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

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)

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.

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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
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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
= ABθ
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

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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...

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Frequently Asked Questions (1)
Q1. What have the authors contributed in "Passification-based decentralized adaptive synchronization of dynamical networks with time-varying delays" ?

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