Table Of ContentNetwork Synchronization with Nonlinear Dynamics and
Switching Interactions∗
Tao Yang†, Ziyang Meng‡, Guodong Shi§, Yiguang Hong¶and Karl Henrik Johanssonk
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2
Abstract
g
u
A This paper considers the synchronization problem for networks of coupled nonlinear dy-
4 namical systems under switching communication topologies. Two types of nonlinear agent
2
dynamics are considered. The first one is non-expansive dynamics (stable dynamics with
]
Y a convex Lyapunov function ϕ(·)) and the second one is dynamics that satisfies a global
S
Lipschitz condition. For the non-expansive case, we show that various forms of joint con-
.
s
c nectivity for communication graphs are sufficient for networks to achieve global asymptotic
[
ϕ-synchronization. We also show that ϕ-synchronizationleads to state synchronizationpro-
3
v vided that certain additional conditions are satisfied. For the globally Lipschitz case, unlike
1
the non-expansive case, joint connectivity alone is not sufficient for achieving synchroniza-
4
5
tion. A sufficient condition for reaching global exponential synchronization is established in
6
. termsoftherelationshipbetweentheglobalLipschitzconstantandthe networkparameters.
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0
We also extend the results to leader-follower networks.
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1
:
v Keywords: Multi-agent systems, nonlinear agents, switching interactions, synchronization.
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∗ThisworkhasbeensupportedinpartbytheKnutandAliceWallenbergFoundationandtheSwedishResearch
r
a
Council.
†T. Yang is with the Pacific Northwest National Laboratory, 902 Battelle Boulevard, Richland, WA 99352
USA(e-mail: [email protected]).
‡Z. Meng is with the Institute for Information-Oriented Control, Technische Universit¨at Mu¨nchen, D-80290
Munich, Germany (e-mail: [email protected]).
§G.ShiiswiththeCollegeofEngineeringandComputerScience,TheAustralianNationalUniversity,Canberra
ACT 0200, Australia (e-mail: [email protected]).
¶Y.HongiswiththeKeyLaboratory ofSystemsandControl,InstituteofSystemsScience,ChineseAcademy
of Science, Beijing 100190, China (e-mail: [email protected]).
kK. H. Johansson is with the ACCESS Linnaeus Centre, School of Electrical Engineering, Royal Institute of
Technology, Stockholm 10044, Sweden (e-mail: [email protected]).
1
1 Introduction
We consider the synchronization problem for a network of coupled nonlinear agents with agent
set V = {1,2,...,N}. Their interactions (communications in the network) are described by
a time-varying directed graph G = (V,E ), with σ : [0,∞) → P as a piecewise constant
σ(t) σ(t)
signal, where P is a finite set of all possible graphs over V. The state of agent i ∈V at time t is
denoted as x (t) ∈ Rn and evolves according to
i
x˙ =f(t,x )+ a (t)(x −x ), (1)
i i ij j i
X
j∈Ni(σ(t))
where f(t,x ) :[0,∞)×Rn → Rn is piecewise continuous in t and continuous in x representing
i i
the uncoupled inherent agent dynamics, N (σ(t)) is the set of agent i’s neighbors at time t, and
i
a (t)> 0 is a piecewise continuous function marking the weight of edge (j,i) at time t.
ij
Systems of the form (1) have attracted considerable attention. Most works focus on the case
where the communication graph G is fixed, e.g., [1–7]. It is shown that for the case where
σ(t)
f(t,x )satisfiesaLipschitzcondition,synchronizationisachievedforaconnectedgraphprovided
i
that the coupling strength is sufficiently large. However, for the case where the communication
graphistime-varying, thesynchronization problembecomesmuchmorechallenging andexisting
literature mainly focuses on a few special cases when f(t,x ) is linear, e.g., the single-integrator
i
case [8–10], the double-integrator case [11], and the neutrally stable case [12,13]. Other studies
assume some particular structures for the communication graph [14–16]. In particular, in [14],
the authors focus on the case where the adjacency matrices associated with all communication
graphs are simultaneously triangularizable. The authors of [15] consider switching communica-
tion graphs that are weakly connected and balanced at all times. A more general case where
the switching communication graph frequently has a directed spanning tree has been considered
in [16]. These special structures on the switching communication graph are rather restrictive
compared to joint connectivity where the communication can be lost at any time.
This paper aims to investigate whether joint connectivity for switching communication
graphs can render synchronization for the nonlinear dynamics (1). We distinguish two classes
depending on whether the nonlinear agent dynamics f(t,x ) is expansive or not. For the non-
i
expansive case, we focus on the case where the nonlinear agent dynamics is stable with a convex
Lyapunov function ϕ(·). We show that various forms of joint connectivity for communication
graphs are sufficient for networks to achieve global asymptotic ϕ-synchronization, that is, the
2
functionϕoftheagent stateconverges toacommonvalue. Wealsoshowthatϕ-synchronization
implies state synchronization provided that additional conditions are satisfied. For the expan-
sive case, we focus on when the nonlinear agent dynamics is globally Lipschitz, and establish
a sufficient condition for networks to achieve global exponential synchronization in terms of a
relationship between the Lipschitz constant and the network parameters.
Theremainderofthispaperisorganized asfollows. Section 2presentstheproblemdefinition
and main results. Section 3 provides technical proofs. In Section 4, we extend the results to
leader-follower networks. Finally, Section 5 concludes the paper.
2 Problem Definition and Main Results
2.1 Problem Set-up
Throughout the paper we make a standard dwell time assumption [17] on the switching signal
σ(t): there is a lower bound τ > 0 between two consecutive switching time instants of σ(t).
D
We also assume that there are constants 0 < a ≤ a∗ such that a ≤ a (t) ≤ a∗ for all t ≥ 0.
∗ ∗ ij
We denote x = [xT,xT,...,xT]T ∈ RnN and assume that the initial time is t = t ≥ 0, and the
1 2 N 0
initial state x(t ) = (xT(t ),...,xT(t ))T ∈ RnN. A digraphis strongly connected if it contains a
0 1 0 N 0
directed path from every node to every other node. The joint graph of G in the time interval
σ(t)
[t ,t ) with t < t ≤ ∞ is denoted as G([t ,t )) = ∪ G(t) = (V,∪ E ). For the
1 2 1 2 1 2 t∈[t1,t2) t∈[t1,t2) σ(t)
communication graph, we introduce the following definition.
Definition 1 (i). G is uniformly jointly strongly connected if there exists a constant T > 0
σ(t)
such that G([t,t+T)) is strongly connected for any t ≥0.
(ii). Assume that G is undirected for all t ≥ 0. G is infinitely jointly connected if
σ(t) σ(t)
G([t,∞)) is connected for any t ≥ 0.
In this paper, we are interested in the following synchronization problems.
Definition 2 The multi-agent system (1) achieves global asymptotic ϕ-synchronization, where
ϕ :Rn → R is a continuously differentiable function, if for any initial state x(t ), there exists a
0
constant d (x(t )), such that lim ϕ(x (t)) = d (x(t )) for any i ∈ V and any t ≥ 0.
⋆ 0 t→∞ i ⋆ 0 0
3
Definition 3 (i) The multi-agent system (1) achieves global asymptotic synchronization if
lim (x (t)−x (t)) = 0 for any i,j ∈ V, any t ≥ 0 and any x(t ) ∈ RnN.
t→∞ i j 0 0
(ii) Multi-agent system (1) achieves global exponential synchronization if there exist γ ≥ 1
and λ > 0 such that
max kx (t)−x (t)k2 ≤ γe−λ(t−t0) max kx (t )−x (t )k2, t ≥ t , (2)
i j i 0 j 0 0
{i,j}∈V×V {i,j}∈V×V
for any t ≥ 0 and any x(t ) ∈ RnN.
0 0
Remark 1 ϕ-synchronization is a type of output synchronization where the output of agent
i ∈V is chosen to be ϕ(x ). It is related to but different from χ-synchronization [18,19] since ϕ
i
is a function of an individual agent state while χ is a function of all agent states.
2.2 Non-expansive Inherent Dynamics
In this section, we focus on when the nonlinear inherent agent dynamics is non-expansive as
indicated by the following assumption.
Assumption 1 ϕ : Rn → R is a continuously differentiable positive definite convex function
satisfying
(i). lim ϕ(η) = ∞;
kηk→∞
(ii). h∇ϕ(η),f(t,η)i ≤ 0 for any η ∈Rn and any t ≥0.
The following lemma shows how Assumption 1 enforces non-expansive dynamics.
Lemma 1 Let Assumption 1 hold. Along the multi-agent dynamics (1), max ϕ(x (t)) is
i∈V i
non-increasing for all t ≥ 0.
We now state main results for the non-expansive case.
Theorem 1 Let Assumption 1 hold. The multi-agent system (1) achieves global asymptotic
ϕ-synchronization if G is uniformly jointly strongly connected.
σ(t)
Theorem 2 Let Assumption 1 hold. Assume that G is undirected for all t ≥ t . The multi-
σ(t) 0
agent system (1) achieves global asymptotic ϕ-synchronization if G is infinitely jointly con-
σ(t)
nected.
4
Remark 2 For the linear time-varying case f(t,x) = A(t)x, if there exists amatrix P = PT > 0
such that
PA(t)+AT(t)P ≤ 0, ∀t ≥ 0, (3)
then ϕ(x) = xTPx for x ∈ Rn satisfies Assumption 1. For the linear time-invariant case
f(t,x)= Ax, the condition (3) is equivalent to that the matrix A is neutrally stable [13].
2.3 ϕ-synchronization vs. State Synchronization
The following result establishes conditions under which ϕ-synchronization may imply state syn-
chronization.
Theorem 3 Let G ≡ G with G being a fixed, strongly connected digraph under which the
σ(t)
multi-agent system (1) achieves global asymptotic ϕ-synchronization for some positive definite
function ϕ: Rn → R. Let Assumption 1 hold. Moreover, assume that
(i). f(t,η) is bounded for any t ≥ 0 and any η ∈ Rn.
(ii). c kηk2 ≤ ϕ(η) ≤ c kηk2 for some 0 < c ≤ c ; and
1 2 1 2
(iii). ϕ(·) is strongly convex.
Then the multi-agent system (1) achieves global asymptotic synchronization.
2.4 Lipschitz Inherent Dynamics
We consider also the case when the nonlinear inherent agent dynamics is possibly expansive.
We focus on when the dynamics satisfies the following global Lipschitz condition.
Assumption 2 There exists a constant L > 0 such that
kf(t,η)−f(t,ζ)k ≤ Lkη−ζk, ∀η,ζ ∈ Rn, ∀t ≥ 0. (4)
Our main result for this case is given below.
Theorem 4 Let Assumption 2 hold. Assume that G is uniformly jointly strongly connected.
σ(t)
Global exponential synchronization is achieved for the multi-agent system (1) if L < ρ /2, where
∗
ρ is a constant depending on the network parameters.
∗
5
Remark 3 Assumption 2 and its variants have been considered in the literature for fixed com-
munication graphs, e.g., [1–7]. Compared with the existing literature, we here study a more
challenging case, where the communication graphs are time-varying. Unlike the fixed case where
the global Lipschitz condition is sufficient to guarantee synchronization, Theorem 4 established a
sufficientsynchronization condition related tothe Lipschitz constant andthe network parameters.
3 Proofs of the Main Results
In this section, we provide proofs of the main results.
3.1 Proof of Lemma 1
Recall that the upper Dini derivative of a function h(t) : (a,b) → R at t is defined as D+h(t) =
h(t+s)−h(t)
limsup . The following lemma from [10,20] is useful for the proof.
s→0+ s
Lemma 2 Let V (t,x) : R×Rn → R (i = 1,...,N) be continuously differentiable and V(t,x) =
i
max V (t,x). If I(t) = {i ∈ {1,2,...,N} : V(t,x(t)) = V (t,x(t))} is the set of indices
i=1,...,N i i
where the maximum is reached at t, then D+V(t,x(t)) = max V˙ (t,x(t)).
i∈I(t) i
Denote I(t) = {i ∈ V : max ϕ(x (t)) = ϕ(x (t))}. We first note that the convexity property
i∈V i i
of ϕ(·) implies that [21, pp.69]
h∇ϕ(η),ζ −ηi ≤ ϕ(ζ)−ϕ(η), ∀η,ζ ∈Rn. (5)
It then follows from Lemma 2, Assumption 1(ii) and (5) that
D+maxϕ(x (t)) = max ∇ϕ(x ),f(t,x )+ a (t)(x −x )
i i i ij j i
i∈V i∈I(t)(cid:10) j∈NXi(σ(t)) (cid:11)
≤ max a (t)(ϕ(x )−ϕ(x )) ≤ 0,
ij j i
i∈I(t) X
j∈Ni(σ(t))
where the last inequality follows from ϕ(x ) ≤ ϕ(x ). This proves the lemma.
j i
3.2 Proof of Theorem 1
It follows from Lemma 1 that for any initial state x(t ) ∈ RnN, there exists a constant d =
0 ∗
d (x(t )) ≥ 0, such that lim max ϕ(x ) = d . We shall show that d is the required
⋆ 0 t→∞ i∈V i ⋆ ⋆
constant in Definition 2 of ϕ-synchronization.
6
We first note that by Lemma 1 that for all i ∈ V, there exist constants 0 ≤ α ≤ β ≤ d ,
i i ⋆
such that
liminfϕ(x (t)) = α , limsupϕ(x (t)) = β .
i i i i
t→∞ t→∞
Also note that it follows from lim max ϕ(x (t)) = d that for any ε > 0, there exists
t→∞ i∈V i ⋆
T (ε) > 0 such that
1
ϕ(x (t)) ∈ [0,d +ε], ∀i∈ V, ∀t ≥ T (ε). (6)
i ⋆ 1
The proof of Theorem 1 is based on a contradiction argument and relies on the following
lemma.
Lemma 3 Let Assumption 1 hold. Assume that G is uniformly jointly strongly connected.
σ(t)
If there exists an agent k ∈ V such that 0 ≤ α < d , then there exists 0 < ρ¯< 1 and t¯such
0 k0 ⋆
that for all i ∈ V, ϕ(x (t¯+(N −1)T )) ≤ ρ¯M +(1−ρ¯)(d +ε), where
i 0 0 ⋆
T , T +2τ , (7)
0 D
with T given in Definition 1(i) and τ is the dwell time.
D
Proof: Let us first define M , αk0+βk0 < d . Then there exists an infinite time sequence
0 2 ⋆
t < t˜ < ... < t˜ < ... with lim t˜ = ∞ such that ϕ(x(t˜ )) = M for all k = 1,2,.... We
0 1 k k→∞ k k 0
then pick up one t˜ , k = 1,2,... such that it is greater than or equal to T (ε) and denote it as
k 1
t˜ .
k0
We now prove the lemma by estimating an upper bound of the scalar function ϕ(x ) agent
i
by agent. The proof is based on a generalization of the method proposed in the proof of [22,
Lemma 4.3] but with substantial differences on the agent dynamics and Lyapunov function.
Moreover, the convexity of ϕ(·) plays an important role.
Step 1. Focus on agent k . By using Assumption 1(ii), (5), and (6), we obtain that for all
0
t ≥ t˜ ,
k0
d
ϕ(x (t)) = ∇ϕ(x ),f(t,x )+ a (t)(x −x )
dt k0 k0 k0 k0j j k0
(cid:10) j∈NXk0(σ(t)) (cid:11)
≤ a (t)(ϕ(x )−ϕ(x ))
k0j j k0
X
j∈Nk0(σ(t))
≤ a∗(N −1)(d +ε−ϕ(x )). (8)
⋆ k0
7
It then follows that for all t ≥ t˜ ,
k0
ϕ(xk0(t)) ≤ e−λ1(t−t˜k0)ϕ(xk0(t˜k0))+ 1−e−λ1(t−t˜k0) (d⋆ +ε), (9)
(cid:16) (cid:17)
where λ = a∗(N −1).
1
Step 2. Consider agent k 6= k such that (k ,k ) ∈ E for t ∈ [t˜ ,t˜ +T ). The existence of
1 0 0 1 σ(t) k0 k0 0
such an agent can be shown as follows. Since G is uniformly jointly strongly connected, it is
σ(t)
not hard to see that there exists an agent k 6= k ∈ V and t ≥ t˜ such that (k ,k ) ∈ E for
1 0 1 k0 0 1 σ(t)
t ∈ [t ,t +τ ) ⊆ [t˜ ,t˜ +T ).
1 1 D k0 k0 0
From (9), we obtain for all t ∈ [t˜ ,t˜ +(N −1)T ],
k0 k0 0
ϕ(x (t)) ≤ κ , ρM +(1−ρ)(d +ε), (10)
k0 0 0 ⋆
where ρ= e−λ1(N−1)T0 = e−a∗(N−1)2T0.
We next estimate ϕ(x (t)) by considering two different cases.
k1
Case I: ϕ(x (t)) > ϕ(x (t)) for all t ∈[t ,t +τ ).
k1 k0 1 1 D
By using Assumption 1(ii), (5), (6), and (10), we obtain for all t ∈ [t ,t +τ ),
1 1 D
d
ϕ(x (t)) ≤ a (t) ϕ(x )−ϕ(x ) +a (t)(ϕ(x )−ϕ(x ))
dt k1 k1j kj k1 k1k0 k0 k1
j∈Nk1(Xσ(t))\{k0} (cid:0) (cid:1)
≤ a∗(N −2)(d +ε−ϕ(x ))+a (κ −ϕ(x )).
⋆ k1 ∗ 0 k1
From the preceding relation, we obtain for t ∈ [t ,t +τ ),
1 1 D
ϕ(x (t)) ≤ e−λ2(t−t1)ϕ(x (t ))+ [a∗(N −2)(d⋆ +ε)+a∗κ0](1−e−λ2(t−t1)),
k1 k1 1 λ
2
where λ = a∗(N −2)+a . Therefore, we have
2 ∗
ϕ(x (t +τ )) ≤ κ , µ(d +ε)+(1−µ)κ , (11)
k1 1 D 1 ⋆ 0
where
λ −a (1−e−λ2τD)
2 ∗
µ = . (12)
λ
2
By applying the same analysis as we obtained (9) to the agent k , we obtain for all t ≥ t +τ ,
1 1 D
ϕ(x (t)) ≤ e−λ1(t−(t1+τD))κ + 1−e−λ1(t−(t1+τD)) (d +ε). (13)
k1 1 ⋆
h i
8
By combining the inequalities (10), (11) and (13), we obtain for all t ∈ [t +τ ,t˜ +(N−1)T ],
1 D k0 0
ϕ(x (t)) ≤ ϕ M +(1−ϕ )(d +ε), (14)
k1 1 0 1 ⋆
where ϕ = (1−µ)ρ2.
1
Case II: There exists a time instant t¯ ∈ [t ,t +τ ) such that
1 1 1 D
ϕ(x (t¯ )) ≤ ϕ(x (t¯ )) ≤ κ . (15)
k1 1 k0 1 0
By applying the similar analysis as we obtained (8) to the agent k , we obtain for all t ≥ t˜ ,
1 k0
d
ϕ(x (t)) ≤ a∗(N −1)(d +ε−ϕ(x (t))).
dt k1 ⋆ k1
This leads to
ϕ(x (t)) ≤ e−λ1(t−t¯1)ϕ(x (t¯ ))+(1−e−λ1(t−t¯1))(d +ε).
k1 k1 1 ⋆
By combining the preceding relation, (10), and (15), and using 0 < ϕ = (1−µ)ρ2 < ρ2 which
1
follows from 0 < µ < 1, we obtain for all t ∈ [t +τ ,t˜ +(N −1)T ],
1 D k0 0
ϕ(x (t)) ≤ ρ2M +(1−ρ2)(d +ε) <ϕ M +(1−ϕ )(d +ε).
k1 0 ⋆ 1 0 1 ⋆
From the preceding relation and (14), it follows that for both cases, we have for all t ∈ [t +
1
τ ,t˜ +(N −1)T ],
D k0 0
ϕ(x (t)) ≤ ϕ M +(1−ϕ )(d +ε).
k1 1 0 1 ⋆
From the preceding relation, (10) and 0 < ϕ < ρ < 1, it follows that for all t ∈ [t +τ ,t˜ +
1 1 D k0
(N −1)T ],
0
ϕ(x (t)) ≤ϕ M +(1−ϕ )(d +ε), j ∈ {k ,k }. (16)
j 1 0 1 ⋆ 0 1
Step 3. Consider agent k ∈/ {k ,k } such that there exists an edge from the set {k ,k } to the
2 0 1 0 1
agent k in E for t ∈ [t ,t +τ ) ⊆ [t˜ +T ,t˜ +2T ). The existence of such an agent k
2 σ(t) 2 2 D k0 0 k0 0 2
and t follows similarly from the argument in Step 2.
2
Similarly, we can bound ϕ(x (t)) by considering two different cases and obtain that for all
k2
t ∈ [t +τ ,t˜ +(N −1)T ],
2 D k0 0
ϕ(x (t)) ≤ ϕ M +(1−ϕ )(d +ε), (17)
k2 2 0 2 ⋆
where ϕ = ((1−µ)ρ2)2.
2
9
By combining (16) and (17), and using 0 < ϕ < ϕ < 1, we obtain that for all t ∈
2 1
[t +τ ,t˜ +(N −1)T ],
2 D k0 0
ϕ(x (t)) ≤ ϕ M +(1−ϕ )(d +ε), j ∈ {k ,k ,k }.
j 2 0 2 ⋆ 0 1 2
Step 4. By repeating the above process on time intervals [t˜ +2T ,t˜ +3T ), ..., [t˜ +(N −
k0 0 k0 0 k0
2)T ,t˜ +(N −1)T ), we eventually obtain that for all i ∈ V,
0 k0 0
ϕ(x (t˜ +(N −1)T )) ≤ ϕ M +(1−ϕ )(d +ε).
i k0 0 N−1 0 N−1 ⋆
where ϕ = ((1−µ)ρ2)N−1. The result of the lemma then follows by choosing ρ¯=ϕ and
N−1 N−1
t¯= t˜ .
k0
We are now ready to prove Theorem 1 by contradiction. Suppose that there exists an agent
k ∈ V such that 0 ≤ α < d . It then follows from Lemma 3 that ϕ(x (t¯+(N−1)T )) < d for
0 k0 ⋆ i 0 ⋆
all i∈ V, provided that ε< ρ¯(d⋆−M0). This contradicts the fact that lim max ϕ(x ) = d .
1−ρ¯ t→∞ i∈V i ⋆
Thus,there doesnot existan agent k ∈ V such that0 ≤ α < d . Hence, lim ϕ(x (t)) = d
0 k0 ⋆ t→∞ i ⋆
for all i ∈ V.
3.3 Proof of Theorem 2
The proof relies on the following lemma.
Lemma 4 Let Assumption 1 hold. Assume that G is infinitely jointly connected. If there
σ(t)
exists an agent k ∈ V such that 0 ≤ α < d , then there exist 0 < ρ˜< 1 and t˜such that
0 k0 ⋆
ϕ(x (t˜+τ )) ≤ ρ˜M +(1−ρ˜)(d +ε), ∀i∈ V.
i D 0 ⋆
Proof: The proof of Lemma 4 is similar to that of Lemma 3 and based on estimating an upper
bound for the scalar quantity ϕ(x ) agent by agent. However, since G is infinitely jointly
i σ(t)
connected, the method that we get the order of the agents based on the intervals induced by
the uniform bound T cannot be used here. We can however apply the strategy in [22] for the
analysis as shown below:
Step 1. In this step, we focus on agent k . Since G is infinitely jointly connected, we can
0 σ(t)
define
tˆ , inf {∃i∈ V |(k ,i) ∈E },
1 0 σ(t)
t∈[t˜k0,∞)
10
