1

Strategic Topology Switching for Security–Part I:
Consensus & Switching Times

arXiv:1711.11183v2 [] 6 Apr 2018

Yanbing Mao, Emrah Akyol, and Ziang Zhang

Abstract—This two-part paper considers the strategic topology
switching for the second-order multi-agent systems under attack.
In Part I, we propose a strategy on switching times that enables
the strategic topology-switching algorithm proposed in Part II [1]
to reach the second-order consensus in the absence of attacks.
The control protocol introduced is governed only by the relative
positions of agents. Based on the stability of switched linear
systems without stable subsystems, and the period of the multiagent systems under fixed topology, the strategy on dwell time
of topology-switching signal is derived. Employing this strategy
and a finite-time consensus algorithm, a decentralized topologyswitching algorithm is proposed. The primary advantages of the
algorithm in achieving consensus are: (i) the control protocol
relies only on relative position measurements, which means that
no velocity measurements are needed; (ii) it has no constraint on
the magnitude of coupling strength. Simulations are provided
to verify the effectiveness of strategic topology switching in
achieving the second-order consensus.
Index Terms—Multi-agent system, second-order consensus,
strategic topology switching, dwell time.

series [1]) to reach the second-order consensus in the absence
of attacks. In the following, we present the definition of
second-order consensus in this context.
Definition 1: [14] The second-order consensus in the multiagent system (1) is achieved if and only if the following holds
for any initial condition:

lim |vi (t) − vj (t)| = 0, ∀i, j = 1, . . . , n.

t→∞

T

HE consensus of multi-agent systems with the firstorder dynamics is a well-studied theoretical problem
(see e.g., [2]–[5]) with many practical applications including
decentralized computation [6], distributed optimization [7],
[8], power sharing for droop-controlled inverters in islanded
microgrids [9], clock synchronization for sensor network [10],
and more. However, current and emerging systems, such
as connected vehicles [2], [5], spacecraft [11], robot [12]
and electrical power networks [13], rely on the second-order
dynamics. This observation, together with the fact that the
consensus algorithms designed for the first-order multi-agent
systems cannot be directly applied to those with the secondorder dynamics, is the main motivation of this work.
A second-order multi-agent system consists of a population
of n agents whose dynamics are governed by the following
equations:
ẋi (t) = vi (t) ,
v̇i (t) = ui (t),

i = 1, . . . , n

(1a)
(1b)

where xi (t) ∈ R is the position, vi (t) ∈ R is the velocity, and
ui (t) ∈ R is the control protocol of agent i.
This first part of the two-part paper proposes a strategy on
switching times that enables the strategic topology-switching
algorithm (explained in detail in the second part of this
Y. Mao, E. Akyol and Z. Zhang are with the Department of Electrical
and Computer Engineering, Binghamton University–SUNY, Binghamton, NY,
13902 USA, (e-mail: {ymao3,eakyol,zhangzia}@binghamton.edu).

(2b)

A. Related Work
Among the few studies [5], [14], [15] on this problem, two
commonly studied control protocols are
ui (t) = α

n
∑
j=1

I. I NTRODUCTION

(2a)

lim |xi (t) − xj (t)| = 0,

t→∞

and
ui (t) = α

n
∑

aij (xj (t) − xi (t)) − βvi (t) ,

aij (xj (t)−xi (t))+β

j=1

n
∑

(3)

aij (vj (t)−vi (t)), (4)

j=1

where aij is the entry of (weighted)adjacency matrix describing the structure of the undirected/directed control network,
and α and β are the coupling strengths. In the past several
years, based on the two consensus protocols (3) and (4), some
more realistic problems are addressed. For example, Mei et
al. [16] propose an adaptive control gain to relax the conditions
on the coupling strengths and obtained a fully distributed consensus algorithm; Qin et al. [17] study the leaderless consensus
and leader-following consensus, and report some lower bounds
for coupling strengths; to deal with the problem of limited
interaction ranges, Song and You [18] propose the rangebased varying weighs along coupling matrix. The conditions
of the well-studied second-order consensus are summarized in
Table I. The control protocols considered therein show that
the control protocols need the necessary individual/relative
velocity measurements. Yu et al. [19] propose a sampledposition based control protocol that does not need velocity
measurements. However, it has a constraint on the magnitudes
of coupling strengths.
B. Motivation of Part-I Paper
In practice, many network topologies are non-static, such
as the communication networks of mobile agents [3] and
brain networks [20]. Recent studies of dynamical networks

2

Table I
C ONDITIONS OF S ECOND -O RDER C ONSENSUS
Reference
[14], [15]
[16], [17]

Constriant on
magnitude of coupling strength
magnitude of coupling strength

[16]

magnitude of coupling strength

[18]

magnitude of coupling strength,
initial conditions
NONE

[5]

Protocol
Directed (3)
Directed
weighted (3)
Directed
weighted (4)
Directed
weighted (3)
Undirected (4)

have highlighted the important role played by the network
topology [21]–[26]. For example, Menck et al. [21] find in
numerical simulations of artificially generated power grids
that tree-like connection schemes, so-called dead ends and
dead trees, can strongly diminish the stability; Schultz et
al. [24] show that how the addition of links can change the
synchronization properties of the network; Mao et al. [25]
propose strategic topology switching to reveal zero-dynamics
attack in the first-order multi-agent systems. In power grids,
a certain group of generators can be cut off or connected to
prevent cascading failure [22]; and in the context of power
systems, topology switching is equivalent to actively tripping
or re-closing transmission lines, or adjusting active power
output or transmission line reactance to improve the linear
stability of the power system [27]. With the advances in
wireless communication networks, it has become more feasible
to set the topology of communication network as a control
variable [28], as carried out in this work.
In past studies, unintentional topology changes in a networked control system are commonly handled as disturbances [3], [29]–[32] to the system. To the best of our knowledge, active/strategic topology switching for networked control
systems has not been systemically studied, with exceptions
being [25], [26], [33]. However, the centralized topologyswitching algorithm studied in [33] and the strategic topologyswitching algorithm proposed in [25] are only applicable to the
first-order multi-agent systems. Moreover, for the second-order
multi-agent systems, the state-dependent topology-switching
algorithm studied in [26] rules out only the Chattering Zeno
behavior, i.e., it still has the Genuinely Zeno behavior. In this
paper, the time-dependent topology switching algorithm derived for the second-order multi-agent systems has neither the
Chattering Zeno behavior nor the Genuinely Zeno behavior.
However, an intriguing question regarding the dynamic
topology is whether the well-studied control protocols can be
simplified significantly, and the advantage of control protocols
under fixed topology can still be maintained. This Part-I paper
answers affirmatively to that.
Security concerns regarding networked cyber-physical systems pose an existential threat to their wide-deployment, see
e.g., Stuxnet malware attack and Maroochy Shire Council
Sewage control incident [34]. The “networked” aspect exacerbates the difficulty of detecting and preventing aforementioned
attacks since centralized measurement (sensing) and control
are not feasible for such large-scale systems [35]. One of the
fundamental problems of security in multi-agent systems is

detection of stealthy attacks. Recent experiments of stealthy
false-data injection attacks on networked control system [36]
showed changes in system dynamics could be used to reveal
stealthy attacks. To make changes in the system dynamics
in order to reveal zero-dynamics attack, Teixeira et al. [36]
considered the method of modifying input or output matrices.
Obviously, topology switching is another way of making
changes in the dynamics of a multi-agent system. Before using
the topology switching to reveal the zero-dynamics attack,
the problem of using the simplified control protocol which
needs only the relative position measurements, whether the
changes in system dynamics can destroy system stability in
the absence of attacks must be investigated. The strategy on
switching times proposed in this Part-I paper addresses this
problem completely.
Another interesting question pertains to the detectability
of zero-dynamics attacks by strategic topology switching is
studied explicitly in Part-II of this two-part paper [1].
C. Contributions of Part-I Paper and Preview of Part-II Results
The two-part paper comprises a study of a strategic
topology-switching algorithm for the second-order multi-agent
system under attack. Part I provides a basis for this strategic
algorithm: when the topology should strategically switch such
that the agents can have the ability of reaching consensus in the
absence of attacks. The contribution of this paper is twofold,
which can be summarized as follows:
•

•

We propose a control protocol using only measurements
of relative positions for the second-order multi-agent
system. Based on the stability of switched linear systems
without stable subsystems, and the period of the multiagent systems under fixed topology, in Section IV we
obtain a strategy on the dwell time of topology-switching
signal that enables consensus in the absence of attacks.
Based on the strategy on switching times, through employing a finite-time consensus network, we propose a
decentralized topology-switching algorithm in Section V.
In achieving the second-order consensus, the algorithm
has no constraint on the magnitude of coupling strength.

Using the obtained strategy on switching times for the simplified control protocol which needs only the measurements
of relative positions, Part II [1] will focus on the strategy on
switching topologies that addresses the problem of switching
to what topologies to reveal zero-dynamics attack. In revealing
zero-dynamics attack, the strategy allows the system operator
or the defender to have no knowledge of the misbehavingagent set or the attack-starting time.
D. Organization of Part-I Paper
Section II presents the notions and terminology. Problem
formulation is given in Section III. In Section IV, we derive the
strategy on switching times. Section V presents a decentralized
topology-switching algorithm. Numerical examples are given
in Section VI. Finally, Section VII concludes this Part-I paper.

3

II. N OTIONS AND T ERMINOLOGY

Lemma 5 (Stability of Switched Systems without Stable
Subsystems [42]): Consider switched linear system

A. Notations
We use P > 0 (≥, <, ≤ 0) to denote a positive definite (positive semi-definite,√negative definite, negative semi-definite)
matrix P . Let i = −1 be the imaginary unit. Rn and Rm×n
denote the set of n-dimensional real vectors and the set of
m × n-dimensional real matrices, respectively. N represents
the set of natural numbers and N0 = N ∪ {0}. Let I and 0
be the identity matrix and the zero matrix with compatible
dimension. 1n ∈ Rn and 0n ∈ Rn denote the vector with all
ones and the vector with all zeros, respectively. The superscript
‘⊤’ stands for matrix transpose.
The interaction among n agents is modeled by an undirected
graph G = (V, E), where V = {1, 2, . . . , n} is the set of
vertices that represent n agents and E ⊂ V × V is the set of
edges of the graph G. The adjacency matrix A = [aij ] ∈ Rn×n
of the undirected graph G is defined as aij = (i, j) ∈ E, where
aij = aji = 1 if agents i and j interact with each other, and
aij = aji = 0 otherwise. Assume that there are no self-loops,
i.e., for any i ∈ V, aii = 0. A path is a sequence of connected
edges in a graph. A graph is a connected graph if there is a
path between every pair of vertices.

where z (t) ∈ R , Aσ(t) ∈ R
and σ(t) ∈ S. Given
scalars α > 0, 0 < β < 1, 0 < τbmin ≤ τbmax , if there exists
a set of matrices Pr,q > 0, q = 0, 1, . . . , L, r ∈ S, such that
∀q = 0, 1, . . . , L − 1, ∀r, s ∈ S, we have
q
A⊤
r Pr,q+1 + Pr,q+1 Ar + Ψr − αPr,q+1 < 0,

Lemma 3 (Proposition 1.3.3 in [40]): Let G be a connected
graph with diameter d. Then G has at least d + 1 distinct
eigenvalues, at least d + 1 distinct Laplace eigenvalues, and at
least d + 1 distinct signless Laplace eigenvalues.
Lemma 4: [41] The determinant of the Vandermonde matrix


1
1
···
1
 a1
a2
···
an 


2
 a21
a
·
·
·
a2n 
2


H ,  a31
∈ Rn×n ,
a32
···
a3n 


 ..
..
..
.. 
 .
.
.
. 
n−1
n−1
n−1
a1
a2
· · · an
is

det (H) = (−1)

n2 −n
2

∏

i<j

(ai − aj ).

(6)

(8)
(9)

A⊤
r Pr,L

(10)
(11)

ln β + αb
τmax < 0,

(12)

+ Pr,L Ar − αPr,L < 0,
Ps,0 − βPr,L ≤ 0, s ̸= r,
−P

L(P

)

r,q+1
r,q
where Ψqr =
. Then, the system (7) is globally
τbmin
uniformly asymptotically stable under any switching signal
σ(t) satisfies (23).
Lemma 6 (Simplified Finite-Time Consensus Algorithm without External Disturbances [43]): Consider the multi-agent
system

n
∑
j=1

The following definition and auxiliary lemmas will be used
throughout this paper.
Definition 2 (Path Graph [37]): The path graph Pn is a tree
with two nodes of vertex degree 1, and the other n − 2 nodes
of vertex degree 2.
Lemma 1: [38] If the undirected graph G is connected, then
its Laplacian L ∈ Rn×n has a simple zero eigenvalue (with
eigenvector 1n ) and all its other eigenvalues are positive and
real.
Lemma 2: [39] The Laplacian of a path graph Pn has the
eigenvalues as
(
)
(k − 1) π
, k = 1, . . . , n.
(5)
λk = 2 − 2 cos
n

m×m

q
A⊤
r Pr,q + Pr,q Ar + Ψr − αPr,q < 0,

ṙi = α̃
B. Preliminaries

(7)

ż (t) = Aσ(t) z (t) ,
m

m̄

bij (rj −ri ) n̄ + β̃

n
∑
j=1

p̄

bij (rj −ri ) q̄ , i = 1,. . ., n (13)

where α̃ > 0 and β̃ > 0 are the coupling strengths, bij is
the element of the coupling matrix that describes topology
of an undirected connected communication network and its
corresponding Laplacian matrix is denoted as LA , the odd
numbers m̄ > 0, n̄ > 0, p̄ > 0 and q̄ > 0 that satisfy m̄ > n̄
and p̄ < q̄. Its global finite-time consensus can be achieved,
i.e.,
n

ri (t) −

1∑
ri (0) = 0, ∀t ≥ P, i = 1, . . . , n.
n i=1

Further, the setting time P is bounded by
( m̄−n̄
)
1
n 2n̄
n̄
1 q̄
P <
+
.
λ2 (LA )
α̃ m̄ − n̄ β̃ q̄ − p̄

(14)

(15)

III. P ROBLEM F ORMULATION
The second-order multi-agent system (1) under the control
protocol, which uses only relative position measurements, that
involves topology switching is described by
(16a)

ẋi (t) = vi (t)
v̇i (t) = ui (t)= γ

n
∑

σ(t)

aij

(xj (t)−xi (t)), i = 1, . . . , n (16b)

j=1

where
(i) γ > 0 is the coupling strength;
(ii) σ(t) : [t0 , ∞) → S , {1, 2, . . . , s}, s ∈ N, is
the switching signal of the interaction topology of the
communication network, i.e., σ(t) = pk ∈ S for
t ∈ [tk , tk+1 ) means the pth topology is activated over
the time interval [tk , tk+1 ), k ∈ N0 ;

4

(iii) apijk is the entry of the adjacency matrix that describes
the activated pth topology of undirected communication
graph over the time interval [tk , tk+1 ), k ∈ N0 .
σ(t)
σ(t)
For the undirected topology, aij = aji , it should be easy
n
∑
to verify from (16b) that
v̇i (t) = 0, ∀t ≥ 0, which implies

IV. S TRATEGY ON S WITCHING T IMES
The objective of this section is to propose a strategy on
switching times that enables the multi-agent system (20) to
achieve the second-order consensus without constraint on the
magnitude of coupling strength γ.

i=1

that the average position
n
n
1∑
1∑
x̄(t) =
xi (0) +
vi (0)t,
n i=1
n i=1
proceeds with the constant velocity
n
n
1∑
1∑
vi (t) =
vi (0).
v̄ =
n i=1
n i=1

(17)

(18)

If the second-order consensus is achieved, the individual
velocities will converge asymptotically to the average of initial
of velocities, i.e., lim |vi (t) − v̄| = 0, i = 1, . . . , n. Based
t→∞
on relations (17) and (18), we define the following fluctuation
terms:
x̃i (t) = xi (t) − x̄(t),
ṽi (t) = vi (t) − v̄.

(19a)
(19b)

The dynamics (16) can now be expressed equivalently as
x̃˙ i (t) = ṽi (t)
n
∑
σ(t)
ṽ˙ i (t) = γ
aij (x̃j (t) − x̃i (t)), i = 1, . . . , n.

(20a)
(20b)

j=1

It follows from (17), (18), and (19) that
1⊤
n x̃ (t) = 0, ∀t ≥ 0
1⊤
n ṽ (t)

= 0, ∀t ≥ 0.

(21)
(22)

In practice, high switching frequency can result in high
switching cost, which is undesirable. In the context of attack
detection by strategic topology switching, low switching frequency is as well undesirable since it can result in attacks
going undetected for a long time. Taking these remarks into
account, we impose the minimum and maximum dwell times
on the topology-switching signal σ(t).
Assumption 1: For the second-order multi-agent system (20), the minimum dwell time τmin and maximum dwell
time τmax of topology-switching signal satisfy
τmin ≤ tk+1 − tk ≤ τmax , ∀k ∈ N0 .

(23)

In Section IV, we first show that the multi-agent system (20)
under fixed topology, i.e, σ(t) = p ∈ S for t ∈ [0, ∞),
is oscillating and has a period.This implies that the multiagent system (16) under fixed topology cannot achieve the
second-order consensus even for large coupling strength. Then
using the stability of switched linear system–Lemma 5 and
the period of the system (20) under fixed topology, a strategy
on switching times is derived, which enables the multi-agent
system (20) to achieve the second-order consensus without
constraint on the magnitude of coupling strength γ. Based on
the strategy on switching times, through employing the finitetime consensus algorithm–Lemma 6, a decentralized topologyswitching algorithm is proposed Section V.

A. Periodically Oscillating under Fixed Topology
The following lemmas show some important properties of
the system (20) under fixed topology, which will be used to
derive feasible decentralized topology-switching algorithm.
Lemma 7: Consider the following system
∫ t
˙x̃ (t) = −γL
x̃ (τ )dτ + ṽ(0), t ≥ 0
(24)
0

where γ > 0, L ∈ R
is the Laplacian matrix of a
connected undirected graph; and x̃ (t) ∈ Rn and ṽ (t) =
x̃˙ (t) ∈ Rn satisfy (21) and (22), respectively. The solutions
of x̃i (t), i = 1, . . . , n, are obtained as
n×n

x̃i (t)
(25)
)
(
n
∑
√
√
ṽ (0)
sin( γλl t) , t ≥ 0
=
qli ql⊤ x̃ (0) cos( γλl t)+ √
γλl
l=2

where λl (λ1 = 0) are the nonzero eigenvalues that are asso⊤
ciated with the orthogonal vectors ql = [ql1 , . . . , qln ] ∈ Rn
of L, l = 2, . . . , n.
Proof: See Appendix A.
Lemma 7 implies that the system (24), with x̃ (t) and
ṽ (t) satisfying (21) and (22), can be viewed as one class
of coupled oscillators. However, given some initial conditions
and topologies, the inadmissible energy functions, i.e., nonzero
constant positive functions, of the system (24) are common.
⊤
⊤
Take the positive function F (t) = γ x̃ (t)Lx̃(t)
+ ṽ (t)ṽ(t)
2
2
as an example. Differentiating it along the solutions of (24)
yields Ḟ (t) = γṽ ⊤ (t) Lx̃ (t)+ṽ ⊤ (t) ṽ˙ (t) = γṽ ⊤ (t) Lx̃ (t)−
γṽ ⊤ (t) Lx̃ (t) ≡ 0, which means that given any nonzero initial
condition, the function is a nonzero scalar over time, which is
inadmissible. Obviously, such inadmissible energy functions
are undesirable since they cannot capture the oscillating property of the system (24). Therefore, it is nontrivial to provide
a guide to construct an admissible energy function.
Lemma 8: Consider the function
F (t) = ϖ

x̃⊤ (t) x̃ (t) ṽ ⊤ (t) ṽ (t)
+
,
2
2

(26)

˙
with x̃(t)
= ṽ(t) ∈ Rn . Along the solutions (25) of the
system (24), if the Laplacian matrix L has distinct eigenvalues
and ϖ satisfies
0 < ϖ ̸= γλi (L) , ∀i = 2, . . . , n,

(27)

where λ1 (L) , . . . , λn (L) are the corresponding eigenvalues
of L. Then, for any nonzero initial condition, there never exists
a nonzero scalar φ such that
F (t) = φ ̸= 0, ∀t ≥ 0.
Proof: See Appendix B.

(28)

5

Remark 1: For the undirected communication network considered in this paper, there indeed exist many topologies
whose associated Laplacian matrices have distinct eigenvalues.
Noticing that 0 ≤ (k−1)π
< π, ∀k = 1, . . . , n, Lemma 2
n
implies that the Laplacian matrix of a path graph has distinct
eigenvalues. Moreover, Lemma 3 provide a guide to design
such topology with distinct Laplacian eigenvalues.

with
(31)

ξ < α,
− ln β
0 < τbmax <
,
α
) L
mT ( − 1
− β L −1
,
0 < τbmax +
2
α−ξ
ξ=
max
{1 − γλi (Lr ), −1 + γλi (Lr )} ,
r∈S,i=1,...,n

B. Dwell Time
The system under switching topology (20) can be modeled
as a switched linear system (7). Lemma 7 shows that each
subsystem of the switched system (20), i.e., the multi-agent
system (20) under each fixed topology, is not stable. The
equilibrium point of the multi-agent system (20) is (x̃∗ , ṽ ∗ ) =
(0n , 0n ). Hence, the problem of strategic topology switching
studied in the following sections would be designing the
stabilizing switching rule for the switched systems without
stable subsystems. Let us first recall a technical lemma regarding the stability of switched linear systems without stable
subsystems–Lemma 5, which will be used to derive a strategy
on switching times that enables agents to achieve consensus
without constraint on the magnitude of coupling strength.
However, without exploring and exploiting additional system
information, the conditions in Lemma 5 are not feasible for
the system (20).
Proposition 1: For the multi-agent system (20), the conditions in Lemma 5 are infeasible.
Proof: See Appendix C.
Let σ(t) = g ∈ S for t ∈ [tk , tk+1 ), k ∈ N0 .∫ Then, the
t
dynamics (20) can be rewritten as x̃˙ (t) = −γLg tk x̃ (τ )dτ
+ ṽ(tk ), t ∈ [tk , tk+1 ). Therefore, Lemma 7 implies that the
multi-agent agent system (20) under each fixed topology has
a period T such that
{
ṽi (t) = ṽi (t + T ) , i = 1, . . . , n
(29)
x̃i (t) = x̃i (t + T ) , ∀t ∈ [tk , tk+1 ) , ∀k ∈ N0 .
Remark 2: The solution (25) implies that to obtain a
small period T , the control designer needs the knowledge
of the eigenvalues and eigenvectors of the Laplacian matrix
that describes the network structure. However, for large-scale
networked systems, the network structure may not be available
to the designer. Fortunately, the well-developed distributed
eigenvalue and eigenvector estimation algorithms [44], [45]
well address this issue.
The period T can be used to make Lemma 5 applicable
to the multi-agent system (20) to derive a strategy on the
switching times.
Theorem 1: Consider the second-order multi-agent system (20). For the given period T satisfying (29), scalars
1 > β > 0, α > 0 and L ∈ N, if the dwell time τ , the
minimum dwell time τmin and the maximum dwell time τmax
satisfy
1

(β − L −1)

mT
L
< τmin ≤ τ ≤ τmax , τbmax +
, m ∈ N (30)
α−ξ
2

(32)
(33)
(34)

where γ is the coupling strength of the multi-agent system (20), and λi (Lr ) , . . . , λn (Lr ) are the corresponding
eigenvalues of the Laplacian matrix Lr . Then, the secondorder consensus is achieved.
Proof: We note that the multi-agent system (20) can
be described by switched system (7) where z (t) =
⊤
[x̃1 (t) , . . . , x̃n (t) , ṽ1 (t) , . . . , ṽn (t)] ∈ R2n and
[
]
0
I
Aσ(t) =
.
(35)
−γLσ(t) 0

For each activated topology of system (20), let us consider
the positive definite matrix
[
]
P̂r,q
0
Pr,q =
> 0,
(36)
0
P̂r,q
where

q

P̂r,q = β − L hI, q = 0, . . . , L, ∀r ∈ S

(37)

with h being a positive scalar. It follows from (37) that
1

P̂r,q = β L P̂r,q+1 , q = 0, . . . , L − 1, ∀r ∈ S,

(38)

Substituting the matrices Pr,q (36) and Aσ(t) (35)
conditions (8), (9) and (10), respectively, yields
[
]
Qr,q
(I − γLr )P̂r,q
∆
Rr,q =
< 0,
(I − γLr )P̂r,q
Qr,q
]
[
Q̆r,q
(I − γLr )P̂r,q+1
∆
< 0,
R̆r,q =
(I − γLr )P̂r,q+1
Q̆r,q
[
]
−αP̂r,L
(I − γLr )P̂r,L
∆
Sr,L =
< 0,
(I − γLr )P̂r,L
−αP̂r,L

into

P̂s,0 = β P̂r,L ,

∀r ̸= s ∈ S.

(39)

(40)
(41)
(42)

where

L
(P̂r,q+1 − P̂r,q ) − αP̂r,q ,
τmin
L
(P̂r,q+1 − P̂r,q ) − αP̂r,q+1 .
=
τmin

Qr,q =

(43)

Q̆r,q

(44)

Let W be the matrix that is orthogonal to the
∆
symmetric matrix Lr , for which W ⊤ Lr W = Λr =
diag {0, λ2 (Lr ) , . . . , λn (Lr )}. Considering the matrices Pr,q
and P̂r,q , in (36) and (37), the conditions (40), (41) and (42)
can be equivalently expressed in term of eigenvalue as
L

(P̂r,q+1 − P̂r,q ) − αP̂r,q ± (1 − γΛr ) P̂r,q < 0,
(45)
τmin
L
(P̂r,q+1 − P̂r,q )−αP̂r,q+1 ±(1 − γΛr ) P̂r,q+1 < 0, (46)
τmin
− αP̂r,L ± (1 − γΛr ) P̂r,L < 0.
(47)

6

In view of Lemma 5, to prove the second-order consensus,
it suffices to verify that the conditions (8)–(12) are satisfied,
as carried out in the following four steps.
Step One: It follows from (36), (38), and (39) that Ps,0 =
βPr,L , r ̸= s ∈ S. Thus, the condition (11) in Lemma 5 holds.
Step Two: To obtain Lemma 5, the considered discretized
Lyapunov function for mode r ∈ S in [42] is
{
(q)
z ⊤(t) Pr (ζ)z(t) , t ∈ Nk,q , q = 0, 1,. . . ,L−1
Vr (t) = ⊤
(48)
z (t) Pr,L z (t) , t ∈ [tk + τmin , tk+1 )
(q)

where Pr (ζ) = (1 − ζ) Pr,q + ζPr,q+1 with 0 < ζ < 1,
min
Nk,q = [tk + θq , tk + θq+1 ), θq+1 = (q+1)τ
, Pr,q > 0,
L
q = 0, 1, . . . , L − 1. In [42], the purposes of the condition (12)
is to ensure that
Vσ(tk ) (tk + τbmax ) ≤ β ∗ Vσ(t− ) (tk ) ,
k

(49)

with 1 > β > 0. Considering the right-hand side of (30), i.e.,
τmax , τbmax + mT
2 , from (48) one has
∗

(50)
T
= Vr (tk + τbmax + m )
2
T
T
⊤
= z (tk + τbmax + m )Pr,L z(tk + τbmax + m ), ∀r ∈ S.
2
2
Lemma 7 shows that system (20) under fixed topology has
period T , for which (29) holds. Then, noting that Pr,L > 0
and the solutions (25), we have
T
T
z ⊤ (tk + τbmax + m )Pr,L z(tk + τbmax + m )
(51)
2
2
= z ⊤ (tk + τbmax )Pr,L z(tk + τbmax ) , Vr (tk + τbmax ), ∀r ∈ S.
Vr (tk + τmax )

Combining (50) with (51) yields

Vr (tk + τmax ) = Vr (tk + τbmax ), ∀r ∈ S.

(52)

We note that the condition (32) is equivalent to αb
τmax +
ln β < 0, which corresponds to the condition (12) in Lemma 5.
Without loss of generality, let σ (tk ) = r ∈ S. Then, it follows
from (49) and (52) that
Vσ(tk ) (tk +τmax ) = Vσ(tk ) (tk +b
τmax ) ≤ β ∗ Vσ(t− ) (tk ) . (53)
k

From (53), we conclude that mT
2 , m ∈ N, which is imposed
on the right-hand side of (30), is to maintain the original
goal of the condition (12) through keeping (53) holding, while
ensuring τmax ≥ τmin . This also means that it is the period T
that makes Lemma 5 applicable to system (20).
Step Three: Since P̂r,L > 0, condition (31) implies
0 > −αP̂r,L + ξ P̂r,L , while condition (34) implies ξ ≥
± (1 − γΛr ). Thus, 0 > −αP̂r,L + ξ P̂r,L > −αP̂r,L ±
(1 − γΛr ) P̂r,L . From (47), the condition (10) in Lemma 5
is satisfied.
Step Four: It follows from (31) and the left-hand side
of (30) that
1
(α − ξ) τmin
+ 1 > β− L .
(54)
L
Considering h > 0, from (38) and (54) one has 1 +
1
P̂
(α−ξ)τmin
> P̂r,q+1 = β − L , which is equivalent to
L

Since 1 > β > 0, relation (38) implies that P̂r,q < P̂r,q+1 .
Condition (31) is equivalent to α − ξ > 0. Therefore, (55)
implies
L
(P̂r,q+1−P̂r,q )−(α−ξ) P̂r,q+1 < 0, q = 0,. . . ,L − 1. (56)
τmin
According to (34), 0 > −αP̂r,q+1 + ξ P̂r,q+1 > −αP̂r,q+1 ±
(1 − γΛr ) P̂r,q+1 and 0 > −αP̂r,q + ξ P̂r,q > −αP̂r,q ±
(1 − γΛr ) P̂r,q , which together with (55) and (56) imply (45)
and (46), respectively. Thus, the conditions (8) and (9) in
Lemma 5 hold and the proof is now completed.
Remark 3: Theorem 1 shows that the strategy on switching
times enables the system (20) that its control protocol uses
only the relative position measurements to achieve the secondorder consensus. Moreover, the strategy has no constraint on
the magnitude of coupling strength in achieving consensus,
which maintains the advantage of the control protocol (4)
studied in [5].
The conditions (30) and (34) in Theorem 1 imply that
the coupling strength affects the minimum dwell time and
maximum dwell time of topology switching signals, which
may further affect the convergence speed to consensus.
Remark 4: For the problem of security in networked systems, in the situation where the defender or the system operator has no knowledge of the attack-starting time, Theorem 1
indicates when the system dynamics should have changes
(caused by topology switching) to reveal zero-dynamics attack [36], so that the changes do not destroy the system
stability in the absence of attacks.
V. D ECENTRALIZED T OPOLOGY S WITCHING
The finite-time consensus algorithm–Lemma 6, can be used
to estimate the global coordinators precisely in finite time.
It is employed to derive a decentralized topology-switching
algorithm. Furthermore, from (15) we can see that through
adjusting the control gains α̃ and β̃, we obtain any desirable
setting time ∞ > P > 0.
Let us adjust parameters α̃ > 0 and β̃ > 0 in the
algorithm (13), such that
( m̄−n̄
)
1
n 2n̄
n̄
1 q̄
+
< τmin .
(57)
λ2 (LA )
α̃ m̄ − n̄ β̃ q̄ − p̄
Therefore, the setting time P in (15) satisfies P < τmin .
Condition (57) together with (14) and (15) show that if we
input individual data
Fi (tk ) =

1
ϖ 2
x̃i (tk ) + ṽi2 (tk ) .
2
2

and Ḟi (tk ) to the corresponding agent i in the algorithm (13)
at time tk , at time tk + τmin each agent in algorithm (13) will
output the exact global coordinators:
n

Fi (tk + τmin ) =

τmin

(P̂r,q+1 − P̂r,q )−(α−ξ) P̂r,q < 0, q = 0, . . . , L−1. (55)

1∑
∆ 1
Fi (tk ) = F (tk ),
n i=1
n

(59)

n

r,q

L

(58)

Ḟi (tk + τmin ) =

1∑
∆ 1
Ḟi (tk ) = Ḟ (tk ).
n i=1
n

(60)

7

Based on Lemma 8 and Theorem 1, through employing the
finite-time consensus algorithm (13), we propose the following
Algorithm 1, which is a decentralized topology-switching
algorithm.
Algorithm 1: Decentralized Topology-Switching Algorithm
Input: Topology set S that includes more than one
connected topology, at least one of which has
distinct Laplacian eigenvalues; individual
functions Fi (tk−1 ) (58) with ϖ satisfying (27);
initial time tk−1 = 0; initial topology Gσ(tk−1 ) ;
dwell time τ generated by Theorem 1;
loop-stopping criteria δ ≥ 0.
1 while F (tk−1 ) > δ do
2
Input individuals Fi (tk ) and Ḟi (tk ) to agent i in the
finite-time consensus algorithm (13) at time tk ;
3
Output F (tk ) by (59) and Ḟ (tk ) by (60) from the
finite-time consensus algorithm (13) to the agents
in (20) at time tk + τmin ;
4
Run until tk+1 ← tk + τ ;
5
if Ḟ (tk ) = 0 then
6
Switch the topology of network (20b) to σ(tk+1 )
that satisfies:
• σ(tk+1 ) ̸= σ(tk ),
• Lσ(tk+1 ) has distinct eigenvalues.
7
else
8
Switch the topology of network (20b) to σ(tk+1 )
that satisfies:
• σ(tk+1 ) ̸= σ(tk ),
9
end
10
Update the topology-switching time: tk−1 ← tk ;
11
Update the topology-switching time: tk ← tk+1 .
12 end
Theorem 2: Consider the system (20). If its topologyswitching signal is generated by Algorithm 1, then the following properties hold:
(i)
if the loop-stopping criteria δ = 0 (in Line 1 of
Algorithm 1), the agents achieve the second-order
consensus;
(ii)
if the loop-stopping criteria δ > 0, the agents achieve
the second-order consensus under admissible consensus error δ through finitely topology switching, i.e.,
F (tk̄ ) ≤ δ with 0 < k̄ < ∞ and F (t) given by (26).
Proof of Theorem 2: We first prove property (i). The
loop-stopping criteria δ = 0 means topology will stop switching when F (tk ) = 0. The definition of F (t) in (26) implies
that lim F (t) = 0 is equivalent to (2). This analysis means
t→∞
the topology switching will not stop until the second-order
consensus is achieved. Since the provided dwell time τ in
Input of Algorithm 1 is generated by Theorem 1, we conclude
that property (i) holds.
We now show that property (ii) holds. Assume the function
F (t) is a non-zero constant over time. If F (0) = φ > δ,
from (28) we have F (t) ≡ φ > δ for ∀t ≥ 0. Thus, Line 1

Table II
C ANDIDATE T OPOLOGY S ET S = {1∗ , 2∗ }
Index σ(t)
1∗
2∗

σ(t)

a12
1
1

σ(t)

a13
0
1

σ(t)

a14
0
0

σ(t)

a23
1
0

σ(t)

a24
0
1

σ(t)

a34
1
1

of Algorithm 1 implies that in this situation the topology will
never stop switching regardless of whether the consensus is
achieved. The objective of Line 5 and Line 6 in Algorithm 1
is to switch to a topology whose associated Laplacian matrix
has distinct eigenvalues when Ḟ (tk ) = 0. Hence, by Lemma 8,
F (t) cannot be a constant over time if Ḟ (tk ) = 0. Obviously,
if Ḟ (tk ) ̸= 0, F (t) cannot be constant over time. Therefore,
by Lemma 8 we conclude property (ii) under Algorithm 1.
Remark 5 (Nonzero Loop-Stopping δ): We note that, in real
world applications of consensus algorithms, such as decentralized computation and distributed optimization, objectives must
be achieved in finite time, rather than in infinite horizon. This
explains motivation behind the nonzero loop-stopping δ (property (ii) in Theorem 2), which achieves “nearly consensus” in
finite time.
VI. S IMULATION
The simulations on a second-order system with n = 4
agents will be presented to demonstrate the effectiveness of
the proposed topology switching algorithm. Initial position and
⊤
velocity conditions are set as x(0) = v(0) = [1, 2, 3, 4] .
To illustrate the effectiveness of the topology-switching
algorithm–Algorithm 1, in achieving the second-order consensus, the topology set S, which is described in Table II,
provided to Algorithm 1 includes only two topologies: the
path graph and the ring graph. The eigenvalues of Laplacian
matrices of the two topologies in Table II are computed as
[λ1 (L1 ) , λ2 (L1 ) , λ3 (L1 ) , λ4 (L1 )] = [0, 0.6, 2, 3.4] , (61)
[λ1 (L2 ) , λ2 (L2 ) , λ3 (L2 ) , λ4 (L2 )] = [0, 2, 2, 4] .

(62)

Lemma 7 implies that the states of the system (20) under
fixed topology keep oscillating, which means that even with
a very large coupling strength, the second-order cannot be
achieved in the situation of fixed topology. This is verified by
the trajectories in Figure 1, where the coupling strength is set
as large as γ = 100. Figure 1 also shows that the states of the
system (20) under each fixed connected topology are periodic.
A. Second-Order Consensus
Let us set the coupling strength as small as γ = 2. Using
the eigenvalues computed in (61) and (62), and the state solutions (25), the period is calculated as T ≈ 6. Considering (61)
and (62), we use (34) to obtain ξ = 7. Following (31), set
α = 7.5 and let β = 0.4 and L = 1. It follows from (32) and
the left-hand side of (30) that τ̂max < 0.1222 and τmin > 3.
Let m = 1. Following (30) and (33), we choose the dwell time
τ = T2 + 0.1 = 3.1. Then, under Algorithm 1, trajectories of
position differences and velocity differences of system (16)
are shown in Figures 2 (a) and (b), respectively. Figure 2

8

(a) Under Fixed Topology 1*, Coupling Strenth

350

(a)

= 100

4

Position Differences

300

V(t)

250
200
150
100
50

2

0

-2

0
0

5

10

Time
(b) Under Fixed Topology 2*, Coupling Strenth

15

-4
0

= 100

50

100

1000
800

6

600

4

400
200
0
0

5

10

15

Time

∑
Figure 1.
Trajectories of V (t) =
(xi (t) − xj (t))2 +
i<j
∑
(vi (t) − vj (t))2 : (i) the second-order consensus cannot be achieved
i<j

under fixed topology, (ii) the system (20) has period under fixed topology.

shows that in the absence of attacks, Algorithm 1 succeeds
in achieving the second-order consensus. Thus, property (i) in
Theorem 2 is verified.
B. Finitely Topology Switching
For the condition (27), we consider ϖ
>
max
{γλi (Lr )}. From γ = 2 and the eigenvalues

i=2,...,n,r=1,2

in (61) and (62), we choose ϖ = 8.1. Let the loop-stopping
criteria δ = 2. Then, under Algorithm 1, the trajectory of
F (t) given in (26) and the topology-switching signal are
shown in Figures 3 (a) and (b), respectively. Figure 3 well
illustrates property (ii) in Theorem 2.
VII. C ONCLUSION
This Part-I paper explains how to take the network topology
as a control variable for the second-order multi-agent system.
The obtained results highlight the merits of topology switching in achieving the second-order consensus: (i) the control
protocol does not need the velocity measurements, and (ii)
the topology-switching algorithm has no constraint on the
magnitude of the coupling strength. The strategy on switching
times provides a basis for the strategic topology switching that
is studied in Part-II paper [1]: when the topology of the multiagent should switch to make changes in the system dynamics

Velocity Differences

V(t)

150

200

250

150

200

250

Time
(b)

2
0
-2
-4
-6
-8
0

50

100

Time
Figure 2. In the absence of attacks, trajectories of positions differences and velocities differences: the second-order consensus are achieved by Algorithm 1.

in order to reveal zero-dynamics attack, such that the changes
do not destroy the agents’ ability of reaching consensus in the
absence of attacks.
A PPENDIX A
P ROOF OF L EMMA 7
By Lemma 1, we arrange the eigenvalues of L in the
increasing order as 0 = λ1 < λ2 ≤ . . . ≤ λn . Since L
is a real symmetric matrix, there exists an orthogonal matrix
⊤
Q , [q1 ; . . . ; qn ] ∈ Rn×n with qi = [qi1 , qi2 , . . . , qin ] ∈ Rn ,
i = 1, . . ., n, such that
Q⊤ = Q−1 ,
q11 = q12 = . . . = q1n ,

(63)
(64)
∆

Q⊤ LQ = diag {0, λ2 , . . . , λn } = Λ.

(65)

We denote X (s) = L {x̃ (t)}, where L(·) stands for the
Laplace transform operator. The Laplace transform of the
dynamics (24) can be obtained as
ṽ (0)
γ
,
sX (s) − x̃ (0) = − LX (s) +
s
s

(66)

9

Combing (70) with (71) and (72) yields

(a)

80

Positive Function F(t)
loop-stopping criteria = 2

Xi (s) =

n
∑
l=2

60

n

∑
s
1
qli ql⊤ x̃ (0) +
qli ql⊤ ṽ (0) .
2
s + γλl
s2 + γλl
l=2

Considering γ > 0 and λl > 0, l = 2, . . . , n, the solution (25)
can be obtained immediately from the inverse Laplace transform of Xi (s).

40

A PPENDIX B
P ROOF OF L EMMA 8

20

Let us assume that (28) holds, from which we obtain
0
0

10

20

30

40

50

Time
(b)

∂d
F (t) = 0, ∀d ∈ N, ∀t ≥ 0.
∂td
It follows from the dynamics (24) that

(73)

¨ (t) = −γLx̃ (t) , ∀t ≥ 0
x̃
ṽ¨ (t) = −γLṽ (t) , ∀t ≥ 0.

2*

(74)
(75)

Signal (t)

The rest of the proof is divided into two steps.
Step One: For the dynamics (24), relations (73), (74),
and (75) imply
1*

0

10

20

30

40

50

Time
Figure 3. Trajectory of F (t) and topology-switching signal σ(t): after fifteen
times topology switching, the energy function F (t) is under preset admissible
error δ = 2, i.e, F (t15 ) < δ = 2.

which is equivalent to
I
X (s) = 2
(sx̃ (0) + ṽ (0)) .
(67)
s I + γL
{
}
s
s
Let Λ̃(s) = diag 1s , s2 +γλ
,
.
.
.
,
and Λ̄(s) =
2 +γλ
s
n
{
}2
1
1
1
diag s2 , s2 +γλ2 , . . . , s2 +γλn . It follows from (63) and (65)
that
sI
sI
=
= QΛ̃(s)Q⊤ ,
s2 I + γL
Q (s2 I + γΛ) Q⊤
I
I
=
= QΛ̄(s)Q⊤ .
2
2
s I + γL
Q (s I + γΛ) Q⊤

(68)
(69)

Let Xi (s), i = 1, . . . , n, be the ith element of X (s).
From (67), (68), and (69),
)
(
Xi (s) = q1i q1⊤ 1s x̃ (0) + s12 ṽ (0)
n
(
)
∑
(70)
s
⊤
+
x̃ (0) + 1s ṽ (0) .
s2 +γλl qli ql
l=2

It follows from (21), (22) and (64) that
q1⊤ x̃ (0) = 0,
q1⊤ ṽ (0)

= 0.

(71)
(72)

∂d
F (t)
∂td
∂ d−1
= ∂t
d−1 Ḟ (t)
∂ d−1 ⊤
= ∂td−1 ṽ (t) (ϖI − γL) x̃ (t)
( ⊤
∂ d−2
(t) (ϖI − γL) ṽ (t)
= ∂t
d−2 2 ṽ
⊤

)
− x̃ (t) γL (ϖI − γL) x̃ (t)
(
∂ d−4
3
⊤
= ∂t
(t) γL (ϖI − γL) ṽ (t)
d−4 (−2 ) ṽ
)
2

− x̃⊤ (t) (γL) (ϖI − γL) x̃ (t)
(
)( ⊤
m−1
∂ d−2m
m−1 2m−1
2
ṽ (t) (γL)
(ϖI − γL)ṽ (t)
= ∂t
d−2m (−1)
)
m
⊤
− x̃ (t) (γL) (ϖI − γL) x̃ (t)
= 0, ∀m ∈ N, ∀d > 2m ∈ N, ∀t ≥ 0

which then implies
m

x̃⊤ (t) (γL) (ϖI − γL) x̃ (t)
⊤

= ṽ (t) (γL)

m−1

(76)

(ϖI − γL) ṽ (t) , ∀m ∈ N, ∀t ≥ 0.

It follows from (76), (63), and (65) that
x̃⊤ (t) Qγ m Λm (ϖI − γΛ) Q⊤ x̃ (t)
⊤

= ṽ (t) Qγ

m−1

m−1

Λ

(77)
⊤

(ϖI − γΛ) Q ṽ (t) , ∀m ∈ N, ∀t ≥ 0

where Λ is given in (65), and Q = [q1 ; . . . ; qn ] ∈ Rn×n is
an orthogonal matrix of the real symmetric matrix L, where
q1 , q2 , . . . , qn are orthogonal vectors that correspond to the
eigenvalues 0 = λ1 , λ2 , . . . , λn of L. Let
x̂ (t) = Q⊤ x̃ (t) ,

(78)

⊤

(79)

v̂ (t) = Q ṽ (t) .

Considering (71) and (72), from (78) and (79), one has
x̂1 (t) = 0 and v̂1 (t) = 0, respectively. Hence, x̂ (t) and v̂ (t)
can be rewritten as
⊤

x̂ (t) = [0, x̂2 (t) , x̂3 (t) , . . . , x̂n (t)] ∈ Rn ,
⊤

n

v̂ (t) = [0, v̂2 (t) , v̂3 (t) , . . . , v̂n (t)] ∈ R .

(80)
(81)

10

Noting that γ > 0 and the matrix Λ given in (65), we
conclude that the relation (77) is equivalent to
n
∑
i=2

=

i=2

where Λ̆ = diag {λ2 , . . . , λn } ∈ R(n−1)×(n−1) and z̄ (t) =
⊤
[z̄2 (t) , . . . , z̄n (t)] ∈ Rn−1 with

λim−1 (ϖ − γλi ) v̂i2 (t), ∀m ∈ N, ∀t ≥ 0,

z̄i (t) = (ϖ − γλi ) x̂2i (t) , i = 2, . . . , n.

which is also equivalent to
n
∑

λm−1
i

(

(ϖ−γλi )

i=2

Let us denote:

ẑi (t) = (ϖ − γλi )

1
 λ2

2

H =  λ2
 ..
 .
λn−2
2

γλi x̂2i

(t)−v̂i2

)
(t) = 0,∀m ∈ N, t≥0. (82)

(

)
γλi x̂2i (t) − v̂i2 (t) , i = 2, . . . , n, (83)

1
···
1
1
λ3
· · · λn−1
λn 

λ23
· · · λ2n−1
λ2n 
(84)
.
..
..
..
.. 

.
.
.
.

λn−2
3

···

λn−2
n−1

λnn−2

Hẑ (t) = 0n−1 ,

(85)

⊤

where ẑ (t) = [ẑ2 (t) , . . . , ẑn (t)] ∈ Rn−1 .
The condition that L has distinct eigenvalues implies that
the elements λl , l = 2, . . ., n, in the Vandermonde matrix
H ∈ R(n−1)×(n−1) given by (84) are also distinct. Hence,
by Lemma 4, det (H) ̸= 0. Therefore, one concludes that the
solution of (85) is ẑ (t) = 0n−1 . Furthermore, considering
the condition (27), one obtains from (83) that γλi x̂2i (t) =
v̂i2 (t) , ∀i = 2, . . . , n, ∀t ≥ 0, which is equivalent to
v̂ (t) ≡ ∆x̂ (t) , ∀t ≥ 0

A PPENDIX C
P ROOF OF P ROPOSITION 1
The proof can be finished by contradiction. Let us assume
the conditions in Lemma 5 are feasible along the solutions
of the dynamics (20). We write the multi-agent system (20)
in the form of switched system (7) where Aσ(t) is given
by
[ (35). We now
] consider a positive definite matrix Pr,q =
Gr,q Vr,q
> 0 where Gr,q , Vr,q , Sr,q ∈ Rn×n . Then,
⊤
Vr,q
Sr,q
from (35), we have
∆

Γr,q = A⊤
Pr,q + Pr,q Ar
[r
⊤
− γVr,q Lr
−γLr Vr,q
=
Gr,q − γLr Sr,q

(86)

where

{
}
√
√
∆ = diag 0, ± γλ2 , . . . , ± γλn ∈ Rn×n .

(87)

Step Two: In view of the dynamics (24), it follows
from (26), (65), (73), (74), (78), (79), and (86) that
∂m
(t)
∂tm F
∂ m−1
= ∂t
m−1 Ḟ (t)
(
)
∂ m−1
= ∂tm−1 ṽ ⊤ (t) x̃ (t) + ṽ ⊤ (t) ṽ˙ (t)
(
)
∂ m−1
= ∂t
ṽ ⊤ (t) (ϖI − γL) x̃ (t)
m−1
)
m−1 (
∂
ṽ ⊤ (t) Q (ϖI − γΛ) Q⊤ x̃ (t)
= ∂t
m−1
)
m−1 (
∂
= ∂t
v̂ ⊤ (t) (ϖI − γΛ) x̂ (t)
m−1
( ⊤
)
∂ m−1
= ∂tm−1 x̂ (t) ∆ (ϖI − γΛ) x̂ (t)
( ⊤
)
∂ m−2
= ∂t
2v̂ (t) ∆ (ϖI − γΛ) x̂ (t)
m−2
)
m−2 (
∂
= ∂t
2x̂⊤ (t) ∆2 (ϖI − γΛ) x̂ (t)
m−2
m−1 ⊤
m

(88)

which can be expressed equivalently in term of the entries of
the matrices Λ in (65) and ∆ in (87) as

i=2

m

Gr,q − γSr,q Lr
⊤
Vr,q + Vr,q

]

.

(89)

(92)

⊤
If Γr,q is negative definite, we have −γLr Vr,q
− γVr,q Lr <
⊤
0 and Vr,q + Vr,q < 0. For undirected topology, it follows
⊤
from Lemma 1 that Lr ≥ 0. Thus, Vr,q + Vr,q
< 0 contradicts
⊤
with −γLr Vr,q − γVr,q Lr < 0. Therefore, Γr,q cannot be
negative definite. Then, noting Γr,q is a real and symmetric
matrix, one concludes that Γr,q has at least one non-negative
real eigenvalue.
Let Φr,q be one orthogonal matrix of Γr,q (92) such that
{
} ∆ q
⊤
Φr,q Γr,q Φr,q = diag λ1r,q , λ2r,q , . . . , λ2n
= Xr and λ1r,q ≥
r,q
1
n
0, where λr,q , . . . , λr,q are the corresponding eigenvalues of
q
matrix Γr,q (92). Then, by denoting Ψ̆qr = Φ⊤
r,q Ψr Φr,q and
⊤
P̆r,q = Φr,q Pr,q Φr,q , the conditions (8) and (11) are expressed
equivalently as

Xqr + Ψ̆qr − αP̆r,q < 0, ∀r ∈ S, ∀q = 0, 1, . . . , L − 1

=2
x̂ (t) ∆ (ϖI − γΛ) x̂ (t)
= 0, ∀m ∈ N, ∀t ≥ 0

(±γλi ) 2 (ω − γλi ) x̂2i (t) ≡ 0, ∀m ∈ N, ∀t ≥ 0.

(91)

As obtained in the previous
( ) step (Step One) of the
( proof,
)
det (H) ̸= (0.) Since det Λ̄ ̸= 0, one has det HΛ̄ =
det (H) det Λ̄ ̸= 0. Therefore, the solution of (90) is z̄ (t) ≡
0n−1 , ∀t ≥ 0. We note that the condition (27) is equivalent to
ϖ − γλi ̸= 0, ∀i = 2, . . . , n, which together with (91) imply
that the obtained solution z̄ (t) = 0n−1 , ∀t ≥ 0, is equivalent
to x̂2i (t) = 0, ∀t ≥ 0, ∀i = 2, . . . , n. Now, considering (86),
one has v̂i2 (t) = 0, ∀t ≥ 0, ∀i = 2, . . . , n. Finally, noting the
orthogonal matrix is full-rank, from (78), (79), (80), and (81)
one concludes that x̃ (t) = 0n and ṽ (t) = 0n , ∀t ≥ 0, which
contradicts with (28). Thus, the proof is complete.

Considering (83) and (84), one obtains from (82) that

n
∑

(90)

HΛ̆z̄ (t) = 0n−1 , ∀t ≥ 0.

2
γλm
i (ϖ − γλi ) x̂i (t)

n
∑

Let m belong to the set of even numbers. Noting the defined
Vandermonde matrix H in (84), from (89) one has

P̆s,0 − β P̆r,L ≤ 0, ∀s ̸= r ∈ S.

(93)
(94)

Let p̆r,q be the element located
[ at the
] first row and the first
column of the matrix P̆r,q , i.e., P̆r,q
= p̆r,q . Considering

Ψ̆qr

=

L(P̆r,q+1 −P̆r,q )
τmin

1,1

and noting the first element of Xqr is

11

It is straightforward to verify from (102) that g̃(0) = 0 and
g̃(1) = 0, which imply that under any fixed L ∈ N < ∞, the
function g̃ (β) has at least one extreme point over the interval
(0, 1) and g̃ (β) > 0 can happen in its interval (0, 1).
The derivative of g̃ (β) (102) with respect to β ∈ (0, 1) is
obtained as

1

0.8

0.6

−1
1
∂g̃ (β)
= eL(1−β L ) β −( L +1) − 1.
∂β

0.4

−1

∂ g̃(β∗ )
L(1−β∗L )
=
∂β∗ = 0, one obtains from (103) that e
1
(L
+1)
, substituting which into (102) yields that at the exβ∗
1
( 1 +1)
treme point: g̃ (β∗ ) = β∗ L
− β∗ = β∗ (β∗L − 1) < 0. Then,

For

0.2

0
0

0.2

0.4

0.6

0.8

1

−1
Figure 4. Take L = 1 as an example, e(1−β ) < β for ∀β ∈ (0, 1).

non-negative, i.e., λ1r,q ≥ 0, from (93) and (94), we conclude
that (8) and (11) imply
L
(p̆r,q+1 − p̆r,q ) < αp̆r,q , ∀r ∈ S, q = 0, 1, . . . , L−1 (95)
τmin
p̆s,0 ≤ β p̆r,L ,
∀s ̸= r ∈ S.
(96)
Since P̆r,q , ∀r ∈ S, ∀q = 0, 1, . . . , L, is positive define,
p̆r,q > 0, ∀r ∈ S, ∀q = 0, 1, . . . , L. Thus, (95) is equivalent
to
(
)
L p̆r,q+1
τmin >
− 1 , ∀q = 0, . . . , L − 1, ∀r ∈ S. (97)
α
p̆r,q
β
Condition (12) is equivalent to τmax < − ln
α , which
together with (97) and τmax ≥ τmin yields
(
)
p̆r,q+1
− ln β > L
− 1 , ∀q = 0, . . . , L − 1, ∀r ∈ S. (98)
p̆r,q

Condition (96) implies
which further implies

p̆r,L
p̆s,0

≥

1
β

> 1 and

p̆s,L
p̆r,0

≥

1
β

> 1,

−1
p̆−1
s,0 p̆r,0 p̆s,L p̆r,L

=

(103)

L−1
∏
q=0

(99)

−1
−2
> 1, ∀r ̸= s ∈ S.
p̆−1
s,q p̆r,q p̆s,q+1 p̆r,q+1 ≥ β

Considering (99), one can pick up one number q̂ ∈
2
−1
−L
{0, 1, . . . , L − 1} such that p̆−1
>
s,q̂ p̆r,q̂ p̆s,q̂+1 p̆r,q̂+1 ≥ β
1, ∀r ̸= s ∈ S, which also implies that one of the indices s
and r, say r, which satisfies
1

−L
p̆−1
> 1, r ∈ S.
r,q̂ p̆r,q̂+1 ≥ β

(100)
1

Combining (98) with (100) yields − ln β > L(β − L − 1)
which is equivalent to
β < eL(1−β

−1
L

)

, β ∈ (0, 1).

(101)

For (101), let us consider the function
g̃ (β) = eL(1−β

−1
L

)

− β, β ∈ (0, 1).

(102)

noting g̃(0) = 0 and g̃(1) = 0, we conclude that under any
fixed L ∈ N < ∞, g̃(β) < 0 for ∀β ∈ (0, 1). Therefore, (101)
never holds (an illustration is given in Figure 4, where we
take L = 1 as an example), which is a contradiction, and this
completes the proof.
ACKNOWLEDGMENT
The authors are grateful to Professor Sadegh Bolouki for
valuable discussions.
R EFERENCES
[1] Y. Mao, E. Akyol, and Z. Zhang, “Strategic topology
switching for security-part ii: detection & switching topologies,”
https://arxiv.org/abs/1711.11181, 2017.
[2] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks
of agents with switching topology and time-delays,” IEEE Transactions
on automatic control, vol. 49, no. 9, pp. 1520–1533, 2004.
[3] A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile
autonomous agents using nearest neighbor rules,” IEEE Transactions on
automatic control, vol. 48, no. 6, pp. 988–1001, 2003.
[4] J. A. Fax and R. M. Murray, “Information flow and cooperative control
of vehicle formations,” IEEE transactions on automatic control, vol. 49,
no. 9, pp. 1465–1476, 2004.
[5] W. Ren and E. Atkins, “Distributed multi-vehicle coordinated control
via local information exchange,” International Journal of Robust and
Nonlinear Control, vol. 17, no. 10-11, pp. 1002–1033, 2007.
[6] J. N. Tsitsiklis, “Problems in decentralized decision making and computation,” MASSACHUSETTS INST OF TECH CAMBRIDGE LAB
FOR INFORMATION AND DECISION SYSTEMS, Tech. Rep., 1984.
[7] A. Nedić and A. Ozdaglar, “Distributed subgradient methods for multiagent optimization,” IEEE Transactions on Automatic Control, vol. 54,
no. 1, pp. 48–61, 2009.
[8] Z. Zhang and M.-Y. Chow, “Convergence analysis of the incremental cost consensus algorithm under different communication network
topologies in a smart grid,” IEEE Transactions on Power Systems,
vol. 27, no. 4, pp. 1761–1768, 2012.
[9] L.-Y. Lu and C.-C. Chu, “Consensus-based droop control synthesis for
multiple dics in isolated micro-grids,” IEEE Transactions on Power
Systems, vol. 30, no. 5, pp. 2243–2256, 2015.
[10] Q. Li and D. Rus, “Global clock synchronization in sensor networks,”
IEEE Transactions on computers, vol. 55, no. 2, pp. 214–226, 2006.
[11] A. Abdessameud and A. Tayebi, “Attitude synchronization of a group
of spacecraft without velocity measurements,” IEEE Transactions on
Automatic Control, vol. 54, no. 11, pp. 2642–2648, 2009.
[12] S.-J. Chung and J.-J. E. Slotine, “Cooperative robot control and concurrent synchronization of lagrangian systems,” IEEE Transactions on
Robotics, vol. 25, no. 3, pp. 686–700, 2009.
[13] B. B. Johnson, S. V. Dhople, A. O. Hamadeh, and P. T. Krein, “Synchronization of nonlinear oscillators in an lti electrical power network,”
IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 61,
no. 3, pp. 834–844, 2014.
[14] W. Yu, G. Chen, and M. Cao, “Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems,”
Automatica, vol. 46, no. 6, pp. 1089–1095, 2010.

12

[15] J. Zhu, Y.-P. Tian, and J. Kuang, “On the general consensus protocol of
multi-agent systems with double-integrator dynamics,” Linear Algebra
and its Applications, vol. 431, no. 5-7, pp. 701–715, 2009.
[16] J. Mei, W. Ren, and J. Chen, “Distributed consensus of second-order
multi-agent systems with heterogeneous unknown inertias and control
gains under a directed graph,” IEEE Transactions on Automatic Control,
vol. 61, no. 8, pp. 2019–2034, 2016.
[17] J. Qin, C. Yu, and B. D. Anderson, “On leaderless and leader-following
consensus for interacting clusters of second-order multi-agent systems,”
Automatica, vol. 74, pp. 214–221, 2016.
[18] X. Ai, S. Song, and K. You, “Second-order consensus of multi-agent
systems under limited interaction ranges,” Automatica, vol. 68, pp. 329–
333, 2016.
[19] W. Yu, W. X. Zheng, G. Chen, W. Ren, and J. Cao, “Second-order
consensus in multi-agent dynamical systems with sampled position
data,” Automatica, vol. 47, no. 7, pp. 1496–1503, 2011.
[20] D. S. Bassett, N. F. Wymbs, M. A. Porter, P. J. Mucha, and S. T.
Grafton, “Cross-linked structure of network evolution,” Chaos: An
Interdisciplinary Journal of Nonlinear Science, vol. 24, no. 1, p. 013112,
2014.
[21] P. J. Menck, J. Heitzig, J. Kurths, and H. J. Schellnhuber, “How dead
ends undermine power grid stability,” Nature communications, vol. 5, p.
3969, 2014.
[22] P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C. Canizares,
N. Hatziargyriou, D. Hill, A. Stankovic, C. Taylor et al., “Definition and
classification of power system stability ieee/cigre joint task force on
stability terms and definitions,” IEEE transactions on Power Systems,
vol. 19, no. 3, pp. 1387–1401, 2004.
[23] S. H. Moghaddam and M. R. Jovanovic, “Topology design for
stochastically-forced consensus networks,” IEEE Transactions on Control of Network Systems, DOI: 10.1109/TCNS.2017.2674962, 2017.
[24] P. Schultz, T. Peron, D. Eroglu, T. Stemler, G. M. R. Ávila, F. A.
Rodrigues, and J. Kurths, “Tweaking synchronization by connectivity
modifications,” Physical Review E, vol. 93, no. 6, p. 062211, 2016.
[25] Y. Mao, E. Akyol, and Z. Zhang, “Strategic topology switching for
security of multi-agent systems,” Accepted by NERCCS 2018: First
Northeast Regional Conference on Complex Systems, 2018.
[26] Y. Mao and Z. Zhang, “Second-order consensus for multi-agent systems
by state-dependent topology switching,” Accepted by 2018 American
Control Conference, 2018.
[27] E. Mallada and A. Tang, “Improving damping of power networks: Power
scheduling and impedance adaptation,” in Decision and Control and
European Control Conference (CDC-ECC), 2011 50th IEEE Conference
on. IEEE, 2011, pp. 7729–7734.
[28] S. K. Mazumder, Wireless networking based control. Springer, 2011.
[29] W. Ren and R. W. Beard, “Consensus seeking in multiagent systems
under dynamically changing interaction topologies,” IEEE Transactions
on automatic control, vol. 50, no. 5, pp. 655–661, 2005.
[30] H. E. Psillakis, “Consensus in networks of agents with unknown highfrequency gain signs and switching topology,” IEEE Transactions on
Automatic Control, vol. 62, no. 8, pp. 3993–3998, 2017.
[31] P. Lin and Y. Jia, “Consensus of a class of second-order multiagent systems with time-delay and jointly-connected topologies,” IEEE
Transactions on Automatic Control, vol. 55, no. 3, pp. 778–784, 2010.
[32] S. Su and Z. Lin, “Distributed consensus control of multi-agent systems
with higher order agent dynamics and dynamically changing directed interaction topologies,” IEEE Transactions on Automatic Control, vol. 61,
no. 2, pp. 515–519, 2016.
[33] G. Xie and L. Wang, “Consensus control for networks of dynamic agents
via active switching topology,” in International Conference on Natural
Computation. Springer, 2005, pp. 424–433.
[34] A. A. Cárdenas, S. Amin, and S. Sastry, “Research challenges for the
security of control systems.” in HotSec, 2008.
[35] F. Pasqualetti, F. Dörfler, and F. Bullo, “Attack detection and identification in cyber-physical systems,” IEEE Transactions on Automatic
Control, vol. 58, no. 11, pp. 2715–2729, 2013.
[36] A. Teixeira, D. Pérez, H. Sandberg, and K. H. Johansson, “Attack models
and scenarios for networked control systems,” in Proceedings of the
1st international conference on High Confidence Networked Systems.
ACM, 2012, pp. 55–64.
[37] J. L. Gross and J. Yellen, Graph theory and its applications. CRC
press, 2005.
[38] C. Godsil and G. F. Royle, Algebraic graph theory. Springer Science
& Business Media, 2013, vol. 207.
[39] D.
A.
Spielman,
“Spectral
graph
theory
(lecture
5
rings, paths, and cayley graphs),” Online lecture notes.
http://www.cs.yale.edu/homes/spielman/561/, September 16, 2014.

[40] A. E. Brouwer and W. H. Haemers, Spectra of graphs. Springer Science
& Business Media, 2011.
[41] R. A. Hom and C. R. Johnson, “Topics in matrix analysis,” Cambridge
UP, New York, 1991.
[42] W. Xiang and J. Xiao, “Stabilization of switched continuous-time
systems with all modes unstable via dwell time switching,” Automatica,
vol. 50, no. 3, pp. 940–945, 2014.
[43] Z. Zuo and L. Tie, “Distributed robust finite-time nonlinear consensus
protocols for multi-agent systems,” International Journal of Systems
Science, vol. 47, no. 6, pp. 1366–1375, 2016.
[44] A. Gusrialdi and Z. Qu, “Distributed estimation of all the eigenvalues and eigenvectors of matrices associated with strongly connected
digraphs,” IEEE control systems letters, vol. 1, no. 2, pp. 328–333, 2017.
[45] A. Y. Kibangou et al., “Distributed estimation of laplacian eigenvalues
via constrained consensus optimization problems,” Systems & Control
Letters, vol. 80, pp. 56–62, 2015.

