# Passification-based decentralized adaptive synchronization of dynamical networks with time-varying delays

01 Jan 2015-Journal of The Franklin Institute-engineering and Applied Mathematics (Pergamon)-Vol. 352, Iss: 1, pp 52-72

TL;DR: This paper is aimed at application of the passification based adaptive control to decentralized synchronization of dynamical networks, and considers Lurie type systems with hyper-minimum-phase linear parts and two types of nonlinearities: Lipschitz and matched.

Abstract: This paper is aimed at application of the passification based adaptive control to decentralized synchronization of dynamical networks. We consider Lurie type systems with hyper-minimum-phase linear parts and two types of nonlinearities: Lipschitz and matched. The network is assumed to have both instant and delayed time-varying interconnections. Agent model may also include delays. Based on the speed-gradient method decentralized adaptive controllers are derived, i.e. each controller measures only the output of the node it controls. Synchronization conditions for disturbance free networks and ultimate boundedness conditions for networks with disturbances are formulated. The proofs are based on Passification lemma in combination with Lyapunov–Krasovskii functionals technique. Numerical examples for the networks of 4 and 100 interconnected Chua systems are presented to demonstrate the efficiency of the proposed approach.

## Summary (2 min read)

Jump to: [1. Introduction] – [2. Problem statement] – [3. Controller design] – [4. Synchronization conditions] – [5. Ultimate boundedness of disturbed system] – [6. Numerical example] and [7. Conclusion]

### 1. Introduction

- It is motivated by a broad area of potential applications: networks of robots, formations of flying and underwater vehicles, control of industrial, electrical, communication, and5 production networks, etc.
- To simplify exposition and make more clear basic ideas the authors do not use shunts in this paper.
- Again, adaptive control laws were derived only for a narrow class of networks, such as fully-controlled and fully-measured agents.
- The last inequality has a simple physical interpretation.

### 2. Problem statement

- Functions ϕ0, ϕij and ψij describe local dynamics of the nodes and their interactions.
- Note that the network model (4) admits delay in local agent dynamics described by the term ψii(t, xi(t− τ(t))).
- (7) Here the authors assume that the controller of the i-th subsystem does not possess any information about other nodes.

### 3. Controller design

- (11) Since the system is uncertain the values of θi are adjusted adaptively using95 the speed-gradient method [33].
- The control law (12) includes undefined terms Γi. Synchronization conditions will be proved for all Γi > 0. At100 the same time, large Γi may cause undesirable oscillations of θi.
- At the same time the terms ϕij , ψij depend on yj105 and may prevent the system from synchronization.
- Therefore, to synchronize the system with (12) one need to ensure that the influence of ϕij , ψij is small enough.
- In what follows the authors derive conditions on110 Lipschitz constants Lij , Mij such that (12) ensures (9) for the network under consideration.

### 4. Synchronization conditions

- Moreover, all tunable parameters θi(t) will tend to constant values.
- Remark 3. Note that the boundedness of xi(t) is not proved in the theorem.
- Now the authors consider the second class of nonlinearities.

### 5. Ultimate boundedness of disturbed system

- An important issue for control system design is providing its robustness with respect to disturbances unmodelled dynamics.
- It is well known however that many adaptive systems do not possess such a property that makes their behavior165 very sensitive to inevitable impreciseness of the plant model.

### 6. Numerical example

- To demonstrate the efficiency of the proposed algorithm the authors make use of a celebrated Chua circuit [39].
- That is, the adaptive controller (12) allows one to ensure ultimate boundedness of a network (4), (8) with a small enough control gain.
- All system parameters are same as previously and the topology of the network was chosen randomly such that L = 0.04 and Mh = 0.04.

### 7. Conclusion

- The authors examined the problem of decentralized adaptive control for dynamical networks with instant and delayed nonlinear interconnections.
- On the other hand, like in the previous designs, the proposed adaptive controllers (12), (22) are decentralized, and therefore, interconnections are required to be weak enough.
- Two types of agent nonlinearity ϕ0 were considered.
- First, for Lipschitz continuous functions it is required that Lipschitz constant is small enough.
- All results255 are formulated for the case of slowly-varying time delay.

Did you find this useful? Give us your feedback

This is a repository copy of Passification-based decentralized adaptive synchronization of

dynamical networks with time-varying delays.

White Rose Research Online URL for this paper:

http://eprints.whiterose.ac.uk/158812/

Version: Accepted Version

Article:

Selivanov, A. orcid.org/0000-0001-5075-7229, Fradkov, A. and Fridman, E. (2015)

Passification-based decentralized adaptive synchronization of dynamical networks with

time-varying delays. Journal of the Franklin Institute, 352 (1). pp. 52-72. ISSN 0016-0032

https://doi.org/10.1016/j.jfranklin.2014.10.007

Article available under the terms of the CC-BY-NC-ND licence

(https://creativecommons.org/licenses/by-nc-nd/4.0/).

eprints@whiterose.ac.uk

https://eprints.whiterose.ac.uk/

Reuse

This article is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs

(CC BY-NC-ND) licence. This licence only allows you to download this work and share it with others as long

as you credit the authors, but you can’t change the article in any way or use it commercially. More

information and the full terms of the licence here: https://creativecommons.org/licenses/

Takedown

If you consider content in White Rose Research Online to be in breach of UK law, please notify us by

emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request.

Passiﬁcation-based decentralized adaptive

synchronization of dynamical networks with

time-varying delays

I

Anton Selivanov

a

, Alexander Fradkov

a,b

, Emilia Fridman

c

a

Department of Theoretical Cybernetics, St.Petersburg State University, St. Petersburg,

Russia

b

Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, St.

Petersburg, Russia

c

Department of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv, Israel

Abstract

This paper is aimed at application of the passiﬁ cat i on based adaptive control to

decentralized synchronization of dy n ami cal networks. We consider Lurie type

systems with hyper-minimum-phase linear parts and two types of nonli n ear i -

ties: Lip schitz and matched. The network is assumed to have both instant and

delayed time-varying interconnections. Agent model may also include d el ays.

Based on the speed-gradient method decentralized adaptive controllers are de-

rived, i. e. each controller measures only the output of the node it controls.

Synchronization conditions for disturb anc e f re e n etworks and ultimate bound-

edness conditions for networks with disturbances are formulated. The proofs are

based on Passiﬁcation lemma in combination with Lyapunov-Krasovskii func-

tionals technique. Numerical ex amp l es for the networks of 4 and 100 intercon-

nected Chua systems are presented to demonstrate the eﬃciency of the proposed

approach.

Keywords: networks, adaptive control, uncertain systems, delays

I

Partially supported by the Russian Foundation for Basic Research (project 14-08-01015),

Israel Science Foundation (grant No. 754/10), Program of basic research of OEMMPU RAS

01 and Government of Russian Federation (Grant 074-U01 to ITMO University). A conference

version of the paper has been presented in [1].

∗

Corresponding author

Email address: antonselivanov@gmail.com (Anton Selivanov)

Preprint submitted to Journal of The Franklin Institute October 22, 2014

1. Introduction

Adaptive synchronization of dynamical networks has attracted a growing

interest during recent years [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. It is motivated by

a broad area of potential applications: networks of robots, formations of ﬂying

and underwater vehicles, control of industrial, electrical, communication, and5

production networks, etc. Although problems of decentralized control for net-

works of coupled systems were studied before, most of the existing works, e. g.

[3, 4, 5, 6, 7, 8], deal with full state feedback and linear interconnections. More-

over, control variables usually appear in all equations of the network model.

Such system models are quite r es tr i ct i ve for appli cat i on s, where uncertainties10

of the system, nonlinear interconnections, switching structure of the network

topology, nonl i n ear dynamics of the local subsystems and incomplete measure-

ment of their states should be taken into account.

The key to solve the above problem is application of the passiﬁcation ap-

proach. It was initially proposed in 1974 for a SIMO plant [12] and later was15

extended to a broad class of MIMO linear and nonlinear systems. Related ver-

sions are also known under names “adaptive systems with implicit reference

models” [13] , “adaptive control based on feedback Kalman-Yakubovich lemma”

[14] and “simple adaptive control” [15, 16]. Adaptive syst em design proposed in

the 1970s was sensitive to disturbances: an arbitrary small disturbance was able20

to destroy boundedness of the traje ct or ie s. Later regularizat i on tricks to over-

come diﬃculties were proposed, e. g. negative parametric feedback used in this

paper. In the early papers on the passiﬁcation based approach the restrictive

hyper-minimum-phase condit i on was imposed. However later the so called “p ar -

allel feedforward compensator” (shunt) was proposed by Barkana in [17, 18] and25

extended in [19] t hat allowed one to relax hyper-minimum-phase condit ion re-

quiring only minimum phaseness, without “relative degree one” property. Thus,

relative degree one restriction has b ee n removed. To simplify exposition and

make more clear basic ideas we d o not use shunts in this paper. The idea

of shunt trick can be found i n [19, 20] while detailed exposition is to appear30

2

elsewhere.

A passiﬁcation based app roach to decentralized adapti ve synchronization of

the Lurie type n etworks with incomplete measurements and incomplete control

was proposed in [21]. Here we extend these re su l t s to the case of t i me -varying

unknown interconnection delays and bounded disturbances.35

For the syn chronization of networks with delayed couplings and disturbances

quite a number of papers have already been published [22, 23, 24, 25, 26, 27,

28, 29]. However, again, adaptive control laws were derived only for a narrow

class of networks, such as fully-controlled and fully-measured agents. Some of

these works deal with non-switching topology or provide non-adaptive control.40

In the cu r re nt work we propose an adaptive decentralized algorithm for syn-

chronization of networks with nonlinear delayed couplings that depend on time.

We consider partly unknown Lurie type nonlinear systems with delayed inter-

connections and bounded disturbances. The controller does not use any infor-

mation about sy s t em parameters, b ut to ensure synchroni zat i on it is requi r ed45

that all subsystems belong to a spe ci al c l ass desc ri bed below (see condit i ons of

Theorems 1, 2, 3, 4). Our approach is based on Passiﬁcation lemma [30] and

Lyapunov-Krasovskii method.

Notations used throughout the paper is fairly standard. The ﬁelds of real

and complex numbers are denoted by R, C. R

n

is n-dimensional Euclidean space50

with Euclidean norm ∥x∥ =

n

i=1

x

2

i

. C[a, b] is a space of continuous func-

tions mapping the interval [a, b] into R

n

with a norm ∥φ∥

C

= max

s∈[a,b]

∥φ(s)∥.

As usual I is an identity matrix, A

T

is transposed matrix A, λ

max

(A) is the

maximum eigen valu e of a square matrix A, sign p = −1 for p < 0, 0 for p = 0

and 1 for p > 0.55

Some preliminary results were presented in [1] .

1.1. Passiﬁcation method

Deﬁnition 1. For given A ∈ R

n×n

, B ∈ R

n

, C ∈ R

l×n

, g ∈ R

l

a trans-

fer function g

T

W (s) = g

T

C(sI − A)

−1

B is called h yper-minimum-phase if the

polynomial g

T

W (s) det(sI − A) is Hurwitz and g

T

CB is a positive number.60

3

To formulate main results we will need Passiﬁcation lemma in the fol lowing

form [31].

Lemma 1 (Passiﬁcation lemma). Let the matrices A ∈ R

n×n

, B ∈ R

n

, C ∈

R

l×n

, g ∈ R

l

be given. Then for existence of a positive-deﬁnite n × n-matrix

P = P

T

> 0 and a vector θ

∗

∈ R

l

such that

P A

∗

+ A

T

∗

P < 0, P B = C

T

g, (1)

where A

∗

= A−Bθ

T

∗

C, it is necessary and suﬃcient that the f unct ion g

T

W (s) =

g

T

C(sI − A)

−1

B is hyper-minimum-phase.

Remark 1. Consider a linear system

˙x = Ax + Bu, y = Cx. (2)

It follows from Passiﬁcation lemma (see [20] for details) that if g

T

C(sI −A)

−1

B

is hyper-minimum-phase then there exists θ

∗

such that the input u = −θ

T

∗

y +

v makes t he sy s tem (2) strictly pas s ive with respect t o a new inpu t v, i. e.

there exist a nonnegative scalar function V (x) and a scalar function ρ(x), where

ρ(x) > 0 for x = 0, such that

V (x) ≤ V (x

0

) +

t

0

v(t)

T

g

T

y(t) − ρ(x(t))

dt

for any solution of the system (2) sati s fy i ng x(0) = x

0

.65

The last inequality has a simple physical interpretation. Function V (x) is an

analog of system total energy. The term v(t)

T

g

T

y(t) can be inter preted as the

power transmitted to the system, meaning that

t

0

v(t)

T

g

T

y(t) dt is the energy

transmitted to the system. The term ρ(x(t)) reﬂects dissipation rate that arises

due to energy loss (friction, for instance). Therefore, the last inequality is an70

energy balance f or a system withou t internal energy sources.

It follows from Passiﬁcation lemma that if g

T

W (s) is hyper-minimum-phase

then there exist P > 0, θ

∗

, ε > 0 such that

P A

∗

+ A

T

∗

P < −εI, P B = C

T

g, (3)

4

##### Citations

More filters

••

TL;DR: The main purpose is to design a sampled-data controller so as to the synchronization error system (SES) is exponentially stable and satisfies a predefined H ∞ / passive performance index simultaneously.

Abstract: This paper investigates the problem of the mixed H ∞ / passive sampled-data synchronization control for complex dynamical networks (CDNs) with distributed coupling delay. The sampled interval is deemed as time-varying. The main purpose is to design a sampled-data controller so as to the synchronization error system (SES) is exponentially stable and satisfies a predefined H ∞ / passive performance index simultaneously. Some novel auxiliary function-based integral inequalities are applied to reduce the conservativeness of the presented results, and some effective synchronization criteria are addressed. The gains for the desired controller can be designed by settling an optimization issue in view of the proposed criteria. Three examples are employed to demonstrate the less conservativeness and superiority of the addressed method.

105 citations

••

TL;DR: This article tackles and solves the problem of cyber-secure tracking for a platoon that moves as a cohesive formation along a single lane undergoing different kinds of cyber threats, that is, application layer and network layer attacks, as well as network induced phenomena.

Abstract: The development of autonomous connected vehicles, moving as a platoon formation, is a hot topic in the intelligent transportation system (ITS) research field. It is on the road and deployment requires the design of distributed control strategies, leveraging secure vehicular ad-hoc networks (VANETs). Indeed, wireless communication networks can be affected by various security vulnerabilities and cyberattacks leading to dangerous implications for cooperative driving safety. Control design can play an important role in providing both resilience and robustness to vehicular networks. To this aim, in this article, we tackle and solve the problem of cyber-secure tracking for a platoon that moves as a cohesive formation along a single lane undergoing different kinds of cyber threats, that is, application layer and network layer attacks, as well as network induced phenomena. The proposed cooperative approach leverages an adaptive synchronization-based control algorithm that embeds a distributed mitigation mechanism of malicious information. The closed-loop stability is analytically demonstrated by using the Lyapunov–Krasovskii theory, while its effectiveness in coping with the most relevant type of cyber threats is disclosed by using PLEXE, a high fidelity simulator which provides a realistic simulation of cooperative driving systems.

103 citations

### Additional excerpts

...Now, as far as limt→∞ θ(t) = 0, by integrating (8), following [77], it is possible to prove that all κij(t) tend to some constant values...

[...]

••

30 Sep 2016

TL;DR: A brief review of distributed consensus algorithms is given, focusing on the basic ideas and relevant mathematical theory, in particular, graph theoretic methods.

Abstract: Many cooperative behaviors of multi-agent teams emerge from local interactions among the agents, where an agent interacts with a few “adjacent” teammates, but has no information about the remaining agents. For instance, the selforganization of many biological populations – including swarms of insects, flocks of birds, and schools of fish – are based on such local interaction rules: the motion and decisions of an individual agent are determined by the behavior of its nearest neighbors in the population. A special case of multi-agent coordination is consensus, that is, the agreement of agents on some quantity of interest or, more generally, the full or partial synchronization of their state trajectories. Establishing consensus is a “benchmark” problem in multi-agent systems study, which allows to reveal the main principles of multi-agent coordination and, in particular, the role of the system’s interaction graph (or topology). Consensus lies in the heart of many natural phenomena (e.g., synchronous oscillation of neural cells, which maintains a stable heart rhythm) and engineering designs (e.g., attitude synchronization of satellites). In this article, we give a brief review of distributed consensus algorithms, focusing on the basic ideas and relevant mathematical theory, in particular, graph theoretic methods.

40 citations

••

TL;DR: A delayed distributed proportional-integral-derivative control is proposed and the overall closed-loop stability is proven by exploiting the Lyapunov-Krasovskii theory.

Abstract: This study addresses the leader-tracking problem for linear multi-agent systems in the presence of both parameter model uncertainties and time-varying communication delays. To solve the robust output consensus problem, a delayed distributed proportional-integral-derivative control is proposed and the overall closed-loop stability is proven by exploiting the Lyapunov-Krasovskii theory. Delay-dependent robust stability conditions are given via linear matrix inequalities which allow the proper tuning of robust control gains. The effectiveness of the theoretical derivation is confirmed through a numerical analysis in the practical application domain of cooperative driving for connected vehicles.

35 citations

••

TL;DR: With the help of a new established differential inequality with distributed time-delay, several useful conditions to guarantee the global exponential synchronization of the complex dynamical network with mixed delays are provided.

Abstract: This paper deals with the synchronization problem of delayed complex dynamical network with hybrid-coupling, which is composed of constant coupling, discrete-delay coupling, and distributed-delay coupling. The internal time-delay of node and coupling time-delays are time-varying and can differ from each other. Based on the aperiodically intermittent control strategy, several appropriate pinning controllers and updating laws are designed in each control period. With the help of a new established differential inequality with distributed time-delay, several useful conditions to guarantee the global exponential synchronization of the complex dynamical network with mixed delays. Moreover, the minimum number of pinned nodes can be determined from the proposed criteria. Two numerical examples are provided to verify the feasibility of theoretical results.

33 citations

### Cites methods from "Passification-based decentralized a..."

...Many control methods, such as adaptive feedback control [11-13], impulsive control [14-15], sampled-data control [16-18], decentralized method [19-20] and so on, have been developed to force the coupled dynamical network to synchronize....

[...]

##### References

More filters

••

26 Jul 2009

TL;DR: This self-contained introduction to the distributed control of robotic networks offers a broad set of tools for understanding coordination algorithms, determining their correctness, and assessing their complexity; and it analyzes various cooperative strategies for tasks such as consensus, rendezvous, connectivity maintenance, deployment, and boundary estimation.

Abstract: This self-contained introduction to the distributed control of robotic networks offers a distinctive blend of computer science and control theory. The book presents a broad set of tools for understanding coordination algorithms, determining their correctness, and assessing their complexity; and it analyzes various cooperative strategies for tasks such as consensus, rendezvous, connectivity maintenance, deployment, and boundary estimation. The unifying theme is a formal model for robotic networks that explicitly incorporates their communication, sensing, control, and processing capabilities--a model that in turn leads to a common formal language to describe and analyze coordination algorithms.Written for first- and second-year graduate students in control and robotics, the book will also be useful to researchers in control theory, robotics, distributed algorithms, and automata theory. The book provides explanations of the basic concepts and main results, as well as numerous examples and exercises.Self-contained exposition of graph-theoretic concepts, distributed algorithms, and complexity measures for processor networks with fixed interconnection topology and for robotic networks with position-dependent interconnection topology Detailed treatment of averaging and consensus algorithms interpreted as linear iterations on synchronous networks Introduction of geometric notions such as partitions, proximity graphs, and multicenter functions Detailed treatment of motion coordination algorithms for deployment, rendezvous, connectivity maintenance, and boundary estimation

1,166 citations

••

TL;DR: In this article, sufficient conditions for an array of linearly coupled systems to synchronize are given for the case where the nonzero eigenvalues of the coupling matrix have real parts that are negative enough.

Abstract: This paper gives sufficient conditions for an array of linearly coupled systems to synchronize. A typical result states that the array will synchronize if the nonzero eigenvalues of the coupling matrix have real parts that are negative enough. In particular, we show that the intuitive idea that strong enough mutual diffusive coupling will synchronize an array of identical cells is true in general. Sufficient conditions for synchronization for several coupling configurations will be considered. For coupling that leaves the array decoupled at the synchronized state, the cells each follow their natural uncoupled dynamics at the synchronized state. We illustrate this with an array of chaotic oscillators. Extensions of these results to general coupling are discussed. >

958 citations

### "Passification-based decentralized a..." refers methods in this paper

...To demonstrate the efficiency of the proposed algorithm we make use of a celebrated Chua circuit [39]....

[...]

••

TL;DR: In this paper, a single controller can pin a coupled complex network to a homogeneous solution, and sufficient conditions are presented to guarantee the convergence of the pinning process locally and globally.

Abstract: In this paper, without assuming symmetry, irreducibility, or linearity of the couplings, we prove that a single controller can pin a coupled complex network to a homogenous solution. Sufficient conditions are presented to guarantee the convergence of the pinning process locally and globally. An effective approach to adapt the coupling strength is proposed. Several numerical simulations are given to verify our theoretical analysis.

945 citations

••

TL;DR: In this article, a time-varying complex dynamical network model is introduced, and the synchronization of such a model is determined by the inner-coupling matrix and the eigenvalues and corresponding eigenvectors of the coupling configuration matrix.

Abstract: Today, complex networks have attracted increasing attention from various fields of science and engineering. It has been demonstrated that many complex networks display various synchronization phenomena. In this note, we introduce a time-varying complex dynamical network model. We then further investigate its synchronization phenomenon and prove several network synchronization theorems. Especially, we show that synchronization of such a time-varying dynamical network is completely determined by the inner-coupling matrix, and by the eigenvalues and the corresponding eigenvectors of the coupling configuration matrix of the network.

937 citations