Synchronization of coupled nonlinear dynamical systemswith time-delayKniknie, T.; Nijmeijer, H.; Oguchi, T.

Published: 01/01/2007

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Citation for published version (APA):Kniknie, T., Nijmeijer, H., & Oguchi, T. (2007). Synchronization of coupled nonlinear dynamical systems withtime-delay: analysis and application. (DCT rapporten; Vol. 2007.014). Eindhoven: Technische UniversiteitEindhoven.

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Synchronization of CoupledNonlinear Dynamical Systems

with Time-delay: Analysisand Application

T. Kniknie (s000810)

DCT 2007.014

DCT external traineeship report

Supervisors:

Prof. Dr. Ing. Toshiki OguchiProf. Dr. Henk Nijmeijer

Tokyo Metropolitan UniversityDepartment of Mechanical EngineeringControl Engineering Laboratory

Eindhoven, March, 2007

Preface

This report treats the results of my internship on the Tokyo Metropolitan University in Japan.I have stayed for 15 weeks in the Control Engineering Laboratory of Prof. Dr. Ing. ToshikiOguchi, part of the Mechanical Engineering Department. Hereby I want to thank ProfessorOguchi for the opportunity to stay in his laboratory, as well as the supervision and usefultips. Moreover I want to thank him for all the effort to make this period a pleasant andinteresting educational and cultural experience, for example by arranging a visa and makinga reservation in the International House of the university.

I also want to thank the students of the Control Engineering Laboratory for helping me get-ting around in the university and helping me with the project.

Thijs Kniknie

Abstract

For a network of four unidirectionally coupled nonlinear systems with time-delay, full and par-tial synchronization conditions are derived. Therefore, the stability of the zero solution of theerror dynamics of the network is investigated. A Lyapunov-Krasovskii functional is definedand the conditions, under which the time-derivative is negative are investigated. However, todo so, the nonlinear terms in the error dynamics have to be rewritten.

To deal with the nonlinear terms in the error dynamics, two methods are employed: using theboundedness of the trajectories and linearization of the nonlinear terms and, alternatively,using the sector condition. Formulating the problem as a generalized eigenvalue optimizationproblem gives the maximum allowable value of time-delay at a specific coupling gain, forwhich synchronization is still stable. A specific choice of formulating the error dynamics givesa condition for full or partial synchronization.

Two examples illustrate the two approaches to deal with the nonlinear terms. In both caseswe can conclude that the approach is too conservative to give results that are comparablewith simulations.

As an application of modeling this network, it is used to simulate synchronous neuron activityin the brain. A neuron model, suitable for analysis and simulations, is chosen. The Lur’e typeneuron model is most suitable in this case. The model is used in the network to simulate theCentral Pattern Generator, which controls animal locomotion in the brain. Again, analysisand simulations differ a lot. However, simulations show that the network is indeed able togenerate a variety of oscillation patterns, necessary for controlling animal gaits. Changingthe time-delay in the coupling results in various phase shifts between the system trajectories,corresponding with animal gaits. Time-delay as a representation of speed is thus a acceptablemodeling choice.

A slightly weaker condition for synchronization is used to obtain a stability diagram wherethe cases in which full and partial synchronization occurs is presented, obtained throughsimulations.

Contents

Preface i

Abstract iii

1 Introduction 1

1.1 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Synchronization of a network of neurons . . . . . . . . . . . . . . . . . 2

1.1.2 Time-delay systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Animal gaits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Outline of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Synchronization of coupled systems with time-delay 5

2.1 Design of the network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Stability of the synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Boundedness of the trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 Extension of the small-gain theorem . . . . . . . . . . . . . . . . . . . 10

2.3.3 Boundedness of the trajectories . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Sector condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6 Generalized eigenvalue optimization . . . . . . . . . . . . . . . . . . . . . . . 15

2.7 Full and partial synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8.1 Four unidirectionally coupled Lorenz systems . . . . . . . . . . . . . . 17

2.8.2 Four unidirectionally coupled Lure type neuron models . . . . . . . . 23

3 A synchronization approach on animal gaits 29

3.1 Animal gaits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Central pattern generator . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.2 Time-delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 A network of neuron models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Synchronization of a network of Lur’e type neuron models . . . . . . . . . . . 34

3.4 Simulations with varying coupling gains . . . . . . . . . . . . . . . . . . . . . 37

3.5 The network as central pattern generator . . . . . . . . . . . . . . . . . . . . 39

vi CONTENTS

4 Conclusions and recommendations 41

4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

A Semi-passivity of the Lorenz system 45

B Derivation of LMI (2.85) 47

C Simulation results 51

Chapter 1

Introduction

1.1 Synchronization

In many situations in nature, the occurrence of synchronization can be observed. Examples arethe synchronous flashing of fireflies, singing crickets and dancing people. The Dutch scientistChristiaan Huygens (1629-1695) was one of the first people to recognize synchronization inmechanical engineering. He discovered the synchronous motion of two pendulum clocks on aship, while searching for a method to find longitude at sea (see Figure 1.1).

Figure 1.1: The original drawing of Christiaan Huygens’ pendulum clocks.

His discovery opened a great opportunity for applications. We can think of assembly andtransportation systems, cooperating robots and remote control of mechanical systems.

Synchronization can be understood as the adaption of objects to eachother’s behavior, in sucha way that their behavior is similar. This influence can be a result of some coupling betweenthe objects, how weak it may be. Synchronization does not always imply that the behavior ofobjects is exactly copied; a rhythm of oscillations or a repeating sequence of events can alsobe considered as some kind of synchronization.

2 Introduction

1.1.1 Synchronization of a network of neurons

The brain is an interesting system to study in this case. The occurrence of synchronous activ-ity of parts of the brain can be observed and since the brain controls the body, synchronousbehavior of the body is a logical consequence. From this point of view, the neurons in thebrain can be considered as separate systems, that synchronize when necessary. This necessityappears for example in the case of locomotion of humans and animals. Locomotion requires acertain sequence of contraction of muscles in the body, and the control of these contractionshappens in the brain.

1.1.2 Time-delay systems

Previous research on neuron models showed that in a network of neurons, neighboring neuronsinfluence eachother’s behavior. A neuron receives signals from its neighbours, and thus it isassumed that a network of neurons can be viewed as a network of coupled oscillators [4]. Anice photograph of a network of neurons is shown in Figure 1.2(a). A graphical interpretationof their interconnections is presented in Figure 1.2(b). The interest in time-delay systemsarises when one considers that the transport of a signal from one neuron to another willtake some time. Synchronization of time-delay systems is thus a useful topic to include,when investigating the behavior of a network of neurons. In this research, a method to findconditions for synchronization of coupled systems with time-delay will be employed.

(a) A network of neurons. (b) A graphical interpretation of a network ofneurons.

Figure 1.2: Neurons.

1.2 Animal gaits 3

1.2 Animal gaits

As stated above, parts of the brain control the movement of the body. It is widely acceptedthat animal locomotion is generated and controlled mostly by a so called central pattern

generator (CPG), which is a network of neurons in the central nervous system, capable ofproducing rhythmic output [6]. The neurons generate a current, that is transported to thelimbs in order to operate the muscles. For different kinds of gaits, different patterns of signalsare necessary to let the limbs move in the desired sequence. For example, if we look atquadrupedal (four-legged) animals, we can distinguish different kinds of gaits at differentspeeds or even at the same speed but under different circumstances (see Figure 1.3 for someexamples).

(a) Trot (b) Gallop

Figure 1.3: Two gaits of a horse (taken from [4]).

Of course one can think of control algorithms that operate in the brain, measuring for examplespeed and difficulty of the terrain, and selecting a gait with this information. But possibly,the brain exhibits some properties that allow it to automatically adapt to changes in theenvironment.

1.3 Problem formulation

This research aims to find conditions, under which a network of four coupled systems withtime-delay synchronizes or partially synchronizes. Analytical stability conditions for the syn-chronization of the network will be derived, using a control theory approach. The analyticalresults will be compared with simulations, using examples.

The second part of this research will treat the possibility of simulating the control of ani-mal gaits with a network of neuron models. The analytical results will be validated withsimulations and a comparison will be made with the gaits of a quadrupedal animal.

1.4 Outline of the report

The outline of the report is as follows. In Chapter 2, a general approach to invesigate thestability of (partial) synchronization is presented, together with an example, concerning thechaotic Lorenz system and a Lur’e type neuron model. Chapter 3 will present the analysis ofthe gaits of some animals, in particular the gaits of a horse. It will also treat the selection of

4 Introduction

a suitable neuron model and the analysis of a network of neurons. Analysis of the stabilityof the synchronization of the network will be done and simulations will be used to validatethe analysis. Furthermore, the ability to simulate animal gaits with this network will beinvestigated. In Chapter 4, conclusions and recommendations will be stated.

Chapter 2

Synchronization of coupled systems

with time-delay

A great deal of research has been done on two coupled oscillators with time-delay [12], [17].In this chapter a network of four coupled nonlinear systems with time-delay will be analyzed.The global structure of the network and the coupling will first be discussed. Secondly, acondition for global asymptotic stability of (partial) synchronization of the network will bederived, using a Lypapunov-Krasovskii functional. Furthermore, the theory will be appliedto two examples, a network of Lorenz systems and a network of Lur’e type neuron models.The results will be validated numerically.

2.1 Design of the network

Consider a network of dynamical systems, as shown in Figure 2.1.

S1 S2

S3S4

Y1

Y2

Y3

Y4

U1

U2

U3

U4

Figure 2.1: A network of dynamical systems with time-delay

The network consists of four identical systems with the input and output, denoted respectivelyas ui(t) and yi(t), for i = 1, 2, 3, 4. The circles indicate that the output signal experiences atime-delay. The systems are unidirectionally coupled in a ring. The choice for unidirectional

6 Synchronization of coupled systems with time-delay

coupling is made for the sake of simplicity.

Next, consider the four continuous-time, nonlinear systems:

xi(t) = Axi + f(xi) + Bui

yi(t) = Cxi

xi(t) = φi(t) t ∈ [−τ, 0]i = 1, 2, 3, 4, (2.1)

where xi ∈ Rn, yi ∈ R

n, ui ∈ Rn, A ∈ R

n×n, B ∈ Rn, C ∈ R

n×n and f : Rn → R

n is a smoothfunction. Suppose we want to investigate full synchronization of this network. Then we haveto define the condition for synchronization:

Definition (2.1). A pair of dynamical systems with state xi(t) and xj(t) respectively (i 6= j),is synchronized if

limt→∞

| xi(t) − xj(t) |= 0. (2.2)

This implies that the error between the trajectories of the systems converges to zero no matterwhat initial conditions are taken. In other words, the solution e = 0 of the error dynamics isglobally asymptotically stable. If the dynamics of (2.1) are used, the error dynamics of thenetwork can be formulated as follows. Denote e(t) = xi(t) − xj(t). Then we obtain

e(t) = Ae(t) + f(xi) − f(xj) + B(ui − uj) (2.3)

e(t) = φi(t) − φj(t) t ∈ [−τ, 0].

In the case of four unidirectionally coupled systems, Definition 2.1 becomes

limt→∞

| xi(t) − xj(t) |= 0,i = 1, 2, 3, 4j = 4, 1, 2, 3.

(2.4)

In order to formulate the error dynamics in terms of the error e, the coupling terms have tobe defined. With the aid of Figure 2.1 and considering the delayed input for each system, weobtain

ui(t) = −KC(xi(t) − xj(t − τ)),i = 1, 2, 3, 4j = 4, 1, 2, 3,

(2.5)

with coupling gain K ∈ Rn×1 > 0, input matrix C ∈ R

1×n and time-delay τ > 0. One caneasily see that in this case the coupling strength for all systems is equal. Combining (2.1),(2.4) and (2.5) and defining eij = xi(t) − xj(t) and eijτ = xi(t − τ) − xj(t − τ), we obtain

et =

e14

e21

e32

e43

=

a0e14 + a1e43τ + f(x1) − f(x1 − e14)a0e21 + a1e14τ + f(x2) − f(x2 − e21)a0e32 + a1e21τ + f(x3) − f(x3 − e32)a0e43 + a1e32τ + f(x4) − f(x4 − e43)

. (2.6)

with

a0 = A − BKC, a1 = BKC

2.2 Stability of the synchronization 7

If we define the vectors

et =

e14

e21

e32

e43

, eτ =

e14τ

e21τ

e32τ

e43τ

, x =

x1

x2

x3

x4

, (2.7)

(2.6) can be rewritten:

et = A0et + A1eτ + f(x) − f(x − et)

= A0et + A1eτ + φ(x, et). (2.8)

with

A0 =

a0 O O OO a0 O OO O a0 OO O O a0

, A1 =

O O O a1

a1 O O OO a1 O OO O a1 O

. (2.9)

It is easy to see that if et = 0 is an asymptotically stable equilibrium point of (2.8), synchro-nization of the network is asymptotically stable.

Linearizing the nonlinear term φ(x, et) around e = 0, we obtain

et = A0et + A1eτ + D(x)et, D(x) =

(

∂φ(x, et)

∂et

)

|et=0 . (2.10)

It is well known that if et = 0 of (2.10) is asymptotically stable, then et = 0 of (2.8) is alsoasymptotically stable.

2.2 Stability of the synchronization

To investigate the asymptotic stability of et = 0 of (2.10), we present Theorem 1 of [16].

Theorem (2.1). The time delay system (2.10) is asymptotically stable for any delay τ sat-isfying 0 < τ < τ , if there exist matrices P > 0, Q > 0, Z > 0, Y and W such that thefollowing Linear Matrix Inequality (LMI) holds:

PA + AT P + Y + Y T + Q PA1 − Y + W T −τY τAT ZAT

1 P − Y T + W −W − W T − Q −τW τAT1 Z

−τY T −τW T −τZ 0τZA τZA1 0 −τZ

< 0. (2.11)

With A0 and A1 as in (2.9) and A = A0 + D(x).

8 Synchronization of coupled systems with time-delay

Proof. Denote

e = e(t + θ) − 2τ < θ < 0

and define a Lyapunov-Krasovskii functional as

V (e) = V1(e) + V2(e) + V3(e) (2.12)

where

V1(e) = e(t)T Pe(t)

V2(e) =

∫ 0

−τ

∫ t

t+β

e(α)T Ze(α)dαdβ

V3(e) =

∫ t

t−τ

e(α)T Qe(α)dα.

By the Newton-Leibniz formula, we have e(t − τ) = e(t) −∫ t

t−τe(α)dα. Then the derivative

of V (e) reads (see [16]):

V (e) = V1(e) + V2(e) + V3(e) (2.13)

V1(e) =1

τ

∫ t

t−τ

[2e(t)T (PA + Y )e(t) + 2e(t)T (PA1 − Y + W T )e(t − τ)

− 2e(t − τ)T We(t − τ) − 2e(t)T τY e(α)dα − 2e(t − τ)T τW e(α)dα]dα

V2(e) =1

τ

∫ t

t−τ

[e(t)T τAT ZAe(t) + 2e(t)T τAT ZA1e(t − τ) + e(t − τ)T τAT1 ZA1e(t − τ)

− e(α)T τZe(α)]dα

V3(e) =1

τ

∫ t

t−τ

[

e(t)T Qe(t) − e(t − τ)T Qe(t − τ)]

dα,

with A = A0 + D(x).Combining these terms, we obtain

V (e) =1

τ

∫ t

t−τ

ζ(t, α)T Λ(τ)ζ(t, α)dα, (2.14)

with

ζ(t, α) =[

e(t)T e(t − τ)T e(α)T]

,

and a symmetric matrix (2.15)

Λ(τ) (2.16)

=

PA + AT P + Y + Y T + τAT ZA + Q ∗ ∗AT

1 P − Y T + W + τAT1 ZA0 −W − W T + τAT

1 ZA1 − Q ∗−τY T −τW T −τZ

.

If Λ(τ) < 0, the system is asymptotically stable. Using a Schur complement we obtain(2.11).

2.3 Boundedness of the trajectories 9

Note that in order to rewrite the error dynamics in state space notation, the nonlinear termφ(x)is linearized around e = 0. However, the linearized term D(x) is still depending on thestate x(t). This means that when solving LMI (2.11), information of the trajectories of thesystems has to be at hand. To obtain this information, we present two approaches. The firstone is using the linearized term D(x) and the boundedness of the trajectories x(t). The secondone is not linearizing the term, but using the sector condition to rewrite the nonlinear termpurely in terms of the error e. This Chapter is concluded with examples of both methods.

2.3 Boundedness of the trajectories

In this section, we present an extension of the small-gain theorem for two bidirectionallycoupled systems, as discussed in [17]. First, semi-passivity and semi-dissipativity are defined.For semi-dissipative systems, the small gain theorem will give proof of the boundedness ofthe trajectories of the systems. Finally, the bounds can be computed using Theorem 2.2.

2.3.1 Preliminaries

Consider the nonlinear system

x(t) = f(x, u), y(t) = h(x) (t ≥ 0), (2.17)

with state x ∈ Rn, input u ∈ R

m, output y ∈ Rm, f : R

n × Rm → R

n and h : Rn → R

m.According to semi-passivity as defined in [14] and [15], we introduce strictly semi-passivityand strictly semi-dissipativity as follows.

Definition (2.2). [strictly semi-passivity]System (2.17) is said to be strictly semi-passive, if there exist a C1-class function V : R

n → R,class-K∞ functions α(·), α(·) and α(·) satisfying

α(‖x‖) ≤ V (x) ≤ α(‖x‖) (2.18)

V (x) ≤ −α(‖x‖) − H(x) + yT u (2.19)

for all x ∈ Rn, u ∈ R

m, y ∈ Rm, where the function H(x) satisfies

‖x‖ ≥ η ⇒ H(x) ≥ 0 (2.20)

for a positive real number η.

Definition (2.3). [strictly semi-dissipativity]System (2.17) is said to be strictly semi-dissipative with respect to the supply rate q(u, y), ifthere exist a C1-class function V : R

n → R, class-K∞ functions α(·), α(·) and α(·) satisfying(2.18) and

V (x) ≤ −α(‖x‖) − H(x) + q(u, y) (2.21)

for all x ∈ Rn, u ∈ R

m, y ∈ Rm, where the function H(x) satisfies (2.20).

10 Synchronization of coupled systems with time-delay

Remark. The system is strictly semi-passive if the supply rate q(u, y) = yT u in (2.21) for allu ∈ R

m.

Furthermore, if the system is strictly semi-dissipative, the following lemma holds:

Lemma (2.1). Suppose that system (2.17) is strictly semi-dissipative with respect to thesupply rate β(‖u‖) for a class-K function β(·). Then there exist class-K functions ρ(·), γ(·)and a real number η > 0 such that the response x(t) of (2.17) with the initial state x(0) = x0

satisfies

‖x‖∞ ≤ maxρ(‖x0‖), γ(‖u‖∞), ρ(η) (2.22)

lim supt→∞

‖x(t)‖ ≤ maxγ(lim supt→∞

‖u(t)‖), ρ(η) (2.23)

for any input u ∈ Lm∞ and any x0 ∈ R

n. Here functions ρ(·) and γ(·) are given by

ρ(r) = α−1 α(r)

γ(r) = α−1 α α−1 κβ(r)(r ≥ 0) (2.24)

where κ > 1, functions α(·), α(·) and α(·) satisfy (2.18) and (2.21) and η satisfies (2.20).

Proof. The proof of this lemma can be found in [17].

2.3.2 Extension of the small-gain theorem

Suppose Lemma 2.1 holds for the network of four unidirectionally coupled systems. Then,similar to (2.22) and (2.23) we can state:

‖xi‖∞ ≤ maxρi(‖x0i‖), γi(‖ui‖∞), ρi(ηi)lim sup ‖x1(t)‖ ≤ maxγ1(lim sup ‖u1(t)‖), ρ1(η1)

, i = 1, 2, 3, 4 (2.25)

with inputs ui(t) = yj(t − τ) = Cjxj(t − τ), i = 1, 2, 3, 4, j = 4, 1, 2, 3.

Consider the following four systems

xi(t) = fi(xi, ui) , yi(t) = Cixi (t ≥ 0), (2.26)

where state xi ∈ Rn, input ui ∈ R

m, output yi ∈ Rm, Ci ∈ R

m×n and fi : Rn × R

m → Rn

with initial conditions

xi(θ) = φi(θ) (−τ ≤ θ ≤ 0)

xi(0) = φi(0) = x0i,(2.27)

for i = 1, 2, 3, 4, respectively, where φi : [−τ, 0] → Rn. Suppose that the systems (2.26)

are strictly semi-dissipative. Then from Lemma 2.1, the systems have the properties (2.25).Now we consider the case in which these four systems are unidirectionally coupled with thefollowing inputs:

ui(t) = yj(t − τ) = Cjxj(t − τ),i = 1, 2, 3, 4j = 4, 1, 2, 3.

(2.28)

2.3 Boundedness of the trajectories 11

Define class-K functions as

πi(r) = γi(σmax(Cj) · r) (r ≥ 0), (2.29)

where γi(·) = ααα−1 κβ(·) and σmax(·) denotes the maximum singular value of a matrix.Then we obtain the following Lemma:

Lemma (2.2). For four systems (2.26), (2.27) coupled by (2.28), if the functions π1(·), π2(·),π3(·) and π4(·) in (2.29) satisfy

π1 π2 π3 π4(r) < r for all r > 0, (2.30)

then the trajectories x1(t), x2(t), x3(t) and x4(t) satisfy

lim supt→∞ ‖x1(t)‖ ≤ maxπ1 π4 π3(ρ2(η2)), π1 π4(ρ3(η3)), π1(ρ4(η4)), ρ1(η1)lim supt→∞ ‖x2(t)‖ ≤ maxπ2 π1 π4(ρ3(η3)), π2 π1(ρ4(η4)), π2(ρ1(η1)), ρ2(η2)lim supt→∞ ‖x3(t)‖ ≤ maxπ3 π2 π1(ρ4(η4)), π3 π2(ρ1(η1)), π3(ρ2(η2)), ρ3(η3)lim supt→∞ ‖x4(t)‖ ≤ maxπ4 π3 π2(ρ1(η1)), π4 π3(ρ2(η2)), π4(ρ3(η3)), ρ4(η4),

(2.31)

where ρ(·) = α−1 α(·) and ηi satisfies (2.20).

Proof. Since the systems (2.26) are strictly semi-dissipative, from Lemma 2.1, there existclass-K functions ρi(·) and γi(·) and positive real numbers ηi such that the trajectories of(2.26) satisfy (2.25) for any inputs u1, u2, u3, u4 ∈ Lm

∞.

First we show boundedness of the trajectories xi(t) for all t ≥ 0. This is proven by contra-diction. Suppose this is not true, then for any real number R > 0, there exists a time T > 0such that the trajectories are defined on an interval [0,T], and satisfy

either ‖x1(T )‖ > R or ‖x2(T )‖ > R or ‖x3(T )‖ > R or ‖x4(T )‖ > R. (2.32)

Here we choose R satisfying

R > Mi, (2.33)

with i = 1, 2, 3, 4 and Mi is defined as:

M1 = max‖φ1‖c, ρ1(‖x01‖), π1(‖φ4‖c, ρ4(‖x04‖), π4(‖φ3‖c, ρ3(‖x03‖),

π3(‖φ2‖c, ρ2(‖x02‖), ρ2(η2)), ρ3(η3)), ρ4(η4)), ρ1(η1)

M2 = max‖φ2‖c, ρ2(‖x02‖), π2(‖φ1‖c, ρ1(‖x01‖), π1(‖φ4‖c, ρ4(‖x04‖),

π4(‖φ3‖c, ρ3(‖x03‖), ρ3(η3)), ρ4(η4)), ρ1(η1)), ρ2(η2) (2.34)

M3 = max‖φ3‖c, ρ3(‖x03‖), π3(‖φ2‖c, ρ2(‖x02‖), π2(‖φ1‖c, ρ1(‖x01‖),

π1(‖φ4‖c, ρ4(‖x04‖), ρ4(η4)), ρ1(η1)), ρ2(η2)), ρ3(η3)

M4 = max‖φ4‖c, ρ4(‖x04‖), π4(‖φ3‖c, ρ3(‖x03‖), π3(‖φ2‖c, ρ2(‖x02‖),

π2(‖φ1‖c, ρ1(‖x01‖), ρ1(η1)), ρ2(η2)), ρ3(η3)), ρ4(η4).

Let T be such that (2.32) holds. Define the truncation of xi to the interval [−τ, T ] by

xi(t) = xi(t) if t ∈ [−τ, T ]

= 0 if t > T,

12 Synchronization of coupled systems with time-delay

and let x1(t) denote the response of the system (2.26) for i = 1 with the initial condition φ1,for the input u1(t). Since

‖u1‖∞ = max−τ≤t≤T−τ

‖C4x4(t)‖ ≤ max−τ≤t≤T

‖C4x4(t)‖ ≤ σmax(C4)‖x4‖[−τ,∞),

and the right hand side of this inequality is assumed to be bounded, we have

‖x1(t)‖ ≤ maxρ1(‖x01‖), π1(‖x4‖[−τ,∞)), ρ1(η1) for allt ≥ 0.

Since, by causality,

x1(t) = x1(t) for all t ∈ [−τ, T ]

holds, we deduce that

‖x1‖∞ = maxt∈[0,T ]

‖x1(t)‖ = maxt∈[0,T ]

‖x1(t)‖ ≤ maxρ1(‖x01‖), π1(‖x4‖[−τ,∞)), ρ1(η1)

‖x1(t)‖ = ‖φ1(t)‖ ≤ ‖φ1‖c for all t ∈ [−τ, 0].

Therefore

‖x1‖[−τ,∞) ≤ max‖φ1‖c, ρ1(‖x01‖), π1(‖x4‖[−τ,∞)), ρ1(η1). (2.35)

For i = 2, 3, 4, similarly we obtain

‖x2‖[−τ,∞) ≤ max‖φ2‖c, ρ2(‖x02‖), π2(‖x1‖[−τ,∞)), ρ2(η2) (2.36)

‖x3‖[−τ,∞) ≤ max‖φ3‖c, ρ3(‖x03‖), π3(‖x2‖[−τ,∞)), ρ3(η3) (2.37)

‖x4‖[−τ,∞) ≤ max‖φ4‖c, ρ4(‖x04‖), π4(‖x3‖[−τ,∞)), ρ4(η4). (2.38)

Now, if a ≤ maxb, c, θ(a) and θ(a) < a, then necessarily maxb, c, θ(a) = maxb, c. Thus,replacing the estimates (2.36),(2.37) and (2.38) into (2.35) and using hypothesis (2.30) yields

‖x1‖[−τ,∞) ≤ M1.

Here we prove that π1, π2, π3, π4 are mutually interchangeable in (2.30). Since the functionπi(·) ∈ K, set r∗i = limr→∞ πi(r).

1. In the case that 0 < r < r∗1, from (2.30), we obtain π2π3π3(r) < π−11 (r). Let s ∈ (0,∞)

such that s = π−11 (r), then π2 π3 π3(π1(π

−11 (r)) < π−1

1 (r) yields π2 π3 π3 π1(s) < sfor all s > 0. One can repeat this, for example with q = π−1

2 (s)

2. For r ≥ r∗1, set p ∈ (0, r∗1) such that π1 π2 π3 π4(r) < p, then we obtain π2 π3 π3(r) < π−1

1 (p). Let s = π−11 (p). Since p < r, π2(p) < π2(r) holds. Therefore

π2 π3 π3(p) < π−11 (p) holds and this yields π2 π3 π3(s) < π−1

1 (s) for all s > 0.Again, this procedure can be repeated to obtain different sequences.

Then, using (2.33), we have

‖x1‖[−τ,∞) ≤ M1 < R and

‖x2‖[−τ,∞) ≤ M2 < R and

‖x3‖[−τ,∞) ≤ M3 < R and

‖x4‖[−τ,∞) ≤ M4 < R,

2.3 Boundedness of the trajectories 13

which contradicts (2.32). Now that we have shown that the trajectories are defined for allt ≥ 0 and bounded, (2.25) yields

‖xi‖∞ ≤ maxρi(‖x0i‖), πi(‖xj‖[−τ,∞)), ρi(ηi). (2.39)

From the initial conditions, we obtain

‖xi(t)‖ = ‖φi(t)‖ ≤ ‖φi‖c for all t ∈ [−τ, 0]. (2.40)

Therefore, (2.39) and (2.40) yield

‖xi‖[−τ,∞) ≤ max‖φi‖c, ρi(‖x0i‖), πi(‖xj‖[−τ,∞)), ρi(ηi)

Then combining these inequalities and using property (2.30), we obtain

‖xi‖[−τ,∞) ≤ Mi.

Similarly, from (2.25), we have

lim supt→∞

‖xi(t)‖ ≤ maxπi(lim supt→∞

‖xj(t)‖), ρi(ηi).

Therefore combining them and using the property (2.30), we obtain (2.31).

2.3.3 Boundedness of the trajectories

Now we have proven the existence of bounds on xi(t), we can state the following theorem:

Theorem (2.2). Define a class-K function as π(r) , γ(σmax(C) · r) for r ≥ 0, where γ(·) isdefined as (2.24). If the function π(·) satisfies

π(r) < r for all r > 0, (2.41)

then the trajectories of the systems (2.6) converge to the bounded set

Ω = x ∈ Rn∣

∣ ‖x‖ ≤ ρ(η). (2.42)

Proof. If (2.41) holds, then π π(r) < r, π π π(r) < r and so on. This implies that (2.30)holds. Furthermore, since the systems are identical, ρi(ηi) = ρj(ηj) for all i, j. Therefore,using Lemma 2.2 and (2.41) the trajectories of the systems (2.6) satisfy

lim supt→∞

‖x1(t)‖ ≤ maxπ1 π4 π3(ρ2(η2)), π1 π4(ρ3(η3)), π1(ρ4(η4)), ρ1(η1)

≤ ρ1(η1). (2.43)

This means all trajectories converge to Ω.

With the above derived conditions, we can calculate the bounds of the trajectories of eachsystem in the network. In order to investigate the global asymptotic stability of the syn-chronization of the network, (2.11) must hold for all trajectories in the bounded set Ω. Ifthis holds for trajectories, starting on the edge of this set, we can conclude that it holds forall trajectories, starting in Ω. To reduce computation time, Ω will be defined by only a fewstrategically chosen points, as we will see in the first example in Section 2.8.1.

14 Synchronization of coupled systems with time-delay

2.4 Sector condition

Another way to deal with the nonlinear term in (2.8) is to use the sector condition. Theapproach is based on the assumption that a nonlinear function is defined in a sector, whichis bounded by linear functions. See for example Figure 2.2. One can easily see that thenonlinear function φ(x) = 1

1+e−2x − 12 belongs to the sector, defined by the linear functions

f1(x) = 0 and f2(x) = 12x.

−3 −2 −1 0 1 2 3−1.5

−1

−0.5

0

0.5

1

1.5

x

f(x)

αx

βx

φ(x)

Figure 2.2: Example of a function satisfying the sector condition.

To make use of this property, we present the sector condition, together with its applicationin a Lyapunov functional [9].

Definition (2.4). A nonlinear function φ(t, x) satisfies a sector condition if

βx ≤ φ(t, x) ≤ αx ∀ x ∈ Ω. (2.44)

If Ω = R, then φ(t, x) satisfies the sector condition globally, in which case it is said thatφ(t, x) belongs to a sector [α, β]. If Ω = [a, b], it is said that φ(t, x) belongs to a sector (α, β).Furthermore, we can easily derive the following condition:

(φ(t, x) − βx)(φ(t, x) − αx) ≤ 0 ∀ x ∈ Ω. (2.45)

Now suppose a simple Lyapunov candidate function is given, V (et) = eTt Pet. For the zero

solution of the error dynamics to be asymptotically stable, we have to prove V (et) < 0. Using(2.8), we obtain:

V (e) = eT Pe + eT P e (2.46)

= 2(A0et + A1eτ + φ(x, et))T Pe. (2.47)

2.5 Discussion 15

Now, using (2.45), the following holds for all λ > 0:

V (e) = 2(A0et + A1eτ + φ(x, et))T Pe − 2λ(φ(x, et) − βet)(φ(x, et) − αet) ≤ 0. (2.48)

This inequality can be rewritten as an LMI, with the nonlinear term φ(x, et) as a state variable.If this LMI holds, the system is asymptotically stable. Of course, the same procedure can befollowed when choosing another Lyapunov candidate function.

2.5 Discussion

In section 2.3 and 2.4 we proposed two methods to simplify the nonlinear term in the errordynamics of the network. Both methods will make the stability criterion more conservative,since an approximation of the nonlinear term is made. In the first case, the bounds onthe trajectories of the systems are approximated, and then used in a linearized term. In thesecond case, the nonlinear function is approximated by a set of linear functions. It is expectedthat these approximations will put more restrictions on the stability of the synchronization.However, the chosen Lyapunov-Krasovskii functional is the best available choice for this typeof time-delay system. The functional should give less conservative results [16]. We can thusconclude that a big part of the benefit of using this functional will be diminished by theconservativeness, introduced by the approximations of the nonlinear term. Furthermore itshould be noted that the choice of (2.12) as most suitable Lyapunov candidate function willnot necessarily lead to the best results. Possibly this choice in combination with the LMIapproach will give a conservative stability condition, so the analytical results will always differfrom numerical or experimental results.

2.6 Generalized eigenvalue optimization

When analyzing the behavior of the network, it is interesting to investigate synchronizationfor varying gain and time-delay. The maximum amount of time-delay, which still ensuressynchronization, can be computed rewriting the LMI as a so-called Generalized Eigenvalue

Optimization Problem (GEVP). The algorithm gives a sufficient condition for synchronizationby solving (2.11) for varying time-delay τ . If the LMI holds, the network is stable for thisvalue of time-delay. First we state the Theorem [12]:

Theorem (2.3). Let η be the optimal solution of the following standard generalized eigenvalueproblem:

minµ>0,P>0,Q>0,Z>0

µ (2.49)

subject to LMI (2.11) for all x ∈ Ω. Then for any τ ∈ [0, 1/µ], the error dynamics (2.10) isasymptotically stable.

To solve the GEVP for (2.11), we formulate the problem as follows:

Minimize µ, subject to

0 < B(x)A(x) < µB(x).

(2.50)

16 Synchronization of coupled systems with time-delay

Below one can easily distinguish the positive definite matrices (2.51) and (2.52), which aregrouped in B(x). The terms in the LMI that contain τ are put in A(x), as can be seen in(2.53) and (2.54).

0 < P, Q, Z (2.51)

0 <

[

−PA0 − AT0 P − Y − Y T − Q ∗

−AT1 P + Y T − W W + W T + Q

]

(2.52)

0 ∗ ∗ ∗0 0 ∗ ∗

−Y T −W T −Z ∗ZA0 ZA1 0 −Z

<

G1 ∗ ∗ ∗G2 G3 ∗ ∗0 0 0 ∗0 0 0 0

(2.53)

[

G1 ∗G2 G3

]

< µ

[

−PA0 − AT0 P − Y − Y T − Q ∗

−AT1 P + Y T − W W + W T + Q

]

. (2.54)

The GEVP can be solved using the LMI toolbox in MATLAB [5] and yields the maximumvalue of µ = 1

τ, for which (2.11) is still satisfied.

2.7 Full and partial synchronization

In the introduction was already briefly noted that synchronization does not imply that thebehavior of each system is exactly copied. For example, Huygens’ pendulum clocks were notswinging completely synchronous, but there was a fixed phase shift of half a period betweenthe clocks. In this particular case this type of synchronization is called anti-phase synchro-nization. In general we can say the systems are partially synchronized.

In the case of four systems in a network, partial synchronization can also be observed. Onecan imagine that for a certain configuration of coupling parameters, a part of the network issynchronized and another part is not. To investigate the parameters for which this situationappears, the formulation of the error dynamics is crucial. For full synchronization, the errorbetween all systems has to converge to zero. For partial synchronization, the error betweenspecific systems has to converge to zero, while other errors do not. In the following exam-ples, the formulation of the error dynamics as a way to distinguish between full and partialsynchronization, will be discussed.

2.8 Examples 17

2.8 Examples

2.8.1 Four unidirectionally coupled Lorenz systems

In this section we will analyze the stability condition for four coupled Lorenz systems withtime-delay. The network is constructed as in Figure 2.1. The Lorenz system is described bythe following set of equations:

Σi =

xi = σ(yi − xi) + uix

yi = rxi − yi − xizi + uiy

zi = −bzi + xiyi + uiz,i = 1, 2, 3, 4 (2.55)

with output v = xi, xi =[

xi yi zi

]T

and σ = 10 r = 28 b = 83 .

Unidirectional coupling is formulated as in (2.5):

ui =

uix

uiy

uiz

= −kC(xi − xj),i = 1, 2, 3, 4j = 4, 1, 2, 3

, (2.56)

with

C =

1 0 00 1 00 0 1

and k > 0.

The Lorenz system with the parameters of (2.55) is a so-called chaotic system. The mathe-matical definition of a chaotic system is:

• It is highly sensitive to initial conditions. A small change in the initial conditions resultsin a large change in the time-response.

• The system is deterministic. This implies that the equations of motion of the systempossess no random inputs. In other words, the irregular behavior of the system arisesfrom non-linear dynamics and not from noisy driving forces.

• The trajectories of the system do converge to fixed points or periodic orbits.

One can imagine that for a network with this properties, synchronization is unlikely. Even ifwe consider a network of identical systems, changes in the initial conditions will completelychange the behavior of the system. However, simulation results show that even chaotic sys-tems can synchronize when coupled. In this example we will try to find a condition for globalasymptotic stability of the synchronization of four Lorenz system with time-delay analytically.

Boundedness of the solutions

It is easy to see that the nonlinear terms xizi and xiyi in (2.55) do not satisfy a sector condi-tion. Therefore, the nonlinear terms have to be linearized and the bounds of the trajectorieshave to be used in order to derive a stability condition. Following the procedure in Section

18 Synchronization of coupled systems with time-delay

2.3, we first investigate semi-passivity of the system. For i = 1, we define a storage functionV1(x1) = xT

1 x1, where x1 = [x1 y1 z1 −σ− r]T . The derivative of V along x1 is given by (seeAppendix A):

V (x1) = −σx21 − y2 − bz2 − (σ + r)z − kx1(x1 − x4τ ) − ky1(y1 − y4τ ) − kz1(z1 − z4τ )

= −α(‖x1‖) − H(‖x1‖) + β(‖x4τ‖), (2.57)

with

H(x1) = (σ − ǫ)x21 + (1 − ǫ)y2

1 + (b − ǫ)(

z −b − 2ǫ

2(b − ǫ)

)2−

b2(σ + r)2

4(b − ǫ)

α(‖x1‖) = (k

2+ ǫ)‖x1‖

2

β(‖x4τ‖) =k

2‖x4τ‖

2,

so, according to Definition 2.2, the system is semi-passive. This implies that Lemma 2.1holds. Since this choice of storage function V1(x1) implies α(r) = α(r), (2.24) is restated:

ρ(r) = α−1 α(r) = rγ(r) = α α α−1 κβ(r) = α−1 κβ(r)

(r ≥ 0). (2.58)

Using the Small-gain theorem and Theorem 2.2, we conclude

γ1(r1) =

√

κk2

k2 + ǫ

r1 < r1. (2.59)

Therefore, for any finite number k > 0, setting κ sufficiently close to 1, π(r) < r holds for allr and all trajectories x1(t) converge to

Ω = x1 ∈ Rn | ‖x1‖ ≤ η1 (2.60)

To find η1, the maximum value of ‖x1‖ for which V (x1) < 0 still holds, has to be found.Therefore, we analyse the terms in (2.57). First we show that −α(‖x1‖) + β(‖x4τ‖) < 0.Since γ(r1) < r1, we can rewrite β(r1) as follows:

α−1 κβ(r1) < r1

κβ(r1) < α(r1)

β(r1) <1

κα(r1)

As a consequence, following the proof of Lemma 2.1, presented in [17],

− α(‖x1‖) + β(‖x4τ‖)

≤− α(‖x1‖) +1

κα(‖x1‖)

≤−κ − 1

κα(‖x1‖) < 0, for all κ > 1

2.8 Examples 19

holds. This implies that V (x1) < 0 if H(x1) ≥ 0. To find the maximum value of ‖x1‖ forwhich this holds, the following optimization problem has to be solved:

max0<ǫ<1

‖x1‖

subject to

H(x1) = 0

Solving the optimization problem for ǫ = 0.01, we obtain

Ω = x1 ∈ Rn | ‖x1‖ ≤ 39.4 (2.61)

A graphical representation of the bounds is given in Figure 2.3. The ball represents thecomputed bounds from (2.61). The stability analysis requires coordinates on this ball to fillin D(x). To reduce computation time, the ball is approximated by a cube, containing the setΩ. If analysis results in stability on the corners of the cube, the network is also stable forpoints inside Ω.

−50

0

50 −50

0

50

0

20

40

60

yx

z

Figure 2.3: Bounds on the trajectories of the Lorenz system.

Synchronization

In order to investigate synchronization of the network, the error dynamics are formulated.First, we distinguish between full and partial synchronization. Consider the network of Lorenzsystems (2.55) with (2.56). Suppose we want to investigate full synchronization of the network.Recall (2.4) for full synchronization of four coupled systems. The error dynamics are nowdefined as follows. Defining the error state vector

e =

e12

e23

e34

e41

with eij =

xi − xj

yi − yj

zi − zj

, (2.62)

20 Synchronization of coupled systems with time-delay

we obtain

e(t) = A0e + A1eτ + φ(e, xi), (2.63)

where

A0 =

At O O OO At O OO O At OO O O At

, At =

−σ − k σ 0r −1 − k 00 0 −b − k

, (2.64)

A1 =

O O O Aτ

Aτ O O OO Aτ O OO O Aτ O

, Aτ =

−k 0 00 −k 00 0 −k

and (2.65)

φ(e, xi) =

0−ex12ez12 − ez12x2 − ex12z2

ex12ey12 + ey12x2 + ex12y2

0−ex23ez23 − ez23x3 − ex23z3

ex23ey23 + ey23x3 + ex23y3

0−ex34ez34 − ez34x4 − ex34z4

ex34ey34 + ey34x4 + ex34y4

0−ex41ez41 − ez41x1 − ex41z1

ex41ey41 + ey41x1 + ex41y1

. (2.66)

Linearizing φ(e, xi) around e = 0 gives:

δφ

δe|e=0 = D(xi) =

D4 O O OO D1 O OO O D2 OO O O D3

with Di =

0 0 0−zi 0 xi

yi xi 0

. (2.67)

Now, suppose we want to investigate partial synchronization. For example, system 1 and 3synchronize and system 2 and 4 synchronize. Therefore, the following error dynamics musthave e = 0 as an asymptotically stable equilibrium:

e =

[

e13

e24

]

with eij =

xi − xj

yi − yj

zi − zj

, (2.68)

e(t) = A0e + A1eτ + φ(e, x3, x4), (2.69)

where

2.8 Examples 21

A0 =

[

At OO At

]

, At =

−σ − k σ 0r −1 − k 00 0 −b − k

, (2.70)

A1 =

[

O −Aτ

Aτ O

]

, Aτ =

−k 0 00 −k 00 0 −k

and (2.71)

φ(e, x3, x4) =

0−ex13ez13 − ez13x3 − ex13z3

ex13ey13 + ey13x3 + ex13y3

0−ex24ez24 − ez24x4 − ex24z4

ex24ey24 + ey24x4 + ex24y4

. (2.72)

Linearizing φ(e, xi) around e = 0 gives:

δφ

δe|e=0 = D(xi) =

[

D3 OO D4

]

with Di =

0 0 0−zi 0 xi

yi xi 0

. (2.73)

Note that if these error dynamics converge to zero, only a sufficient condition for partialsynchronization is found. It can still happen that the network synchronizes fully. This is adrawback of this approach. However, since the condition for full synchronization can also bedetermined, we can still ’extract’ a condition for partial synchronization. How this is done,will be explained in Chapter 3.

To investigate the stability of e = 0, recall (2.11):

PA + AT P + Y + Y T + Q PA1 − Y + W T −hY hAT ZAT

1 P − Y T + W −W − W T − Q −hW hAT1 Z

−hY T −hW T −hZ 0hZA hZA1 0 −hZ

< 0.

Using A = A0 + D(xi) and A1 as stated above, we solve the Generalized Eigenvalue Op-timization Problem for partial and full synchronization and for varying coupling gain K.Consequently, we obtain the stability diagrams in Figure 2.4.

22 Synchronization of coupled systems with time-delay

0 20 40 60 80 100 120

0

5

10

15

20

25

Coupling gain K

τ max

[m

s]

Partial synchronizationFull synchronization

(a) Computations

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

Coupling gain K

τ max

[m

s]

Full synchronization

(b) Simulations

Figure 2.4: Stability region of 4 unidirectionally coupled Lorenz systems with time-delay.

2.8 Examples 23

Discussion of the results

Clearly, Figure 2.4(a) shows that there is a minimum value of the coupling gain K, for whichpartial synchronization occurs. For K < 10, the maximum amount of time-delay τmax consid-erably decreases. Around K = 30, τmax increases a little till a maximum value of τ = 20[ms].The stability of the synchronization of the network is hardly affected by a further increase ofthe coupling gain. Calculating the stability region for full synchronization, the GEVP resultsshow no value of time-delay τ for which the network synchronizes. However, simulation resultscontradict this outcome. As shown in Figure 2.4(b), we see that there is a region, for whichthe network synchronizes. The simulations show that at K = 5, the network synchronizeseven for a time-delay τ = 70[ms]. There is a minimum value K, for which the network syn-chronizes, but is it even lower than the one, calculated in the case of partial synchronization.Furthermore, partial synchronization is not observed in the simulations.

Apparently, The Lyapunov-Krasovskii functional in combination with the small-gain theoremresults in a too conservative condition for stability. As discussed in section 2.5, Lyapunov sta-bility in general is a strict condition for stability. Furthermore, linearization of the nonlinearterms in the Lorenz equations also introduces a difference between simulations and analysisof the network. Using another approach, possibly based on symmetries of the network, asdiscussed in [6] and [14], may result in a less conservative condition.

2.8.2 Four unidirectionally coupled Lure type neuron models

As a second example, consider a network of Lure type neuron models [8]

Vi =

vi(t) = cφ(avi) − bvi + u0 − wi + uex + uiv

wi(t) = ρ[φ(d(vi + v0)) − wi] + uiwi = 1, 2, 3, 4, (2.74)

with

Vi =

[

vi

wi

]

, (2.75)

yi(t) = CVi(t), (2.76)

φ(x) =1

1 + e2−4x.

Assume full state coupling

ui = −kC(Vi − Vjτ ), k > 0, C = [1 1],i = 1, 2, 3, 4j = 4, 1, 2, 3.

(2.77)

The parameters are chosen as follows:

a = 1.8 b = 3 c = 2.2 d = 5u0 = −0.2 v0 = −0.35 ρ = 0.3 uex = 0.04.

24 Synchronization of coupled systems with time-delay

Sector condition

First, we check whether the nonlinear terms satisfy the sector condition. There are twononlinear functions:

φi1(vi) = φ(avi)

φi2(v) = φ(d(vi + v0)).

In Figure 2.5 both nonlinear terms are plotted as function of v.

−1 −0.5 0 0.5 1−3

−2

−1

0

1

2

3

V1

φ 1(V1)

(a) φi1(vi)

−1 −0.5 0 0.5 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

V1

φ 1(V1)

(b) φi2(vi)

Figure 2.5: The nonlinear functions φi1 and φi2.

On first sight, the functions do not satisfy a sector condition. A coordinate transformationwill help in this case. The coordinate transformation will be conducted as follows:

vi = vi +5

18

wi = wi +5

18

φi(vi) = φi(vi) −1

2.

Substituting this transformation into (2.74) for i = 1, we obtain:

˙v1(t) = c[φ(a(v1 −5

18)) −

1

2] − b(v1 −

5

18) + u0 − w1 −

5

18+ uex − K(v1 − v2τ )

˙w1(t) = ρ[φ(d(v1 −5

18+ v0)) −

1

2− w1 −

5

18].

Now, concerning the nonlinear functions, the following sector condition holds: there exists anα > 0, such that 0 < φ(y) ≤ α(y) for all y ∈ y = Cx | x ∈ Ω. Then

0 ≤ c[φ(a(v1 −5

18)) −

1

2] = φ1(v1) ≤ cav1

0 ≤ ρ[φ(d(v1 −5

18+ v0)) −

1

2] = φ2(v1) ≤ ρdv1 −

7

40d.

2.8 Examples 25

One can see that φ2 still does not satisfy a sector condition. However, when formulating theerror dynamics, all constant terms will disappear. Consider for example the error dynamicsv1 − v2. Since v1 − v2 = v1 − v2 = ev, we obtain:

0 ≤ φ1(v1) − φ1(v2) ≤ caev (2.78)

0 ≤ φ2(v1) − φ2(v2) ≤ ρdev. (2.79)

Furthermore, the following inequality holds:[

φ1(v1) − φ1(v2)

φ2(v1) − φ2(v2)

]T [φ1(v1) − φ1(v2) − caev

φ2(v1) − φ2(v2) − ρdev

]

≤ 0. (2.80)

To check if condition (2.78) and (2.78) hold, the behavior of the nonlinear terms in the errordynamics is simulated. In Figure 2.6 it can be seen that the sector conditions are satisfied.

−0.5 0 0.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

eV

φ 1(V1)−

φ 1(V2)

(a) φ1(v1) − φ1(v2)

−0.5 0 0.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

eV

φ 2(V1)−

φ 2(V2)

(b) φ2(v1) − φ2(v2)

Figure 2.6: The nonlinear error functions for V1 − V2.

Error dynamics

The error dynamics for this network are formulated similarly as for the Lorenz network. Forpartial synchronization we obtain:

e(t) = A0e + A1eτ + Ψ(vi), i = 1, 2, 3, 4, (2.81)

where

A0 =

[

At OO At

]

At =

[

−b − k −10 −ρ − k

]

(2.82)

A1 =

[

O −Aτ

Aτ O

]

Aτ =

[

−k 00 −k

]

(2.83)

Ψ(vi) =

φ1(v1) − φ1(v3)

φ2(v1) − φ2(v3)

φ1(v2) − φ1(v4)

φ2(v2) − φ2(v4)

. (2.84)

To investigate full synchronization, the errors between all systems are included.

26 Synchronization of coupled systems with time-delay

Stability analysis

Again the following Lyapunov-Krasovskii functional will be used [16]:

V (e) = V1(e) + V2(e) + V3(e)

where

V1(e) = e(t)T Pe(t)

V2(e) =

∫ 0

−τ

∫ t

t+β

e(α)T Ze(α)dαdβ

V3(e) =

∫ t

t−τ

e(α)T Qe(α)dα.

From the Newton-Leibniz formula, we have e(t − τ) = e(t) −∫ t

t−τe(α)dα. Furthermore the

sector condition is formulated as in (B.3). Then, the time derivative of V (et) reads (seeAppendix B):

V (e) =1

τ

∫ t

t−τ

ζ(t, α, vi)T Λ(τ)ζ(t, α, vi)dα,

with

ζ(t, α, vi) =[

e(t)T e(t − τ)T Ψ(vi) e(α)T]

,

and a symmetric matrix

Λ(τ) =

PA0 + AT

0P + Y + Y T + τAT

0ZA0 + Q ∗ ∗ ∗

AT

1P − Y T + W + τAT

1ZA0 −W − WT + τAT

1ZA1 − Q ∗ ∗

P + R + τZA0 + ληC τZA1 − R −2λ ∗−τY T −τWT −τRT −τZ

,

for any λ ≥ 0.

Note that the nonlinear terms in Ψ(vi) are now state variables. The system is stable whenΛ(τ) < 0. Using the Schur complement [3], we obtain the following LMI concerning thestability of the network:

PA0 + AT0 P + Y + Y T + Q ∗ ∗ ∗ ∗

AT1 P − Y T + W −W − W T − Q ∗ ∗ ∗

P + R + τZA0 + ληC τZA1 − R −2λ + τZ ∗ ∗−τY T −τW T −τRT −τZ ∗τZA0 τZA1 0 0 −τZ

< 0, (2.85)

With system matrices A0 and A1 and ηC (see Appendix B) as the implementation of thesector condition in the Lyapunov stability approach. Formulating this LMI as a GeneralizedEigenvalue Problem and solving for µ = 1

τgives the stability diagram as shown in Figure 2.7.

Again, we conclude that there is a big difference between the simulations and the calcula-tions. Similar to the first example concerning the Lorenz systems, the stability diagram forpartial synchronization gives a lower bound for the coupling gain. Full synchronization doesnot occur, according to the calculations. Again, full synchronization is observed for variousvalues of coupling gain K and high values of time-delay τ in the simulations. There is a peak

2.8 Examples 27

in the stability region at K = 0.4. Why this peak appears in the simulations is unknown.Furthermore, partial synchronization is only observed under very strict conditions, i.e. spe-cific initial conditions and long simulation times. The combination of the sector conditionwith the Lyapunov stability approach also proves to be too conservative to obtain an analyticstability bound for full synchronization of the network.

28 Synchronization of coupled systems with time-delay

0 1 2 3 4 5 6 7 8 9 10

0

50

100

150

200

250

300

Coupling gain K

τ max

[m

s]

Full synchronizationPartial synchronization

(a) Computations

0 2 4 6 8 10

0

100

200

300

400

500

600

700

800

900

1000

Coupling gain K

τ max

[m

s]

Full synchronization

(b) Simulations

Figure 2.7: Stability region of 4 unidirectionally coupled Lure systems with time-delay.

Chapter 3

A synchronization approach on

animal gaits

In this chapter, the approach, explained in Chapter 2 will be used to investigate the synchro-nous behavior of a network of neurons. First, a view on animal locomotion will be given.Second, a representation of the control of this locomotion as a network of four neurons willbe given. A suitable model of a neuron will be chosen and the synchronization of a networkof neurons will be investigated. Finally, the results will be compared with simulations, whichresemble the control of animal gaits in the brain.

3.1 Animal gaits

Previous studies on animal locomotion resulted in the observation of a series of gaits, de-pending on the animal’s body and the walking speed. A nice and clear example is the gait ofa horse. With increasing walking speed, a different gait is required to maintain the desiredspeed. The diagram in Figure 3.1 shows the phase relations of the legs of a horse. The phaserelations of the legs indicate the sequence of leg placement, which resembles a particulargait. Studies on these phase relations showed that they can be obtained naturally via Hopfbifurcation in small networks [4], [6]. This implies that the control of animal gaits can berepresented by a simple network of oscillators, and changing parameters represents changingthe phase relations and thus the gait. In fact, this leads to a widely accepted theory aboutanimal locomotion, treated in the following section.

3.1.1 Central pattern generator

Animal locomotion is generated and controlled mostly by a so called Central Pattern Genera-

tor (CPG), which is a network of neurons in the central nervous system, capable of producingrhythmic output [6]. The neurons generate a current, that is transported to the limbs in orderto operate the muscles. For different kinds of gaits, a different pattern of signals is necessaryto let the limbs move in the desired sequence. It is assumed that the neurons have notionof neurons in their neighborhood when they pulse a signal. Moreover, the pulsing of neigh-boring neurons influences the state of the neuron itself. It may cause the neuron to pulse aswell. In previous research a model of a network of oscillators, capable of generating differentsignal patterns, has been investigated [4], [6]. The neurons are coupled with a coupling term,

30 A synchronization approach on animal gaits

Figure 3.1: Phase relations in the gaits of a horse (taken from [4]). The number at every legindicates the phase shift of the leg with respect to the left front leg.

simulating the influence of the states of neighboring neurons. With the coupling strengthas bifurcation parameter, a variety of oscillation patterns is obtained, representing animalgaits. In this research, we also consider the transport time between neurons as a bifurcationparameter. It is obvious that the transport of a signal from one neuron to another will takesome time. Despite the time-delay being very small, simulations show that this parametercan have a large influence on the oscillation patterns.

3.1.2 Time-delay

The time-delay between two neurons is assumed to be constant. This implies that whateverthe circumstances, the transport of a current from one neuron to another will always costa constant amount of time. However, when the speed of the animal increases, the reactiontime of the network will be smaller. We can say that the brain has to think faster in orderto keep up with the required gait at this speed. The time-scale of the actions of the networkwill change. This means that the influence of the time-delay on the reaction time changes.We can illustrate this with an example. Consider a linear continuous-time time-delay system

x(t) = Ax(t) + Bu(t − τ). (3.1)

We can apply a time-scale transformation θ = st, t = θs

as a representation of the demand to

3.2 A network of neuron models 31

think faster. Now (3.1) becomes

x(θ

s) = Ax(

θ

s) + Bu(

θ

s− τ). (3.2)

The influence of the time-delay τ on the system will change under changing time-scale para-meter s. One can also use another approach and scale τ :

x(θ) = Ax(θ) + Bu(θ − sτ), (3.3)

which should give the same results as (3.2). A graphical representation of the time-scalinginfluence is shown in Figure 3.2.

t

τ

Before time-scaling

After time-scaling

θ

sτ

Figure 3.2: Effect of time-scaling on the influence of time-delay.

The example illustrates that, while the time-delay in a network of neurons is constant, wecan simulate a change in operation speed by varying it. We can see this operation speed as ameasure of overall speed of the animal, since the animal has to react faster on changes in itsenvironment.

3.2 A network of neuron models

A lot of scientific contributions treat the modeling of neurons. A model that nowadays isstill the most complete one, has been proposed by Hodgkin and Huxley [7]. However, themodel is quite complex. It is described by a set of four differential equations and has a largenumber of parameters. Numerical simulations thus are time-consuming and analysis of themodel is difficult. As a result, a variety of reduced-order models is developed. These modelsmay not be a perfect resemblance of real neurons, but most of them exhibit the same dynamicproperties.

All models are based on the electrochemical properties of the neuron. In Figure 3.3, the partsof a neuron, responsible for its behavior, are named. The axon is responsible for generatingpotential difference V = Vi − Veq, by activation of sodium Na+ and potassium K+ ions,which results in a current flow through the membrane of the axon. Furthermore, there exists

32 A synchronization approach on animal gaits

Figure 3.3: The neuron (taken from [1]).

a resting potential Veq and a leakage current Il. Using their famous squid axon voltage clampexperiments, Hodgkin and Huxley proposed a model of a neuron, based on an electrical cir-cuit. The parameters have been estimated using the experiments. Most of the other modelsare simplifications of this model, using the dynamical properties of the differential equations.

Since this research concerns the synchronization of oscillations of a network of neurons, themodel should have some specific properties. First of all, simulations have to show whether thenetwork can produce various oscillation patterns, resembling animal gaits. Since the analysisof the network requires the formulation of error dynamics, the model should be as simple aspossible. This way, excessive computation times and complex error dynamics are avoided.We briefly discuss four models, with their advantages and disadvantages:

FitzHugh-Nagumo model

The FitzHugh-Nagumo model [12] is a second-order representation of the Hodgkin-Huxleymodel. This reduction is achieved by introducing an external random forcing input I(t).

v(t) = −v(v − a)(v − 1) − w + I(t)

w(t) = ǫ(v − bw) (3.4)

y(t) = v(t)

3.2 A network of neuron models 33

with constants a,b, and ǫ. This model shows nonperiodic oscillations for I(t) ∈ [0.02, 0.04].However, the random input introduces some difficulties when formulating the error dynamics.One either has to set I(t) equal in all systems, so it will vanish in the error dynamics, or keepa random term in the error dynamics. In the first case, the oscillatory behavior of the networkwill not be observed and in the second case, investigating Lyapunov stability of the networkwill require knowledge of the stability of stochastic systems. Investigating the stability ofstochastic systems is out of the scope of this research.

Morris-Lecar model

The Morris-Lecar model [11], like the Hodgkin-Huxley model, is a very complete model. Alot of oscillation patterns can be generated, but unfortunately the model is very complex, dueto a large number of nonlinear terms.

v(t) =1

C[−gKn(v − Vk) − gCam(v − VCa) − gl(v − Vl) + I)]

n(t) = λn(v)(n∞(v) − n)

m(t) = λm(v)(m∞(v) − m)

y(t) = v(t)

with (3.5)

n∞(v) =1

2(1 + tanh((v − V3)/V4))

λn(v) = λn cosh((v − V3)/2V 4)

m∞(v) =1

2(1 + tanh((v − V1)/V2))

λm(v) = λm cosh((v − V1)/2V2)

and constants gK , Vk, gCa, VCa, gl, and I.

The error dynamics of a network of Morris-Lecar models will be very difficult to analyze.

Lur’e type neuron model

The Lur’e type neuron model [8] is a simplification of the Morris-Lecar model. The number ofnonlinear terms is reduced to two, which can be chosen either piecewise linear or continuous.The model is also reduced to second-order.

v(t) = cφ(av) − bv + u0 − w + uex

w(t) = ρ[φ(d(v + V0)) − w] (3.6)

y(t) = v(t)

with

φ(x) =1

1 + e2−4x

with constants a, b, c, d, u0, uex, ρ and V0. The model has the same dynamics properties of themore complicated Morris-Lecar model [8]; it shows oscillatory behavior in various manners.

34 A synchronization approach on animal gaits

In Figure 3.4 the response of one system is plotted against time for the following parameters:

a = 1.8 b = 3 c = 2.2 d = 5u0 = −0.2 v0 = −0.35 ρ = 0.3 uex = 0.04,

. (3.7)

Clearly the pulsing character of the system can be seen. The Lur’e model will be used inmodeling a network of neurons.

0 10 20 30 40 50 60 70 80 90 100−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t

v(t)

,w(t

)

v(t)w(t)

Figure 3.4: Response of the Lur’e neuron model.

3.3 Synchronization of a network of Lur’e type neuron models

Now we have chosen a model, we construct a network as in the example of Chapter 2. Recallthe dynamics of the Lur’e model in (3.6). For one system in the network we write:

Vi =

vi(t) = cφ(avi) − bvi + u0 − wi + uex + uiv

wi(t) = ρ[φ(d(vi + v0)) − wi] + uiw

yi(t) = CVi(t)i = 1, 2, 3, 4 (3.8)

with (3.9)

φ(x) =1

1 + e2−4x. (3.10)

3.3 Synchronization of a network of Lur’e type neuron models 35

Since only the voltage v(t) is an output of the system, the network is coupled as follows:

ui = −K(yi − yjτ ), K > 0, C = [1 0],i = 1, 2, 3, 4j = 4, 1, 2, 3

(3.11)

= −KC(Vi − Vjτ )

= −K(vi − vjτ )

Using the parameters from (3.7) and choosing initial conditions for both states randomlybetween [0, 0.5], the network shows full and partial synchronization for varying coupling gainK and time-delay τ . To validate these simulations, we follow the same procedure as in Section2.8.2, but now with only the voltage v(t) as output. The error dynamics are

e(t) = A0e + A1eτ + Ψ(vi), i = 1, 2, 3, 4,

where

A0 =

[

At OO At

]

At =

[

−b − k −10 −ρ

]

A1 =

[

O −Aτ

Aτ O

]

Aτ =

[

−k 00 0

]

Ψ(vi) =

φ1(v1) − φ1(v3)

φ2(v1) − φ2(v3)

φ1(v2) − φ1(v4)

φ2(v2) − φ2(v4)

,

for partial synchronization. For full synchronization the errors between all systems are in-cluded in the error dynamics. Using the Lyapunov-Krasovskii functional from (2.12) andcondition (2.79), we obtain LMI (2.85). Formulating the Generalized Eigenvalue Problemand solving for varying coupling gain K, we obtain Figure 3.5. Again we conclude what wasseen in the example in Section 2.8.2: the calculations show a difference with the simulations.The Lyapunov stability approach in combination with the sector condition results in a toostrict condition for full synchronization of the network.

Moreover, if we take a look at partial synchronization from a symmetry point of view [14],an interesting observation arises. Since all systems are coupled exactly the same with exactlythe same coupling parameters, the network is highly symmetric. A permutation of systemsshould give the same results, since all systems are identical with identical couplings. Lookingat the synchronization problem, this means that if we investigate synchronization of system1 and 3, we also investigate synchronization of system 1 and 2 or any other combination.In the analysis this approach would mean that actually we investigate full synchronization.However, since the time-delay introduces a phase shift in between the systems, there still issome difference in the couplings between the systems. From simulations we can conclude thata certain value of time-delay introduces a phase lag of 180 in the trajectories of the coupledsystem and a phase lag of 360 for the system that is coupled after that. This implies that eventhough the network consists of identically coupled identical systems, the time-delay disturbsthe input such, that we can not speak of exact symmetry in the network. It is however stillinteresting to look at synchronization from a symmetry point of view to obtain better results.

36 A synchronization approach on animal gaits

0 1 2 3 4 5 6 7 8 9 10

0

50

100

150

200

250

300

Coupling gain K

τ max

[m

s]

Full synchronizationPartial synchronization

(a) Computations

0 1 2 3 4 5 6 7

0

100

200

300

400

500

600

700

800

900

1000

Coupling gain K

τ max

[m

s]

Full synchronization

(b) Simulations

Figure 3.5: Stability region of 4 unidirectionally coupled Lure systems with time-delay, outputyi = vi.

3.4 Simulations with varying coupling gains 37

3.4 Simulations with varying coupling gains

In order to get an overview on full and partial synchronization of the network, simulationsare done with varying coupling gains and time-delay. The coupling gains in the network arenow split into K0 for u1 and u3, and K1 for u2 and u4 (see Figure 3.6).

S1 S2

S3S4

Y1

Y2

Y3

Y4

U1

U2

U3

U4

K0

K1

K1

K0

Figure 3.6: The network with varying coupling gains.

In these simulations, a weaker form of synchronization is stated, named practical synchro-nization:

Definition 1 (3.1). A pair of dynamical systems with state xi(t) and xj(t) respectively(i 6= j), is practically synchronized if there exists a δ > 0 such that

limt→∞

| xi(t) − xj(t) |≤ δ, (3.12)

for arbitrary initial conditions.

This definition is used, because in some cases, the error dynamics do not exactly convergeto zero, but in practice we still can speak of synchronized systems. These results can not becompared with analytical results, because mathematically speaking, synchronization does notoccurs in these cases. It is used only to look at the behavior of the network under varyingcoupling gain and time-delay.

In Figure 3.7 a 3-dimensional plot of the maximum allowable time-delay τmax for K0 and K1

is shown. To investigate full synchronization, the errors between all systems are taken intoaccount (Figure 3.7(a)). For partial synchronization, all combinations of pairs of systems arelooked at. As discussed earlier, this condition for partial synchronization also contains thesituation of full synchronization. Therefore we extract the results for full synchronizationfrom the results for partial synchronization, resulting in Figure 3.7(b). Clearly, it can beseen that when K0 equals K1, full synchronization occurs. When they are different, partialsynchronization is observed. In both cases, there is a minimum coupling gain for whichany form of synchronization occurs. This graph shows that when the coupling gains differfrom each other, partial synchronization is likely to happen. Full synchronization however, isrestricted to equal coupling gains (K0 = K1).

38 A synchronization approach on animal gaits

00.5

11.5

2

0

0.5

1

1.5

2

0

500

1000

1500

K0K

1

τ max

[m

s]

(a) Full synchronization

00.5

11.5

2

0

0.5

1

1.5

2

0

500

1000

1500

K0K

1

τ max

[m

s]

(b) Partial synchronization

Figure 3.7: Synchronization, investigated through simulations, δ = 1 · 10−5.

3.5 The network as central pattern generator 39

3.5 The network as central pattern generator

In this section we focus on numerical simulations of the network and the way it resemblesthe Central Pattern Generator (CPG). In animal locomotion, the limbs have to move in aparticular manner at a particular speed. For example, when a horse is performing the ’walk’gait, each leg move with a quarter period phase shift with respect to its neighboring legs. InFigure 3.1, the phase relations of the legs of a horse are shown for all gaits. The CPG gener-ates the pulsing patterns that the neurons have to send to the muscles, to let the legs performthe desired gait. These pulsing patterns are of course very complex, due to the large numberof muscles that are involved in animal locomotion. However, the basic patterns should be thesame for all separate muscles.

We want to check whether the constructed network of 4 systems, unidirectionally coupled withtime-delay, can generate these patterns. In [4] and [6], the various patterns are generated byvarying the coupling gains in the network separately. Furthermore, they conclude that for thenetwork to generate all gaits patterns, it has to consist of 8 systems, coupled unidirection-ally and bidirectionally in a particular manner. We try to obtain the same patterns with anetwork of 4 unidirectionally coupled systems with time-delay. The coupling gain is constantat K0 = K1 = 1. When varying only the time-delay τ , a variety of oscillation patterns canbe generated, due to the partial synchronization that occurs in the network. The simulationresults are presented in Appendix C.

As seen in Appendix C, the network of 4 unidirectionally coupled systems with time-delayis able to generate most of the oscillation patterns that are required for controlling animalgaits by only varying the time-delay. The phase shifts between the legs of a horse, shown inFigure 3.1 can be distinguished in the simulations as well. For example, the ’walk’ gait can beobtained by setting τ = 1.5[ms] and the ’trot’ or ’bound’ gait can be observed at τ = 4[ms],depending on which system represents which leg. As discussed in section 3.1.2, we thus canlook at the time-delay as a measure of increasing speed, since the gaits are obtained in orderof increasing speed. Of course this network is very simple and the brain is a much morecomplex network of neurons. However, we can conclude that the time-delay is an importantfactor in the dynamics of the network and has to be included in modeling the brain. It shouldbe noted that when varying the coupling gains of the network separately, as done in [4] and[6], even more patterns could be generated.

Chapter 4

Conclusions and recommendations

4.1 Conclusions

A network of four unidirectionally coupled nonlinear systems with time-delay is thoroughlyanalyzed to find conditions, under which the network fully or partially synchronizes.

From the analysis of the network, we can conclude that using the Lyapunov-Krasovskii func-tional in combination with the linearization of the nonlinear terms and the boundedness of thetrajectories or in combination with the sector condition is not a suitable method for investigat-ing the stability of the synchronization. According to the analysis, full synchronization doesnot occur, while simulations show that it does for various coupling gains and time-delays. Thereason for this difference is that the used method is too conservative to give a stability bound.

The network is used to simulate output of the Central Pattern Generator, a part in the brainthat is responsible for the control of animal locomotion. The Lur’e type neuron model is avery suitable model to use in this case, because is has relatively simple equations of motion,which makes formulating error dynamics more easy. This simplification does not affect thedynamical properties of the neuron model much, so it still resembles real neuron behavior.Unfortunately, analysis of the network gives little information about the stability of synchro-nization of four of these models in a network.

Simulations of the network’s behavior under varying amounts of time-delay show that it cangenerate a variety of oscillation patterns. These patterns correspond well to the oscillationpatterns, that are generated by a Central Pattern Generator. This observation is an indicationthat time-delay is an important parameter in modeling a network of neurons. Furthermore,the assumption that increasing time-delay is equivalent to changing the operation speed inthe network, is justified.

42 Conclusions and recommendations

4.2 Recommendations

More methods to investigate (partial) synchronization of a network of nonlinear systems withtime-delay have to be investigated. In [14], synchronization, based on symmetries in thenetwork, is investigated. This method can be of much use in this case, since the network ishighly symmetric.

Using other models that the Lur’e type neuron model could be interesting, since the Lur’etype neuron model is a quite exotic model. More realistic neuron models like the Hodgkin-Huxley or Morris-Lecar model can give a more realistic representation of a network of neurons.

The network of unidirectionally coupled neuron models exhibits the properties to simulateanimal gaits. However, one can expect that the influence of neurons on each-others behavior isbidirectional and nonlinear. Investigating the properties of a network of four bidirectionallycoupled systems is an interesting research topic. It is also interesting to investigate thebehavior of the network with all coupling gains varying and with varying time-delays forevery coupling.

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to Boris Pavlovich Demidovich, Systems and Control Letters 52, pp. 257-261, Elsevier(2004).

44 BIBLIOGRAPHY

[14] Alexander Pogromsky, Giovanni Santoboni, Henk Nijmeijer, Partial synchronization:

from symmetry towards stability, Physica D 172, pp. 65-87, Elsevier (2002).

[15] A. Yu. Pogromsky, Passivity based design of synchronizing systems, Int. Journal of Bi-furcation and Chaos, Vol. 8, No.2, pp. 295-319, World Scientific Publishing Company(1998).

[16] Shengyuan Xu and James Lam, Improved Dalay-Dependent Stability Criteria for Time-

Delay Systems, IEEE Transactions on Automatic Control, Vol. 50, No.3, pp. 384-387(March 2005).

[17] T. Yamamoto, T. Oguchi and H. Nijmeijer, Synchronization of coupled nonlinear systems

with time-delay, Proceedings of European Control Conference 2007, Kos, Greece, toappear (July 2007).

Appendix A

Semi-passivity of the Lorenz system

Recall the definition of semi-passivity:

Definition (strictly semi-passivity). System (2.17) is said to be strictly semi-passive, if thereexist a C1-class function V : R

n → R, class-K∞ functions α(·), α(·) and α(·) satisfying

α(‖x‖) ≤ V (x) ≤ α(‖x‖)

V (x) ≤ −α(‖x‖) − H(x) + yT u

for all x ∈ Rn, u ∈ R

m, y ∈ Rm, where the function H(x) satisfies

‖x‖ ≥ η ⇒ H(x) ≥ 0

for a positive real number η.

Consider the Lorenz system for i = 1:

Σ1 =

x1 = σ(y1 − x1) − k(x1 − x4τ )y1 = rx1 − y1 − x1z1 − k(y1 − y4τ )z1 = −bz1 + x1y1 − k(z1 − z4τ ),

(A.1)

We define a storage function V (x1) = xT1 x1 with x1 = [x1 y1 z1 − σ − r]T . The derivative

of V (x1) is

V (x1) = x1x1 + y1y1 + (z1 − σ − r)x1 (A.2)

= − σx21 − y2

1 − bz21 + (σ + r)z1

− kx1(x1 − x4τ ) − ky1(y1 − y4τ ) − kz1(z1 − z4τ )

+ ǫx21 − ǫx2

1 + ǫy21 − ǫy2

1 + ǫz21 − ǫz2

1

With slack variable 0 < ǫ < 1. Here the derivative is split into two parts. The first part isrewritten as follows.

− σx21 − y2

1 − bz21 + (σ + r)z1 + ǫx2

1 + ǫy21 + ǫz2

1

= − (σ − ǫ)x21 − (1 − ǫ)y2

1 − (b − ǫ)

[

z21 −

b(σ + r)

(b − ǫ)z1

]

= − (σ − ǫ)x21 − (1 − ǫ)y2

1 − (b − ǫ)

[

z1 −b(σ + r)

2(b − ǫ)

]2

+b2(σ + r)2

4(b − ǫ)

= − H(x1)

46 Semi-passivity of the Lorenz system

For the second part we use the following expressions.

(x − y)T K(x − y) =xT Kx + yT Ky − 2xT Ky

xT Ky =1

2xT Kx +

1

2yT Ky −

1

2(x − y)T K(x − y)

xT K(x − y) =1

2xT Kx −

1

2yT Ky +

1

2(x − y)T K(x − y).

If K ≤ 0, then1

2(x − y)T K(x − y) ≤ 0

and xT K(x − y) ≤1

2xT Kx −

1

2yT Ky.

Using the above derived expressions, the second part of (A.2) becomes

− kx1(x1 − x4τ ) − ky1(y1 − y4τ ) − kz1(z1 − z4τ ) − ǫx21 − ǫy2

1 − ǫz21

= − kx1(x1 − x4τ ) − ǫx21

= − (K

2+ ǫ)‖x1‖

2 +K

2‖x4τ‖

2

= − α(‖x1‖) + β(‖x4τ‖).

Combining these terms again, we conclude that the Lorenz system is semi-passive with respectto the supply rate β(‖x4τ‖).

Appendix B

Derivation of LMI (2.85)

Set

et = e(t + θ), −2τ < θ < 0

and define a Lyapunov-Krasovskii functional as

V (et) = V1(et) + V2(et) + V3(et) (B.1)

where

V1(et) = e(t)T Pe(t)

V2(et) =

∫ 0

−τ

∫ t

t+β

e(α)T Ze(α)dαdβ

V3(et) =

∫ t

t−τ

e(α)T Qe(α)dα.

By the Newton-Leibniz formula, we have e(t − τ) = e(t) −∫ t

t−τe(α)dα. The error dynamics

are formulated as et = A0et + A1eτ + Ψ(vi). Furthermore we use the sector condition:

Ψ(vi)T (Ψ(vi) − ηCe(t)) ≤ 0, (B.2)

with

ηC =

ca 0 0 0 0 0 0 0ρd 0 0 0 0 0 0 00 0 ca 0 0 0 0 00 0 ρd 0 0 0 0 00 0 0 0 ca 0 0 00 0 0 0 ρd 0 0 00 0 0 0 0 0 ca 00 0 0 0 0 0 ρd 0

and ηC =

ca 0 0 0ρd 0 0 00 0 ca 00 0 ρd 0

, (B.3)

for full and partial synchronization respectively. The derivatives of the three terms in (B.1)then are:

48 Derivation of LMI (2.85)

V1 = e(t)P e(t) + e(t)Pe(t)

= e(t)P (Ae(t) + A1e(t − τ) + Ψ(v)) + (Ae(t) + A1e(t − τ) + Ψ(v))Pe(t)

= 2e(t)T (PA)e(t) + 2e(t)T (PA1)e(t − τ)

+ 2e(t)T PΨ(v) − 2λΨ(v)T (Ψ(v) − ηCe(t))

= 2e(t)T (PA0)e(t) + 2e(t)T (PA1)e(t − τ)

+ 2e(t)T PΨ(v) − 2λΨ(v)T (Ψ(v) − ηCe(t))

+ 2e(t)T Y e(t) − 2e(t)T Y e(t − τ) − 2e(t)T Y

∫ t

t−τ

e(α)dα

+ 2e(t − τ)T We(t) − 2e(t − τ)T We(t − τ) − 2e(t − τ)T W

∫ t

t−τ

e(α)dα

+ 2Ψ(v)T Re(t) − 2Ψ(v)T Re(t − τ) − 2Ψ(v)T R

∫ t

t−τ

e(α)dα

= 2e(t)T (PA0 + Y )e(t) + 2e(t)T (PA1 − Y + W T )e(t − τ) + 2e(t)T (P + RT + λCT ηT )Ψ(v)

− 2e(t − τ)T (W )e(t − τ) − 2e(t − τ)T (RT )Ψ(v)

− Ψ(v)T (2λ)Ψ(v)

− 2e(t)T (Y )

∫ t

t−τ

e(α)dα − 2e(t − τ)T (W )

∫ t

t−τ

e(α)dα − 2Ψ(v)T (R)

∫ t

t−τ

e(α)

=1

τ

∫ t

t−τ

[2e(t)T (PA0 + Y )e(t) + 2e(t)T (PA1 − Y + W T )e(t − τ)

+ 2e(t)T (P + RT + λCT ηT )Ψ(v) − 2e(t − τ)T (W )e(t − τ) − 2e(t − τ)T (RT )Ψ(v)

− Ψ(v)T (2λ)Ψ(v)

− 2e(t)T (τY )e(α)dα − 2e(t − τ)T (τW )e(α)dα − 2Ψ(v)T (τR)e(α)]dα

V2 =

∫ −τ

0[e(t)T Ze(t) − e(t + β)T Ze(t + β)]

=

∫ −τ

0[e(t)T Ze(t) − e(α)T Ze(α)]

=

∫ −τ

0[(A0e(t) + A1e(t − τ) + Ψ(v))T Z(A0e(t) + A1e(t − τ) + Ψ(v)) − e(α)T Ze(α)]

=1

τ

∫ t

t−τ

[e(t)T τAT0 ZA0e(t) + 2e(t)T τAT

0 ZA1e(t − τ)

+ e(t − τ)T τAT1 ZA1e(t − τ) − e(α)T τ e(α)

+ 2e(t)T τAT0 ZΨ(v) + 2e(t − τ)T τAT

1 ZΨ(v) + Ψ(v)T τZΨ(v)]dα

V3 = e(t)Qe(t) − e(t − τ)Qe(t − τ)

1

τ

∫ t

t−τ

[

e(t)T Qe(t) − e(t − τ)T Qe(t − τ)]

dα

49

Combining these three terms, we obtain

V (et) =1

τ

∫ t

t−τ

ζ(t, α, v)T Λ(τ)ζ(t, α, v)dα (B.4)

with

ζ(t, α, v) =[

e(t)T e(t − τ)T Ψ(v) e(α)T]

,

and a symmetric matrix

Λ(τ) =

PA0 + AT0 P + Y + Y T + τAT

0 ZA0 + Q ∗ ∗ ∗AT

1 P − Y T + W + τAT1 ZA0 −W − W T + τAT

1 ZA1 − Q ∗ ∗P + R + τZA0 + ληC τZA1 − R −2λ ∗

−τY T −τW T −τRT −τZ

.

Applying a Schur complement as in [16], we obtain LMI (2.85).

Appendix C

Simulation results

Here, some simulation results are shown for the unidirectionally coupled network of Lur’etype neuron models with partial state coupling. For specific values of time-delay, the phaseshifts, as shown in Figure 3.1, are compared with the ones shown here. The left graph showsthe time response v(t) of the four systems. The graph to the right shown the time responseof the errors of the two states between the systems.

In Figure C.1 the network is simulated for a time-delay τ = 0.5 [ms]. Clearly, it can be seenthat the network synchronizes fully for this value of time-delay. Only after 20 [ms] all systemspulse exactly similar. In the lower figure it can be seen that the errors between all systemsconverge to zero after 20 [ms]. If we assume that the amount time-delay represents the speedof the horse, low values of time-delay represent low speeds. Looking at the possible gaits atthis speed in Figure 3.1, we can only conclude that for these values of time-delay, the horsestands still.

In Figure C.2 the time-delay in the coupling is increased to τ = 1.5 [ms]. Now the systems donot synchronize fully, as can be seen very clearly if we look at the errors between the systems.However, we can see that the oscillations are similar, but only with a phase shift of a quarterof the period of the oscillation. This phase shift corresponds with the ’walk’ gait from Figure3.1, where we can also see the phase shifts of a quarter of the period.

If the time-delay is increased to τ = 4 [ms], the pattern changes again, as we see in FigureC.3. Because the frequency of the oscillation increases, we zoom in on the last part of thesimulation for a better view on the results. In this case, system 1 and 3 synchronize andsystem 2 and 4 synchronize as well. Between these two pairs, a phase shift of half a periodappears. This oscillation pattern corresponds to three possible gaits in Figure 3.1, the ’trot,’pace’ and ’bound’ gait.

Further increasing the time-delay, the ’gallop’ gait is obtained for τ = 8 [ms], as can be seenin Figure C.4.

52 Simulation results

0 10 20 30 40 50 60 70 80 90 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

V

S1S2S3S4

(a) States vi(t)

0 10 20 30 40 50 60 70 80 90 100

−0.05

0

0.05

S1−

S2

0 10 20 30 40 50 60 70 80 90 100

−0.05

0

0.05

S2−

S3

0 10 20 30 40 50 60 70 80 90 100

−0.05

0

0.05

S3−

S4

0 10 20 30 40 50 60 70 80 90 100−0.05

0

0.05

0.1

S4−

S1

0 10 20 30 40 50 60 70 80 90 100−0.1

−0.05

0

0.05

S1−

S3

0 10 20 30 40 50 60 70 80 90 100

−0.1−0.05

00.05

t

S2−

S4

(b) Errors

Figure C.1: K = 1, τ = 0.5 [ms], Stand.

53

0 5 10 15 20 25 30 35 40 450

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

V

S1S2S3S4

1 2 3 4

(a) States vi(t)

0 5 10 15 20 25 30 35 40 45−0.4

−0.2

0

0.2

S1−

S2

0 5 10 15 20 25 30 35 40 45−0.4−0.2

00.20.4

S2−

S3

0 5 10 15 20 25 30 35 40 45−0.4

−0.2

0

0.2

S3−

S4

0 5 10 15 20 25 30 35 40 45−0.4

−0.2

0

0.2

S4−

S1

0 5 10 15 20 25 30 35 40 45−0.4−0.2

00.20.4

S1−

S3

0 5 10 15 20 25 30 35 40 45−0.4−0.2

00.20.4

t

S2−

S4

(b) Errors

Figure C.2: K = 1, τ = 1.5 [ms], Walk.

54 Simulation results

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

t

v i(t)

375 380 385 390 395 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

v i(t)

S1S2S3S4

S1S2S3S4

1,3 2,4

(a) States vi(t)

0 50 100 150 200 250 300 350 400−0.4−0.2

00.20.4

S1−

S2

0 50 100 150 200 250 300 350 400−0.4−0.2

00.20.4

S2−

S3

0 50 100 150 200 250 300 350 400−0.4−0.2

00.20.4

S3−

S4

0 50 100 150 200 250 300 350 400−0.4−0.2

00.20.4

S4−

S1

0 50 100 150 200 250 300 350 400

−0.2

0

0.2

S1−

S3

0 50 100 150 200 250 300 350 400

−0.2

0

0.2

t

S2−

S4

(b) Errors

Figure C.3: K = 1, τ = 4 [ms], Trot/Pace/Bound.

55

0 50 100 150 200 250 300 350 400

0

0.1

0.2

0.3

0.4

0.5

t

v i(t)

375 380 385 390 395 4000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

v i(t)

S1S2S3S4

S1S2S3S4

1 4 3 2

(a) States vi(t)

0 50 100 150 200 250 300 350 400−0.4−0.2

00.20.4

S1−

S2

0 50 100 150 200 250 300 350 400−0.4−0.2

00.20.4

S2−

S3

0 50 100 150 200 250 300 350 400−0.4−0.2

00.20.4

S3−

S4

0 50 100 150 200 250 300 350 400−0.4−0.2

00.20.4

S4−

S1

0 50 100 150 200 250 300 350 400−0.5

0

0.5

S1−

S3

0 50 100 150 200 250 300 350 400−0.5

0

0.5

t

S2−

S4

(b) Errors

Figure C.4: K = 1, τ = 8 [ms], Gallop.