# Synchronization via impulsive deadbeat coupling

S. Emre Tuna111The author is with Department of Electrical and Electronics
Engineering, Middle East Technical University, 06800 Ankara, Turkey. Email:
<EMAIL_ADDRESS>
###### Abstract

For linear networks, where the coupling between the agents takes place through
periodic impulses, a simple method is proposed for synchronization. It is
shown that closing the loop by (normalized) deadbeat feedback gain produces
synchronous behavior if the coupling strength $\mu$ is large enough. With such
choice of control law, in the limiting case ($\mu\to\infty$) exact
synchronization is achieved after $n$ periods, where $n$ is the order of
individual agent dynamics.

##  1 Introduction

Figure 1: Coupled LC oscillators.

Consider the circuit in Fig. 1 where a pair of identical LC oscillators are
linked by a resistor and a switch. The position of the switch is represented
by the function $\sigma:{\mathbb{R}}\to\\{0,\,1\\}$. At a given time $t$,
$\sigma(t)=1$ indicates that the switch is closed (i.e., the oscillators are
connected) and $\sigma(t)=0$ means the switch is open (i.e., the oscillators
are decoupled). Letting $x_{1}=[v_{1}\ i_{1}]^{\top}$ and $x_{2}=[v_{2}\
i_{2}]^{\top}$ this system enjoys the model

$\displaystyle{\dot{x}}_{i}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{rr}0&-1\\\
1&0\end{array}\right]x_{i}+\left[\begin{array}[]{c}1\\\
0\end{array}\right]u_{i}\,,\qquad i=1,\,2$

with the input currents

$\displaystyle
u_{1}=-u_{2}=\frac{1}{r}\,\sigma(t)(v_{2}-v_{1})=\frac{1}{r}\,\sigma(t)[1\ \
0](x_{2}-x_{1})\,.$

Introducing the matrices

$\displaystyle A=\left[\begin{array}[]{rr}0&-1\\\
1&0\end{array}\right]\,,\quad B=\left[\begin{array}[]{c}1\\\
0\end{array}\right]\,,\quad K=[1\ \
0]\,,\quad\Gamma=\left[\begin{array}[]{rr}1&-1\\\ -1&1\end{array}\right]$

and defining the overall state $x=[x_{1}^{\top}\ x_{2}^{\top}]^{\top}$ we can
represent the network by

$\displaystyle{\dot{x}}=[I\otimes A]x-\frac{1}{r}\,\sigma(t)[\Gamma\otimes
BK]x$

where $I$ is the identity matrix and “$\otimes\,$” is the Kronecker product
symbol. Suppose now the switching signal $\sigma$ is a pulse train with period
$T$ and pulse width $d$ as shown in Fig. 2. Further suppose that the ratio of
the pulse width $d$ to the coupling resistance $r$ is fixed. That is,
$d/r=\mu$ for some $\mu\in{\mathbb{R}}$. Then the case $r\ll 1$ can be
approximated by the following impulsive differential equation

$\displaystyle{\dot{x}}=[I\otimes
A]x-\mu\sum_{k=1}^{\infty}\delta(t-kT)[\Gamma\otimes BK]x$ (3)

where $\delta$ is the Dirac delta function. It is intuitively clear that the
harmonic oscillators of our simple network will asymptotically synchronize for
any $\mu$ and (almost) any $T$. However, a remarkable thing happens when $\mu$
is large and the switching period is set to $T=\frac{\pi}{2}\,$sec.
Synchronization then is (nearly) exact. Namely, $x_{1}(t)=x_{2}(t)$ for
$t>2T$. Our aim in this paper is to provide a formal explanation to this
deadbeat behavior in a general setting, and thereby obtain a control design
method. Such synchronization method may find applications in certain networks
modeled by impulsive differential equations.

Figure 2: Switching signal.

The network dynamics (3), which can also be expressed as (16), make a special
case of the so-called impulsive system [12, 7, 2]. An interesting area of
application for impulsive control theory has been synchronization over
networks. In one of the earliest works on the subject [13], Yang and Chua
establish synchronization of a pair of chaotic systems via an impulsive
control law. In the same vein, Zhou and coauthors study the collective
behavior of harmonic oscillators subject to impulsive velocity coupling in
[16]. More general setups have also been of interest. For instance, in [8, 9,
5, 11] sufficient conditions for synchronization are presented for a certain
class of nonlinear networks under the assumption that the inner coupling
matrix is positive definite. A leader-follower scheme is employed via
impulsive coupling in [6]; a method is proposed in [14] by which the agents
achieve fixed-time synchronization through signals communicated from an
isolated node. The impulsive synchronization literature is indeed rich and
many other important results have been reported. For a comprehensive list of
recent research effort we refer the reader to the surveys [15, 4].

Despite their generality the nonlinear impulsive network models studied in the
related literature do not seem to cover as a special case the linear impulsive
network (3), which is what we intend to investigate in this paper. The
coupling we consider being periodic, it is clear that the behavior of the
network (3) can be understood through a discrete-time linear time-invariant
(LTI) model. For our analysis therefore we will not require any tools from
impulsive control theory. For the case with scalar inputs
($u_{i}\in{\mathbb{R}}$) and under a standard connectivity assumption on the
communication graph represented by the Laplacian $\Gamma$, we first show that
the units (e.g., oscillators) of the network exactly synchronize in finite
time if the matrix $K$ is chosen as the normalized deadbeat feedback gain and
the coupling strength is arbitrarily large ($\mu\to\infty$). Then, based on
this initial observation, we establish our main result (Theorem 1) where we
present a lower bound on $\mu$ for asymptotic synchronization.

##  2 Problem statement

Consider a network of $q$ identical units

$\displaystyle{\dot{x}}_{i}=Ax_{i}+Bu_{i}$ (4)

where $x_{i}\in{\mathbb{R}}^{n}$ is the state and $u_{i}\in{\mathbb{R}}$ the
(scalar) input of the $i$th unit. The matrices $A\in{\mathbb{R}}^{n\times n}$
and $B\in{\mathbb{R}}^{n\times 1}$ make a controllable pair $(A,\,B)$.
Available to each unit is a certain signal of the form

$\displaystyle z_{i}=\sum_{j=1}^{q}\gamma_{ij}(x_{j}-x_{i})$ (5)

with weights $\gamma_{ij}\geq 0$ (we let $\gamma_{ii}=0$). These signals are
used to generate impulsive control inputs

$\displaystyle u_{i}=\mu\sum_{k=1}^{\infty}\delta(t-kT)Kz_{i}$ (6)

where $\mu>0$ is the coupling strength, $T>0$ is the separation between two
consecutive impulses, and $K\in{\mathbb{R}}^{1\times n}$ is the feedback gain.
Combining (4), (5), and (6) produces the closed-loop network dynamics

$\displaystyle{\dot{x}}_{i}=Ax_{i}+\mu\sum_{k=1}^{\infty}\delta(t-kT)BK\left(\sum_{j=1}^{q}\gamma_{ij}(x_{j}-x_{i})\right)\,,\qquad
i=1,\,2,\,\ldots,\,q\,.$ (7)

The associated Laplacian $\Gamma\in{\mathbb{R}}^{q\times q}$ of this network
reads

$\displaystyle\Gamma=\left[\begin{array}[]{cccc}\sum_{j}\gamma_{1j}&-\gamma_{12}&\cdots&-\gamma_{1q}\\\
-\gamma_{21}&\sum_{j}\gamma_{2j}&\cdots&-\gamma_{2q}\\\
\vdots&\vdots&\ddots&\vdots\\\
-\gamma_{q1}&-\gamma_{q2}&\cdots&\sum_{j}\gamma_{qj}\\\ \end{array}\right]\,.$
(12)

Standing assumptions. We make the usual connectivity assumption that the
directed graph of the network (7) contains a spanning tree. This is equivalent
to saying that $\Gamma$ has a simple eigenvalue at the origin and the
remaining $q-1$ eigenvalues have strictly positive real parts; see [1]. We
further assume that the pair $(e^{AT},\,B)$ is controllable. Note that this
assumption is satisfied for almost all $T$ since $(A,\,B)$ is controllable.

In this paper we study the collective behavior of the network (7) for the
specific feedback gain $K$ chosen as $K=Ge^{-AT}$ where $G$ is the deadbeat
gain for the pair $(e^{AT},\,B)$, meaning the matrix $\left[e^{AT}-BG\right]$
is nilpotent. In other words, our goal here is to analyze the impulsive
dynamics (7) under the constraint

$\displaystyle\left[(I-BK)e^{AT}\right]^{n}=0\,.$ (13)

We note that the example network in Section 1 satisfies the condition (13) for
$T=\frac{\pi}{2}$.

In the following section we present three lemmas. Those preliminary results
will be invoked later when we set out to establish our main theorem.

##  3 Preliminaries

Consider the network (7) whose Laplacian $\Gamma$ is given in (12). Let
$\lambda_{1},\,\lambda_{2},\,\ldots,\,\lambda_{q}$ be the eigenvalues of
$\Gamma$ ordered as $0=\lambda_{1}<{\rm Re}(\lambda_{2})\leq\cdots\leq{\rm
Re}(\lambda_{q})$. Clearly, $\Gamma{\mathbf{1}}=0$, where
${\mathbf{1}}\in{\mathbb{R}}^{q}$ is the vector of all ones. That is,
${\mathbf{1}}$ is the eigenvector for $\lambda_{1}=0$. We let
$\ell\in{\mathbb{R}}^{q}$ denote the left eigenvector for $\lambda_{1}$ (i.e.,
$\ell^{\top}\Gamma=0$) normalized to satisfy $\ell^{\top}{\mathbf{1}}=1$. Note
that $e^{-\Gamma t}\to{\mathbf{1}}\ell^{\top}$ as $t\to\infty$. Given
$Q\in{\mathbb{C}}^{n\times n}$, its conjugate transpose is denoted by $Q^{H}$
and its (induced) 2-norm by $\|Q\|$.

Let us construct the (square) controllability matrix $\mathcal{C}=\left[B\ \
e^{AT}B\ \cdots\ e^{A(n-1)T}B\right]$ which is nonsingular since
$(e^{AT},\,B)$ is controllable. Recall that $G\in{\mathbb{R}}^{1\times n}$
denotes the deadbeat gain for the pair $(e^{AT},\,B)$. It turns out that $G$
enjoys the closed-form expression $G=e_{n}^{\top}\mathcal{C}^{-1}e^{AnT}$
where the unit vector $e_{n}\in{\mathbb{R}}^{n}$ is the $n$th column of the
identity matrix; see [10]. Then, since $K=Ge^{-AT}$, we have
$K=e_{n}^{\top}\mathcal{C}^{-1}e^{A(n-1)T}$. This allows us to claim the
following.

###### Lemma 1

The constraint (13) implies $KB=1$.

Proof.
$KB=e_{n}^{\top}\mathcal{C}^{-1}e^{A(n-1)T}B=e_{n}^{\top}\mathcal{C}^{-1}\left[B\
\ e^{AT}B\ \cdots\
e^{A(n-1)T}B\right]e_{n}=e_{n}^{\top}\mathcal{C}^{-1}\mathcal{C}e_{n}=e_{n}^{\top}e_{n}=1.$
$\blacksquare$

Alternative proof. Note that (13) implies (by Cayley-Hamilton Theorem) that
the characteristic polynomial of the matrix $\left[(I-BK)e^{AT}\right]$ is
$d(s)=s^{n}$. That is, all the eigenvalues are located at the origin. Let
$w\in{\mathbb{R}}^{n}$ be a left eigenvector of the matrix, i.e.,
$w^{\top}\left[(I-BK)e^{AT}\right]=0$. This implies, since $e^{AT}$ is
nonsingular, $w^{\top}(I-BK)=0$. Solving the last equation for $K$ yields
$K=(w^{\top}B)^{-1}w^{\top}$. Then $KB=(w^{\top}B)^{-1}w^{\top}B=1$.
$\blacksquare$

The next result is evident. Still we provide a demonstration for the sake of
completeness.

###### Lemma 2

Given $M\in{\mathbb{R}}^{m\times m}$ and $\tau\in{\mathbb{R}}$, let
$y:{\mathbb{R}}\to{\mathbb{R}}^{m}$ be a solution to the impulsive
differential equation ${\dot{y}}=\delta(t-\tau)My$. Then
$y(\tau^{+})=e^{M}y(\tau^{-})$.

Proof. Without loss of generality let $\tau=0$. For $\varepsilon>0$ let the
pulse function $p_{\varepsilon}:{\mathbb{R}}\to{\mathbb{R}}$ be defined as

$\displaystyle
p_{\varepsilon}(t)=\left\\{\begin{array}[]{cl}\displaystyle\frac{1}{2\varepsilon}&\mbox{for}\
\displaystyle t\in\left[-\varepsilon,\,\varepsilon\right]\vspace{0.1in}\\\
0&\mbox{elsewhere}\,.\end{array}\right.$

Note that $\lim_{\varepsilon\to 0}p_{\varepsilon}(t)=\delta(t)$. Now, given
some $y(\cdot)$ satisfying ${\dot{y}}=\delta(t)My$ let
$y_{\varepsilon}(\cdot)$ be the solution to the approximate differential
equation ${\dot{y}}_{\varepsilon}=p_{\varepsilon}(t)My_{\varepsilon}$ starting
from the initial condition $y_{\varepsilon}(-\varepsilon)=y(0^{-})$. It is
easy to see that
$y_{\varepsilon}(\varepsilon)=e^{M}y_{\varepsilon}(-\varepsilon)=e^{M}y(0^{-})$.
Therefore

$\displaystyle y(0^{+})=\lim_{\varepsilon\to
0}y_{\varepsilon}(\varepsilon)=e^{M}y(0^{-})$

which was to be shown. $\blacksquare$

The following is a result about the Schur decomposition of the matrix
$\left[(I-BK)e^{AT}\right]$. We will need it for our main theorem.

###### Lemma 3

There exists a strictly upper triangular $N\in{\mathbb{C}}^{n\times n}$
satisfying

$\displaystyle Q^{H}\left[(I-BK)e^{AT}\right]Q=N$ (15)

for some unitary $Q\in{\mathbb{C}}^{n\times n}$.

Proof. Recall that all the eigenvalues of $\left[(I-BK)e^{AT}\right]$ are at
the origin by (13). The result then follows by [3, Thm. 7.1.3]. $\blacksquare$

We end this section with the definition of a synchronous network.

###### Definition 1

The network (7) is said to be synchronous if $\|x_{i}(t)-x_{j}(t)\|\to 0$ as
$t\to\infty$ for all pairs $(i,\,j)$ and all initial conditions.

##  4 Main result

In this section we analyze the collective behavior of the network (7) under
the condition (13) with respect to the coupling strength $\mu$. The
(continuous-time) system (7) under study is linear time-varying, yet periodic.
This means we can represent it by an LTI model in discrete time. We now obtain
this model. Let $x=[x_{1}^{\top}\ x_{2}^{\top}\ \cdots\ x_{q}^{\top}]^{\top}$.
Then (7) can be cast into (3) which yields for $k=1,\,2,\,\ldots$

${\dot{x}}(t)&=[I\otimes A]x(t)\,,\quad t\neq kT\\\
x(kT^{+})=e^{-\mu[\Gamma\otimes BK]}x(kT^{-})$ (16)

where (16b) follows from Lemma 2. Observe that (16a) allows us to write
$x(kT^{-})=e^{T[I\otimes A]}x((k-1)T^{+})$. Combining this with (16b) we
obtain

$\displaystyle x(kT^{+})=e^{-\mu[\Gamma\otimes BK]}e^{T[I\otimes
A]}x((k-1)T^{+})\,.$ (17)

###### Lemma 4

We can write $e^{-\mu[\Gamma\otimes BK]}e^{T[I\otimes
A]}=\left[e^{-\Gamma\mu}\otimes
e^{AT}\right]+\left[(I-e^{-\Gamma\mu})\otimes(I-BK)e^{AT}\right]$.

Proof. Recall $KB=1$ by Lemma 1. This at once yields $(BK)^{k}=BK$ for
$k=1,\,2,\,\ldots$ (Namely, $[BK]$ is a projection matrix.) Using the series
definition of the matrix exponential we can then write

$\displaystyle e^{-\mu[\Gamma\otimes BK]}$ $\displaystyle=$ $\displaystyle
e^{[-\Gamma\mu\otimes BK]}$ $\displaystyle=$ $\displaystyle
I+[-\Gamma\mu\otimes BK]+[-\Gamma\mu\otimes BK]^{2}+[-\Gamma\mu\otimes
BK]^{3}+\cdots$ $\displaystyle=$ $\displaystyle I+[-\Gamma\mu\otimes
BK]+\left[(-\Gamma\mu)^{2}\otimes(BK)^{2}\right]+\left[(-\Gamma\mu)^{3}\otimes(BK)^{3}\right]+\cdots$
$\displaystyle=$ $\displaystyle I+\left[-\Gamma\mu\otimes
BK\right]+\left[(-\Gamma\mu)^{2}\otimes
BK\right]+\left[(-\Gamma\mu)^{3}\otimes BK\right]+\cdots$ $\displaystyle=$
$\displaystyle
I+\left[(-\Gamma\mu+(-\Gamma\mu)^{2}+(-\Gamma\mu)^{3}+\cdots)\otimes
BK\right]$ $\displaystyle=$ $\displaystyle I+\left[(e^{-\Gamma\mu}-I)\otimes
BK\right]\,.$

And it is easy to check that

$\displaystyle I+\left[(e^{-\Gamma\mu}-I)\otimes
BK\right]=\left[e^{-\Gamma\mu}\otimes
I\right]+\left[(I-e^{-\Gamma\mu})\otimes(I-BK)\right]\,.$

Since $e^{T[I\otimes A]}=e^{[I\otimes AT]}=\left[I\otimes e^{AT}\right]$ it
follows that

$\displaystyle e^{-\mu[\Gamma\otimes BK]}e^{T[I\otimes A]}$ $\displaystyle=$
$\displaystyle\left(\left[e^{-\Gamma\mu}\otimes
I\right]+\left[(I-e^{-\Gamma\mu})\otimes(I-BK)\right]\right)\left[I\otimes
e^{AT}\right]$ $\displaystyle=$ $\displaystyle\left[e^{-\Gamma\mu}\otimes
e^{AT}\right]+\left[(I-e^{-\Gamma\mu})\otimes(I-BK)e^{AT}\right]$

which was to be shown. $\blacksquare$

For a cleaner presentation we introduce the notation $x_{i}[k]=x_{i}(kT^{+})$
with $x_{i}[0]=x_{i}(0)$. Then by (17) and Lemma 4 we obtain

$\displaystyle x[k+1]=\Big{(}\left[e^{-\Gamma\mu}\otimes
e^{AT}\right]+\left[(I-e^{-\Gamma\mu})\otimes(I-BK)e^{AT}\right]\Big{)}x[k]$
(18)

for $k=0,\,1,\,\ldots$ This is the discrete-time LTI representation of the
network (7) when (13) holds.

###### Remark 1

Note that if $\|x_{i}[k]-x_{j}[k]\|\to 0$ as $k\to\infty$ for all pairs
$(i,\,j)$ and all initial conditions then the network (7) is synchronous.
Moreover, if for some $k$ we have $x_{i}[k]=x_{j}[k]$ for all $(i,\,j)$ then
$x_{i}(t)=x_{j}(t)$ for all $(i,\,j)$ and $t>kT$.

Below is our main result.

###### Theorem 1

Consider the network (7) under the condition (13). Suppose

$\displaystyle\mu>\frac{1}{{\rm
Re}(\lambda_{2})}\ln\left(\left\|BKe^{AT}\right\|\times\sum_{k=0}^{n-1}\|N\|^{k}\right)$
(19)

where $N$ comes from the Schur decomposition (15). Then the network is
synchronous.

Prior to demonstrating Theorem 1 we find it worthwhile to present its
adumbration.

###### Theorem 2

Consider the network (7) under the condition (13). Suppose $\mu=\infty$. Then
the solutions satisfy $x_{i}(t)=x_{j}(t)$ for all $(i,\,j)$ and $t>nT$.

Proof. Consider (18). Recall $e^{-\Gamma\mu}\to{\mathbf{1}}\ell^{\top}$ as
$\mu\to\infty$ where $\ell^{\top}{\mathbf{1}}=1$. Hence

$\displaystyle x[k+1]=\Big{(}\left[{\mathbf{1}}\ell^{\top}\otimes
e^{AT}\right]+\left[(I-{\mathbf{1}}\ell^{\top})\otimes(I-BK)e^{AT}\right]\Big{)}x[k]\,.$
(20)

Observe that
${\mathbf{1}}\ell^{\top}(I-{\mathbf{1}}\ell^{\top})=(I-{\mathbf{1}}\ell^{\top}){\mathbf{1}}\ell^{\top}=0$.
Moreover, $({\mathbf{1}}\ell^{\top})^{k}={\mathbf{1}}\ell^{\top}$ and
$(I-{\mathbf{1}}\ell^{\top})^{k}=I-{\mathbf{1}}\ell^{\top}$ for $k\geq 1$.
Therefore

$\displaystyle x[k]$ $\displaystyle=$
$\displaystyle\Big{(}\left[{\mathbf{1}}\ell^{\top}\otimes
e^{AT}\right]+\left[(I-{\mathbf{1}}\ell^{\top})\otimes(I-BK)e^{AT}\right]\Big{)}^{k}x[0]$
$\displaystyle=$ $\displaystyle\Big{(}\left[{\mathbf{1}}\ell^{\top}\otimes
e^{AT}\right]^{k}+\left[(I-{\mathbf{1}}\ell^{\top})\otimes(I-BK)e^{AT}\right]^{k}\Big{)}x[0]$
$\displaystyle=$ $\displaystyle\Big{(}\left[{\mathbf{1}}\ell^{\top}\otimes
e^{AkT}\right]+\left[(I-{\mathbf{1}}\ell^{\top})\otimes\left[(I-BK)e^{AT}\right]^{k}\right]\Big{)}x[0]\,.$

Then, thanks to (13), setting $k=n$ yields

$\displaystyle x[n]=\left[{\mathbf{1}}\ell^{\top}\otimes
e^{AnT}\right]x[0]={\mathbf{1}}\otimes\left(e^{AnT}\left[\ell^{\top}\otimes
I\right]x[0]\right)$

which means $x_{i}[n]=e^{AnT}\left[\ell^{\top}\otimes I\right]x[0]$ for all
$i$. The result follows by Remark 1. $\blacksquare$

An interesting observation is in order here: The deadbeat character of
synchronization established in Theorem 2 is not lost under time-varying
communication topology (as long as $\mu=\infty$). That is, even under a time-
varying Laplacian matrix, the individual solutions still converge to a common
trajectory in finite time, provided that $\Gamma$ is continuous and the
associated graph admits a spanning tree at times $t=T,\,2T,\,\ldots$ Here is
why. Under time-varying $\Gamma$, the difference equation (20) gets
generalized to

$\displaystyle x[k+1]=\Big{(}\left[{\mathbf{1}}\ell_{k}^{\top}\otimes
e^{AT}\right]+\left[(I-{\mathbf{1}}\ell_{k}^{\top})\otimes(I-BK)e^{AT}\right]\Big{)}x[k]$

where $\ell_{k}$ is the left eigenvector of $\Gamma$ at $t=kT$ for the
eigenvalue at the origin, with the property $\ell_{k}^{\top}{\mathbf{1}}=1$.
By inspection one realizes $\left[{\mathbf{1}}\ell_{k}^{\top}\otimes
e^{AT}\right]+\left[(I-{\mathbf{1}}\ell_{k}^{\top})\otimes(I-BK)e^{AT}\right]=\left[I\otimes(I-BK)e^{AT}\right]+\left[{\mathbf{1}}\ell_{k}^{\top}\otimes
BKe^{AT}\right]$. This allows us to write

$\displaystyle x[k+1]$ $\displaystyle=$
$\displaystyle\Big{(}\left[I\otimes(I-BK)e^{AT}\right]+\left[{\mathbf{1}}\ell_{k}^{\top}\otimes
BKe^{AT}\right]\Big{)}x[k]$ $\displaystyle=$
$\displaystyle\left[I\otimes(I-BK)e^{AT}\right]x[k]+{\mathbf{1}}\otimes\Big{(}\left[\ell_{k}^{\top}\otimes
BKe^{AT}\right]x[k]\Big{)}$

which implies for all $i$

$\displaystyle
x_{i}[k+1]=\left[(I-BK)e^{AT}\right]x_{i}[k]+\left[\ell_{k}^{\top}\otimes
BKe^{AT}\right]x[k]\,.$

Since for all $i$ the second term $\left[\ell_{k}^{\top}\otimes
BKe^{AT}\right]x[k]$ is the same, for any pair $(i,\,j)$ the error
$e_{ij}=x_{i}-x_{j}$ must evolve through

$\displaystyle e_{ij}[k+1]=\left[(I-BK)e^{AT}\right]e_{ij}[k]$

which, by (13), yields $e_{ij}[n]=0$. That is, $x_{i}[n]=x_{j}[n]$ and the
deadbeat behavior is hence preserved.

Proof of Theorem 1. Let $U\in{\mathbb{C}}^{q\times(q-1)}$ be such that

$\displaystyle\Gamma[{\mathbf{1}}\ \ U]=[{\mathbf{1}}\ \
U]\left[\begin{array}[]{cc}0&0\\\ 0&\Lambda\end{array}\right]$

where $\Lambda\in{\mathbb{C}}^{(q-1)\times(q-1)}$ is an upper block triangular
matrix whose diagonal entries are the nonzero eigenvalues of $\Gamma$. Namely,

$\displaystyle\Lambda=\left[\begin{array}[]{cccc}\lambda_{2}&\ast&\cdots&\ast\\\
0&\lambda_{3}&\cdots&\ast\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\lambda_{q}\end{array}\right]\,.$

Then let $V\in{\mathbb{C}}^{q\times(q-1)}$ be such that $[{\mathbf{1}}\ \
U]^{-1}=[\ell\ \ V]^{\top}$. Note that this implies $V^{\top}\Gamma=\Lambda
V^{\top}$. Introduce the auxiliary variables $\eta\in{\mathbb{R}}^{n}$ and
$\xi\in{\mathbb{C}}^{(q-1)n}$ as $\eta=\left[\ell^{\top}\otimes I\right]x$ and
$\xi=\left[V^{\top}\otimes I\right]x$. Since
${\mathbf{1}}\ell^{\top}+UV^{\top}=[{\mathbf{1}}\ \ U][\ell\ \ V]^{\top}=I$,
we can decompose $x$ as

$\displaystyle x=[{\mathbf{1}}\otimes I]\eta+[U\otimes I]\xi\,.$

This decomposition tells us that if $\xi[k]\to 0$ as $k\to\infty$ then
$\|x_{i}[k]-\eta[k]\|\to 0$ for all $i$; meaning the network (7) is
synchronous by Remark 1. Now we establish that the condition (19) guarantees
the convergence $\xi[k]\to 0$. Observe that $\left[e^{-\Gamma\mu}\otimes
e^{AT}\right]+\left[(I-e^{-\Gamma\mu})\otimes(I-BK)e^{AT}\right]=\left[I\otimes(I-BK)e^{AT}\right]+\left[e^{-\Gamma\mu}\otimes
BKe^{AT}\right]$. Also note that $V^{\top}\Gamma=\Lambda V^{\top}$ implies
$V^{\top}e^{-\Gamma\mu}=e^{-\Lambda\mu}V^{\top}$. Hence by (18) we can write

$\displaystyle\xi[k+1]$ $\displaystyle=$ $\displaystyle\left[V^{\top}\otimes
I\right]x[k+1]$ (23) $\displaystyle=$ $\displaystyle\left[V^{\top}\otimes
I\right]\Big{(}\left[e^{-\Gamma\mu}\otimes
e^{AT}\right]+\left[(I-e^{-\Gamma\mu})\otimes(I-BK)e^{AT}\right]\Big{)}x[k]$
$\displaystyle=$
$\displaystyle\Big{(}\left[V^{\top}\otimes(I-BK)e^{AT}\right]+\left[V^{\top}e^{-\Gamma\mu}\otimes
BKe^{AT}\right]\Big{)}x[k]$ $\displaystyle=$
$\displaystyle\Big{(}\left[V^{\top}\otimes(I-BK)e^{AT}\right]+\left[e^{-\Lambda\mu}V^{\top}\otimes
BKe^{AT}\right]\Big{)}x[k]$ $\displaystyle=$
$\displaystyle\Big{(}\left[I\otimes(I-BK)e^{AT}\right]+\left[e^{-\Lambda\mu}\otimes
BKe^{AT}\right]\Big{)}\left[V^{\top}\otimes I\right]x[k]$ $\displaystyle=$
$\displaystyle\underbrace{\Big{(}\left[I\otimes(I-BK)e^{AT}\right]+\left[e^{-\Lambda\mu}\otimes
BKe^{AT}\right]\Big{)}}_{\displaystyle\Phi}\xi[k]\,.$

In the light of (23) the convergence $\xi[k]\to 0$ is equivalent to that all
the eigenvalues of the matrix $\Phi$ are in the open unit disc. Since
$\Lambda$ is upper triangular, so is $e^{-\Lambda\mu}$ with diagonal entries
$e^{-\lambda_{2}\mu},\,e^{-\lambda_{3}\mu},\,\ldots,\,e^{-\lambda_{q}\mu}$.
Thus the matrix $\Phi$ is upper block triangular with diagonal block entries
$\left[(I-BK)e^{AT}+e^{-\lambda_{i}\mu}BKe^{AT}\right]=:D_{i}$ for
$i=2,\,3,\,\ldots,\,q$. Let now $\varphi\in{\mathbb{C}}$ be an arbitrary
eigenvalue of $\Phi$. Then $\varphi$ must be an eigenvalue of some block
$D_{i}$. Recall that all the eigenvalues of $\left[(I-BK)e^{AT}\right]$ are
zero due to (13). Combining this with the result [3, Thm. 7.2.3] from
perturbation theory we reach to the following implication

$\displaystyle\left\|e^{-\lambda_{i}\mu}BKe^{AT}\right\|\times\sum_{k=0}^{n-1}\|N\|^{k}<1\implies|\varphi|<1$

where $N$ satisfies (15). It is easy to check that the left-hand side of the
implication is satisfied for all $i\in\\{2,\,3,\,\ldots,\,q\\}$ if $\mu$
satisfies (19) thanks to the ordering ${\rm Re}(\lambda_{2})\leq\cdots\leq{\rm
Re}(\lambda_{q})$. Hence the result.  $\blacksquare$

##  5 Conclusion

In this paper we investigated the linear network (7) where the coupling
between the units is realized through periodic impulses. We showed that if the
state feedback gain $K$ is chosen to satisfy the deadbeat condition (13) then
for large enough coupling strength $\mu$ the units throughout the network
asymptotically synchronize. The identity $KB=1$, which comes for free under
the deadbeat condition, renders $[BK]$ a projection matrix. This observation
was essentially what made our analysis here possible. This is also the flower
pot under which is the key for generalization. Namely, if one replaces the
condition (13) with the weaker assumption:

All the eigenvalues of $\left[(I-BK)e^{AT}\right]$ are in the open unit disc
and $[BK]$ is a projection matrix.

then it should still hold that synchronization is achieved for $\mu$ large
enough.

## References

  * [1] F. Bullo, J. Cortes, and S. Martinez.  Distributed Control of Robotic Networks.  Princeton University Press, 2009.
  * [2] M. Fiacchini and I.-C. Morarescu.  Constructive necessary and sufficient condition for the stability of quasi-periodic linear impulsive systems.  IEEE Transactions on Automatic Control, 61:2512–2517, 2016.
  * [3] G.H. Golub and C.F. Van Loan.  Matrix Computations (Third Edition).  The Johns Hopkins University Press, 1996.
  * [4] X.P. Han, Y.S. Zhao, and X.D. Li.  A survey on complex dynamical networks with impulsive effects.  Frontiers of Information Technology & Electronic Engineering, 21:199–219, 2020.
  * [5] W. He, F. Qian, and J. Cao.  Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control.  Neural Networks, 85:1–9, 2017.
  * [6] W. He, F. Qian, J. Lam, G. Chen, Q.-L. Han, and J. Kurths.  Quasi-synchronization of heterogeneous dynamic networks via distributed impulsive control: Error estimation, optimization and design.  Automatica, 62:249–262, 2015.
  * [7] J.P. Hespanha, D. Liberzon, and A.R. Teel.  Lyapunov conditions for input-to-state stability of impulsive systems.  Automatica, 44:2735–2744, 2008.
  * [8] J. Lu, D.W.C. Ho, and J. Cao.  A unified synchronization criterion for impulsive dynamical networks.  Automatica, 46:1215–1221, 2010.
  * [9] J. Lu, D.W.C. Ho, J. Cao, and J. Kurths.  Single impulsive controller for globally exponential synchronization of dynamical networks.  Nonlinear Analysis: Real World Applications, 14:581–593, 2013.
  * [10] S.E. Tuna.  A dual pair of optimization-based formulations for estimation and control.  Automatica, 51:18–26, 2015.
  * [11] Y. Wang, J. Lu, J. Liang, J. Cao, and M. Perc.  Pinning synchronization of nonlinear coupled Lure networks under hybrid impulses.  IEEE Transactions on Circuits and Systems II: Express Briefs, 66:432–436, 2019.
  * [12] T. Yang.  Impulsive Control Theory.  Springer, 2001.
  * [13] T. Yang and L.O. Chua.  Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication.  IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 44:976–988, 1997.
  * [14] X. Yang, J. Lam, D.W.C. Ho, and Z. Feng.  Fixed-time synchronization of complex networks with impulsive effects via nonchattering control.  IEEE Transactions on Automatic Control, 62:5511–5521, 2017.
  * [15] X. Yang, D. Peng, X. Lv, and X. Li.  Recent progress in impulsive control systems.  Mathematics and Computers in Simulation, 155:244–268, 2019.
  * [16] J. Zhou, H. Zhang, L. Xiang, and Q. Wu.  Synchronization of coupled harmonic oscillators with local instantaneous interaction.  Automatica, 48:1715–1721, 2012.