control signal is uniformly bounded with

respect to this controller parameter, and therefore by increasing it to achieve an arbitrarily

small synchronization error does not increase the magnitude of the control signal. Unlike the

previous chapters, we realize the controller using a reduced-order high-gain observer. This

is done because we do not use additional controller dynamics, as in the previous chapter.

We show that the synchronization error can be made arbitrarily small by tuning a controller

and observer parameter, respectively. We show that the control signal is uniformly bounded

with respect to these parameters. We demonstrated the efficacy of the proposed controller

with two examples i) a network of oscillators on the IEEE-300 bus system and ii) a platoon

of vehicles.

Fourth, we design an extended high gain observer-based controller to achieve synchroniza-

tion in a leader-follower network of nonlinear multi-agent systems having the same relative

degree r. The agents have access to only relative output information from their neighbors.

The class of systems considered in this chapter is more general than the previous chapters.

But unlike the previous chapters, we focus on synchronization with reduced information

exchange. An extended high-gain observer based on the relative output exchange is used

to estimate the uncertain terms and then using feedback control to cancel them. In other

words, we compensate for the heterogeneous dynamics of each agent using extended high-

167

gain observer and feedback control. Finally, simulations were done on the example from [55]

and with a network of power systems to show the efficacy of the proposed controller.

Fifth, we design a load-estimator-based consensus algorithm that achieves practical fre-

quency synchronization in the presence of unknown time-varying power demand in a network

of lossless, connected and network-reduced power systems. We assume the availability fre-

quency measurement. The controller design procedure is different from the previous chapter

as instead of canceling the power system dynamics, we only compensate for the time-varying

power demand. We show that the frequency synchronization error can be made arbitrar-

ily small by tuning a controller and observer parameter. Finally, simulations on a network

reduced four area network is performed to show the performance of the proposed controller.

Throughout the thesis, we assumed that there is no noise in the measurements. However,

the presence of measurement noise puts a constraint on how high the observer gain could

be, which results in a trade-off between fast convergence to the state estimates and the error

due to measurement noise. In practice, the choice of 
 is bounded from below by the level

of measurement noise in the system. In experimental applications of high-gain observers,

typically, a low-pass filter is used to filter out the high-frequency content of the measurement

noise before using the measurement in the observer [122], [123].

One of the basic ideas to mitigate the effect of measurement noise is by adjusting the

observer gain so that a higher gain is used during the transient period to converge to the state

estimate quickly. Then the gain is lowered as the estimation error approaches the steady-state

since the effect of measurement noise is notable when the estimation error is small. Based

on this idea, some techniques have been developed in the literature, including switching gain

between two values [124], adaptive law to adjust gain [125], and using nonlinear gain where

the gain reduces when the estimation error is small [126].

168

7.2 Future Directions

7.2.1 Stability of Networks with Signed Laplacians

Signed graphs appear in a wide range of applications like social networks, power systems,

and biological networks. The graphs primarily contain edges where the weights on these

edges can be positive or negative. The negative weights have physical interpretations. For

example, antagonistic relations in social relations and the presence of critical transmission

lines in power systems across which the bus angles difference is greater than π/2. The

current research in this field focuses on the conditions required on signed Laplacian, such

that consensus is achieved.

The future research in this area will mainly focus on achieving consensus for any signed

graph. Since the scalable consensus controller designed in this thesis can change the closed-

loop Laplacian matrix using only relative information. This can help in changing the closed-

loop Laplacian of signed networks whose underlying signed Laplacian has positive eigenvalues

and, as a result, help in achieving consensus. The main features of using this scalable

controller will be:

• The distributed controller will use only local relative information

• The closed-loop system under the proposed controller will behave as a virtual Lapla-

cian, where the non-zero eigenvalues can be assigned to be negative, thus achieving

consensus.

One of the main applications of this algorithm will be in the stability analysis of power

systems in the presence of critical transmission lines where the bus angles differ by greater

169

than π/2 and thereby providing solutions to those scenarios where the signed Laplacian is

not positive semi-definite.

7.2.2 Scalable Formation Control of Mobile Robots

Formation control of mobile robots is one of the well-studied distributed control problems

due to its wide applications in surveillance and searching operations. The mobile robots are

generally equipped with omnidirectional range-based proximity sensors, which can measure

the relative position or velocity of a neighboring agent if it lies within its proximity disk.

The main objective of the formation control algorithm is to maintain a pre-specified distance

between the robots while they move or meet at a certain goal point.

In the literature,

nonlinear consensus algorithms have been developed to maintain connectivity among a group

of agents while achieving consensus.

The performance of this algorithm, when the network size increases, needs to be in-

vestigated. This is essential since converging to a certain goal within a specific time is very

important for applications such as search and rescue. Therefore, there is a need to investi-

gate the performance of these algorithms in large sparse graphs and develop new distributed

scalable controllers that can maintain a specific convergence rate with an increase in network

size. The presence of proximity sensors, which make consensus algorithm nonlinear, make

this work challenging and can explain the lack of research done in this area. Future research

in this direction will be the design of new scalable controllers such that performance does

not degrade with an increase in network size. The scalable controllers designed in this thesis

can help in the design of the new controllers.

170

7.2.3 Frequency Synchronization in Higher Order Power System

Models

The second-order power systems model considered in the thesis was lossless, network-reduced,

and the voltage fluctuations were assumed to be constant. As a future direction to this

work, a fourth-order model, including the turbine-governor dynamics, will be considered.

This research direction will help solve the current practical challenges faced in the frequency

synchronization of power systems. The other important directions in this area will include:

• Effects of uncertain mass and damping coefficients

• Effects of loss-of-communication/packet and dropouts/delay.

171

APPENDIX

172

Kronecker Product

The Kronecker product of matrices ¯A ∈ Rm×n and ¯B ∈ Rp×q is defined as



¯A ⊗ ¯B =



¯a11 ¯B . . .
...

...

¯a1n ¯B

...

¯am1 ¯B . . . ¯amn ¯B

where [ ¯A]ij = ¯aij and it satisfies the following properties [6]:

( ¯A ⊗ ¯B)( ¯C ⊗ ¯D) = ( ¯A ¯C ⊗ ¯B ¯D),

( ¯A ⊗ ¯B)T = ¯AT ⊗ ¯BT

¯A ⊗ ( ¯B + ¯C) = ¯A ⊗ ¯B + ¯A ⊗ ¯C

Moreover if ¯A and ¯B are nonsingular matrices then

( ¯A ⊗ ¯B)−1 = ¯A−1 ⊗ ¯B−1

Consensus Algorithm

Consider a group of N agents where each agent is labeled by the index vi and vi ∈ V. Let
G = {V,E,A} encode the communication topology connecting the agents. The information

173

available to each agent is given by

N(cid:88)

ζi =

aij(xj − xi)

i = 1, . . . , N

j=1

The above form of information is available to the agents implies that it can only measure

the relative difference of the states between itself and its neighbors. Each of the agents has

single integrator dynamics represented by

˙xi = ui

i = 1, . . . , N.

We define the total energy of the graph associated with its edges as [19]

N(cid:88)

i,j=1

EG =

1
2

aij(xj − xi)2.

The controller is now chosen as the gradient-based feedback law

ui = −1
2

∂EG
∂xi

=

aij(xj − xi)

N(cid:88)

j=1

for i = 1, . . . , N . The gradient-based feedback law is the standard decentralized controller

used for solving the consensus problem as considered in [19].

174

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