neu]College of Information Science and Engineering, Northeastern University,
Shenyang 110819, China wsu]School of Electrical Engineering and Computer
Science, Washington State University, Pullman, WA 99164, USA ut]Department of
Electrical Engineering, Mathematics and Computer Science, University of
Twente, Enschede, The Netherlands shfc]School of Statistics and Mathematics,
Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China

# Delayed state synchronization of continuous-time multi-agent systems in the
presence of unknown communication delays

Zhenwei Liuneu  Ali Saberiwsu  Anton A. Stoorvogelut  Rong Lishfc  [
<EMAIL_ADDRESS>[<EMAIL_ADDRESS>[<EMAIL_ADDRESS>[<EMAIL_ADDRESS>
###### Abstract

This paper studies delayed synchronization of continuous-time multi-agent
systems (MAS) in the presence of unknown nonuniform communication delays. A
_delay-free_ transformation is developed based on a communication network
which is a directed spanning tree, which can transform the original MAS to a
new one without delays. By using this transformation, we design a static
protocol for full-state coupling and a dynamic protocol for delayed state
synchronization for homogeneous MAS via full- and partial-state coupling.
Meanwhile, the delayed output synchronization is also studied for
heterogeneous MAS, which is achieved by using a low-gain and output regulation
based dynamic protocol design via the delay-free transformation.

###### keywords:

Multi-agent systems, Delayed state synchronization, Continuous-time,
Communication delays

##  1 Introduction

The problem of synchronization among agents in a multi-agent system has
received substantial attention in recent years, because of its potential
applications in cooperative control of autonomous vehicles, distributed sensor
network, swarming and flocking and others. The objective of synchronization is
to secure an asymptotic agreement on a common state or output trajectory
through decentralized control protocols (see [1, 7, 9, 11, 13, 17, 24] and
references therein).

Recently synchronization in a network with time delay has attracted a great
deal of interest. As clarified in [3], we can identify two kinds of delay.
Firstly there is _communication delay_ , which results from limitations on the
communication between agents. Secondly we have _input delay_ , which is due to
computational limitations of an individual agent. Many works have focused on
dealing with input delay (see e.g. [10, 12, 16, 20, 22, 23, 26, 27]), but
communication delay is much less understood at this moment. In the case of
communication delay, only for a constant synchronization trajectory do we
preserve the diffusive nature of the network. This diffusive nature is an
intrinsic part of the currently available design techniques and hence only
this case has been studied. Some works in this area can be seen in [2, 8, 21,
22].

References [4] and [5] solved the synchronization problem for nonlinear
heterogeneous MAS with unknown non-uniform constant communication delays. Some
other results for non-uniform communication delays can also be found in [8,
14, 15, 21, 25].

On the other hand, the concept of delayed synchronization was introduced in
[4] and [5]. Compared with general synchronization problem, delayed
synchronization allows a fixed signal lag from parents node to their son node
when constant communication delay is considered. It means that there exists a
fixed distance (or some other physical quantities) between two agents to keep
moving. But, due to the restriction of strongly connected network, the
synchronized trajectory must converse to a constant.

In this paper, we study delayed synchronization problems of MAS in the
presence of unknown communication delays. The communication network is assumed
to be a directed spanning tree (i.e., it has one root node and the other non-
root nodes have indegree one). The contribution of this paper is threefold:

  * •

A _delay-free_ transformation is established to remove the effect from unknown
communication delays, and obtain a transformed MAS without communication
delays.

  * •

We develop the delayed state synchronization results of homogeneous MAS based
on the delay-free transformation, and obtain a dynamic synchronized
trajectory. Static and dynamic protocol designs are provided for both full-
and partial-state coupling cases respectively.

  * •

We also develop the delayed output synchronization results of heterogeneous
MAS. A low-gain and output regulation based dynamic protocol design is
provided via the delay-free transformation.

We will see that, compared to earlier work, our approach is limited to a graph
which is a directed spanning tree. However, the intrinsic advantage is that
the synchronized trajectory is not limited to a constant but will follow the
trajectory of the root agent.

Notations and definitions: Given a matrix $A\in\mathbb{R}^{m\times n}$,
$A^{\mbox{\tiny T}}$ and $A^{*}$ denote the transpose and conjugate transpose
of $A$, respectively while $\|A\|$ denotes the induced 2-norm of $A$. A square
matrix $A$ is said to be Hurwitz stable if all its eigenvalues are in the open
left half complex plane. $A\otimes B$ depicts the Kronecker product between
$A$ and $B$. $I_{n}$ denotes the $n$-dimensional identity matrix and $0_{n}$
denotes $n\times n$ zero matrix; we will use $I$ or $0$ if the dimension is
clear from the context.

A _weighted directed graph_ $\mathcal{G}$ is defined by a triple
$(\mathcal{V},\mathcal{E},\mathcal{A})$ where $\mathcal{V}=\\{1,\ldots,N\\}$
is a node set, $\mathcal{E}$ is a set of pairs of nodes indicating connections
among nodes, and $\mathcal{A}=[a_{ij}]\in\mathbb{R}^{N\times N}$ is the
weighting matrix, and $a_{ij}>0$ iff $(j,i)\in\mathcal{E}$. Each pair in
$\mathcal{E}$ is called an _edge_. A _path_ from node $i_{k}$ to $i_{1}$ is a
sequence of nodes $\\{i_{1},\ldots,i_{k}\\}$ such that
$(i_{j},i_{j+1})\in\mathcal{E}$ for $j=1,\ldots,k-1$. A _directed tree_ is a
subgraph (subset of nodes and edges) in which every node has exactly one
parent node except for one node, called the _root_ , which has no parent node.
In this case, the root has a directed path to every other node in the tree. A
_directed spanning tree_ is a subgraph which is a directed tree containing all
the nodes of the original graph. An agent is called a _root agent_ if it is
the root of some directed spanning tree of the associated graph. Let
$\Pi_{\mathcal{G}}$ denote the set of all root agents for a graph. For a
weighted graph $\mathcal{G}$, a matrix $L=[\ell_{ij}]$ with

$\ell_{ij}=\left\\{\;\begin{array}[]{cl}\sum_{k=1}^{N}a_{ik},&i=j,\\\
-a_{ij},&i\neq j,\end{array}\right.$ (1)

is called the _Laplacian matrix_ associated with the graph $\mathcal{G}$. In
the case where $\mathcal{G}$ has non-negative weights, $L$ has all its
eigenvalues in the closed right half plane and at least one eigenvalue at zero
associated with right eigenvector 1.

##  2 Problem formulation

We will study a MAS consisting of $N$ identical agents:

$\begin{cases}\dot{x}_{i}(t)=Ax_{i}(t)+Bu_{i}(t),\\\
y_{i}(t)=Cx_{i}(t),\end{cases}$ (2)

where $x_{i}(t)\in\mathbb{R}^{n}$, $u_{i}(t)\in\mathbb{R}^{m}$ and
$y_{i}(t)\in\mathbb{R}^{p}$ are the state, input and the output, respectively,
of agent $i$ for $i=1,\ldots,N$.

###### Assumption 1

We assume that

  * •

($A$, $B$, $C$) is stabilizable and detectable.

  * •

All eigenvalues of $A$ are in the closed left half complex plane.

The communication network provides agent $i$ with the following information,

$\zeta_{i}(t)=\sum_{j=1}^{N}a_{ij}\left[y_{i}(t)-y_{j}(t-\tau_{ij})\right]$
(3)

where $a_{ij}\geqslant 0$ and $a_{ii}=0$. This communication topology of the
network can be described by a weighted graph $\mathcal{G}$ with weighting
matrix $\mathcal{A}=[a_{ij}]$. We can obtain the associated Laplacian matrix
$L$ via (1).

Here $\tau_{ij}\in\mathbb{R}^{+}$ represents an unknown constant communication
delays from agent $j$ to agent $i$. This communication delay implies that it
takes $\tau_{ij}$ seconds for agent $j$ to transfer its state information to
agent $i$. Furthermore, we assume Agent 1 is root of graph in this paper.

###### Definition 1

For any $\beta>0$, let $\mathbb{G}_{\beta}^{N}$ denote the set of directed
graphs with $N$ nodes which are equal to a directed spanning tree for which
the corresponding Laplacian matrix $L$ is lower triangular with the top row
identical to zero which has the property that $\ell_{ii}\geqslant\beta$ for
$i=2,\ldots,N$ while agent $1$ is the root agent. Similarly for any
$\alpha>\beta>0$, let $\mathbb{G}_{\alpha,\beta}^{N}$ denote the set of
directed graphs with $N$ nodes which are equal to a directed spanning tree for
which the corresponding Laplacian matrix $L$ is lower triangular with the
first row equal to zero with the property that
$\beta\leqslant\ell_{ii}\leqslant\alpha$ for $i=2,\ldots,N$.

###### Remark 1

Note that any graph which is a directed spanning tree will have a lower
triangular Laplacian matrix after a possible reordering of the agents.

For the graph defined by Definition 1, we know the Laplacian matrix $L$ is
lower triangular with the first row identical to zero. Therefore, we have

$L=\begin{pmatrix}0&0&0&\cdots&0\\\ \ell_{21}&\ell_{22}&0&\cdots&0\\\
\ell_{31}&\ell_{32}&\ell_{33}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&0\\\
\ell_{N1}&\cdots&\ell_{N,N-2}&\ell_{N,N-1}&\ell_{N,N}\end{pmatrix}.$

Since the graph is equal to a directed spanning tree, there are in every row
(except the first one) exactly two elements unequal to $0$.

Our goal is to achieve delayed state synchronization among agents in a MAS,
that is

$\lim_{t\to\infty}\,\left[x_{i}(t)-x_{j}(t-\tau_{ij})\right]=0,$ (4)

for all $i,j\in\\{1,\ldots,N\\}$.

For the MAS (2), we formulate delayed state synchronization problems as
follows.

###### Problem 1

Consider a MAS described by agents (2) and (3) associated with a directed
graph $\mathcal{G}\in\mathbb{G}_{\beta}^{N}$ is equal to a spanning tree,
where $\mathbb{G}_{\beta}^{N}$ is defined in Definition 1. The _delayed state
synchronization problem_ with a set of graphs $\mathbb{G}_{\beta}^{N}$ in the
presence of unknown, nonuniform, arbitrarily large communication delays is to
find a distributed static protocol of the type,

$u_{i}(t)=F\zeta_{i}(t),\quad(i=1,\ldots,N)$ (5)

for each agent such that (4) is satisfied for all $i,j\in\\{1,\ldots,N\\}$,
for any directed graph $\mathcal{G}\in\mathbb{G}_{\beta}^{N}$ and for any
communication delay $\tau_{ij}\in\mathbb{R}^{+}$.

###### Problem 2

Consider a MAS described by agents (2) and (3) associated with a directed
graph $\mathcal{G}\in\mathbb{G}_{\alpha,\beta}^{N}$ is equal to a spanning
tree, where $\mathbb{G}_{\alpha,\beta}^{N}$ is defined in Definition 1. The
_delayed state synchronization problem_ with a set of graphs
$\mathbb{G}_{\alpha,\beta}^{N}$ in the presence of unknown, nonuniform,
arbitrarily large communication delays is to find a distributed dynamic
protocol of the type,

$\left\\{\;\begin{array}[]{cl}\dot{\chi}_{i}(t)&=A_{c}\chi_{i}(t)+B_{c}\zeta_{i}(t),\\\
u_{i}(t)&=C_{c}\chi_{i}(t)+D_{c}\zeta_{i}(t),\end{array}\right.$ (6)

for each agent such that (4) is satisfied for all $i,j\in\\{1,\ldots,N\\}$,
for any directed graph $\mathcal{G}\in\mathbb{G}_{\beta}^{N}$ and for any
communication delay $\tau_{ij}\in\mathbb{R}^{+}$.

##  3 Delayed state synchronization for homogeneous MAS with communication
delays

In this section, we will give delayed state synchronization results based on
algebraic Riccati equation for full- and partial-state coupling.

###  3.1 Full-state coupling

Firstly, we consider full-state coupling (i.e., $C=I$). We define

$\tilde{x}_{i}(t)=x_{i}(t+\bar{\tau}_{i,1})$

where $\bar{\tau}_{i,1}$ denotes the sum of delays from agent $i$ to the root
(agent $1$) based on its path, and
$\tau_{ij}=\bar{\tau}_{i,1}-\bar{\tau}_{j,1}$. Then, we have

$\displaystyle\tilde{\zeta}_{i}(t)$
$\displaystyle=\zeta_{i}(t+\bar{\tau}_{i,1})=\sum_{j=1}^{N}a_{ij}\left[x_{i}(t+\bar{\tau}_{i,1})-x_{j}(t+\bar{\tau}_{i,1}-\tau_{ij})\right]$
$\displaystyle=\sum_{j=1}^{N}a_{ij}(\tilde{x}_{i}(t)-\tilde{x}_{j}(t))$

and $\tilde{u}_{i}(t)=u_{i}(t+\bar{\tau}_{i,1})$. Especially, we have
$\bar{\tau}_{1,1}=0$, i.e., $\tilde{x}_{1}(t)=x_{1}(t)$ and
$\tilde{u}_{1}(t)=0$ (since $\tilde{\zeta}_{1}(t)=\zeta_{1}(t)=0$). Thus, (2),
(3) and (5) yield:

$\dot{\tilde{x}}_{i}(t)=A{\tilde{x}}_{i}(t)+\sum_{j=1}^{N}\ell_{ij}BF\tilde{x}_{j}(t)$
(7)

Thus, our synchronization objective can be expressed as

$\lim_{t\to\infty}\left[\tilde{x}_{i}(t)-\tilde{x}_{j}(t)\right]=0.$ (8)

We define

${\tilde{x}}(t)=\begin{pmatrix}{\tilde{x}}_{1}(t)\\\ \tilde{x}_{2}(t)\\\
\vdots\\\ \tilde{x}_{N}(t)\end{pmatrix}.$

We have

$\dot{\tilde{x}}(t)=\left(I\otimes A+L\otimes(BF)\right)\tilde{x}(t).$ (9)

This is referred to as our _delay-free transformation_.

Further, let

$\eta_{1}(t)=\tilde{x}_{1}(t),\text{ and
}\eta_{i}(t)=\tilde{x}_{i}(t)-\tilde{x}_{1}(t)\text{ with }i=2,\ldots,N,$

we have

$\eta=\begin{pmatrix}\eta_{1}\\\ \eta_{2}\\\ \vdots\\\
\eta_{N}\end{pmatrix}=\begin{pmatrix}\tilde{x}_{1}(t)\\\
\tilde{x}_{2}(t)-\tilde{x}_{1}(t)\\\ \vdots\\\
\tilde{x}_{N}(t)-\tilde{x}_{1}(t)\end{pmatrix}=(T\otimes I)\tilde{x}(t).$

for some suitable defined matrix $T$. Thus, based on the above transformation
$T$, we obtain new expression of (9),

$\dot{\eta}(t)=\left(I_{N}\otimes A+L_{Q}\otimes(BF)\right)\eta(t)$ (10)

where

$\displaystyle L_{Q}=TLT^{-1}$ $\displaystyle=\begin{pmatrix}0&0\\\
0&L_{Q}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}0&0&0&\cdots&0\\\
0&\ell_{22}&0&\cdots&0\\\ 0&\ell_{32}&\ell_{33}&\ddots&\vdots\\\
\vdots&\vdots&\ddots&\ddots&0\\\
0&\ell_{N2}&\cdots&\ell_{N,N-1}&\ell_{N,N}\end{pmatrix},$ (11)

where $L_{Q}$ is a positive-definite lower triangular matrix.

Obviously, due to the structure of $L_{Q}$, the synchronization of (9) is
equivalent to the asymptotic stability of the following $N-1$ subsystems,

$\dot{\eta}_{i}(t)=(A+\ell_{ii}BF)\eta_{i}(t),\,\ i=2,\ldots,N.$ (12)

If (12) is globally asymptotically stable for $i=2,\ldots,N$, we see from the
above that $\eta_{i}(t)\rightarrow 0$ for $i=2,\ldots,N$. This implies that

$\tilde{x}(t)-(T^{-1}\otimes I)\begin{pmatrix}\eta_{1}(t)\\\ 0\\\ \vdots\\\
0\end{pmatrix}\rightarrow 0.$

Note that the first column of $T^{-1}$ is equal to the vector $\mathbf{1}$ and
therefore

$\tilde{x}_{i}(t)-\eta_{1}(t)\rightarrow 0$

for $i=1,\ldots,N$. This implies that we achieve state synchronization.

Conversely, suppose that the network (12) reaches state synchronization. In
this case, we shall have

$\tilde{x}(t)-\mathbf{1}\otimes\tilde{x}_{1}(t)\to 0$

for all initial conditions. Then
$\eta(t)-(T\mathbf{1})\otimes\tilde{x}_{1}(t)\to 0$. Since $\mathbf{1}$ is the
first column of $T$, we have

$T\mathbf{1}=\begin{pmatrix}1\\\ 0\\\ \vdots\\\ 0\end{pmatrix}.$

Therefore, $\eta(t)-(T\mathbf{1})\otimes\tilde{x}_{1}(t)\to 0$ implies that
$\eta_{1}(t)-\tilde{x}_{1}(t)\to 0$ and $\eta_{i}(t)\to 0$ for $i=2,\ldots,N$
for all initial conditions.

Thus, we obtain the following lemma.

###### Lemma 1

The MAS (9) achieves state synchronization if and only if the system (12) is
globally asymptotically stable for $i=2,3,\ldots,N$. The synchronized
trajectory converses to the trajectory of the root agent.

We let

$w=\begin{pmatrix}1&0&\cdots&0\end{pmatrix}$

be the normalized left eigenvector associated with the zero eigenvalue of $L$.
Then, from Lemma 1, we have

$\dot{\eta}_{1}(t)=A\eta_{1}(t),\quad\eta_{1}(0)=\tilde{x}_{1}(0),$ (13)

In other words, $\eta_{1}(0)$ is the initial condition of the root agent.
Therefore, the synchronized trajectory given by (13) yields that the
synchronized trajectory is given by

$x_{s}(t)=e^{At}\tilde{x}_{1}(0)=e^{At}x_{1}(0),$ (14)

which is the trajectory of the root agent and delay-free. Therefore, the root
agent is sometimes referred to as the leader.

#### Protocol design:

For full-state coupling, we design a parameterized static protocol of the
form:

$u_{i}(t)=-\rho B^{\mbox{\tiny T}}P\zeta_{i}(t),$ (15)

where $P>0$ is the unique solution of the continuous-time algebraic Riccati
equation,

$A^{\mbox{\tiny T}}P+PA-PBB^{\mbox{\tiny T}}P+Q=0,$ (16)

with $Q>0$, and $\rho\geqslant\frac{1}{2\beta}$ with
$\ell_{ii}\geqslant\beta>0$.

Using an algebraic Riccati equation we can design a suitable protocol provided
$(A,B)$ is stabilizable. The synchronization based on protocol (15) is as
follows.

###### Theorem 1

Consider a MAS described by (2) and (3). Let any $\beta>0$ be given, and
consider the set of network graphs $\mathbb{G}_{\beta}^{N}$ with
$\ell_{ii}\geqslant\beta$.

If $(A,B)$ is stabilizable, then the state synchronization problem stated in
Problem 1 with $\mathbf{G}=\mathbb{G}_{\beta}^{N}$ is solvable. In particular,
the protocol (15) solves the state synchronization problem for any graph
$\mathcal{G}\in\mathbb{G}_{\beta}^{N}$ and $\tau_{ij}\in\mathbb{R}^{+}$.
Moreover, the synchronized trajectory is given by (14).

######  Proof 3.2.

For protocol (15), we can obtain

$\tilde{u}_{i}(t)=-\rho B^{\mbox{\tiny T}}P\tilde{\zeta}_{i}(t).$

By using the delay-free transformation, it means that we only need to prove
that the system

$\dot{z}(t)=(A-\ell_{ii}\rho BB^{\mbox{\tiny T}}P)z(t)$ (17)

is asymptotically stable for any $\ell_{ii}$ that satisfies
$\ell_{ii}\geqslant\beta$.

We observe that

$\displaystyle(A-\ell_{ii}$ $\displaystyle\rho BB^{\mbox{\tiny
T}}P)^{\mbox{\tiny T}}P+P(A-\ell_{ii}\rho BB^{\mbox{\tiny T}}P)$
$\displaystyle=-Q-(2\ell_{ii}\rho-1)PBB^{\mbox{\tiny T}}P$
$\displaystyle\leqslant-Q.$

Therefore, $(A-\ell_{ii}\rho BB^{\mbox{\tiny T}}P)$ is Hurwitz stable for any
$\ell_{ii}\geqslant\beta>0$. Based on Lemma 1, the delayed state
synchronization result can be proved.

###  3.2 Partial-state coupling

In the following, we give the transformation for partial state coupling.
Similar to the case of full-state coupling, we have the following expression
for (2) and (3) by using our delay-free transformation,

$\left\\{\;\begin{array}[]{ll}\dot{\tilde{x}}_{i}(t)=A{\tilde{x}}_{i}(t)+B\tilde{u}_{i}(t)\\\
\tilde{\zeta}_{i}(t)=\sum_{j=1}^{N}a_{ij}(\tilde{y}_{i}(t)-\tilde{y}_{j}(t))\end{array}\right.$
(18)

with $\tilde{x}_{i}(t)=x_{i}(t-\bar{\tau}_{i,1})$,
$\tilde{y}_{i}(t)=y_{i}(t-\bar{\tau}_{i,1})$.

The MAS described by (2) and (3) after implementing the dynamic protocol (6)
is described by

$\left\\{\;\begin{array}[]{ccl}\dot{\bar{x}}_{i}(t)&=&{\begin{pmatrix}A&BC_{c}\\\
0&A_{c}\end{pmatrix}}\bar{x}_{i}(t)+{\begin{pmatrix}BD_{c}\\\
B_{c}\end{pmatrix}}\tilde{\zeta}_{i}(t),\\\
\tilde{y}_{i}(t)&=&{\begin{pmatrix}C&0\end{pmatrix}}\bar{x}_{i}(t),\\\
\tilde{\zeta}_{i}(t)&=&{\displaystyle\sum_{j=1}^{N}a_{ij}(\tilde{y}_{i}(t)-\tilde{y}_{j}(t)),}\end{array}\right.$
(19)

for $i=1,\ldots,N$, where

$\bar{x}_{i}(t)=\begin{pmatrix}\tilde{x}_{i}(t)\\\
\chi_{i}(t)\end{pmatrix},\qquad\bar{x}(t)=\begin{pmatrix}\bar{x}_{1}(t)\\\
\vdots\\\ \bar{x}_{N}(t)\end{pmatrix}$

Define

$\bar{A}=\begin{pmatrix}A&BC_{c}\\\
0&A_{c}\end{pmatrix},\bar{B}=\begin{pmatrix}BD_{c}\\\
B_{c}\end{pmatrix},\bar{C}=\begin{pmatrix}C&0\end{pmatrix}.$

Then, the overall dynamics of the $N$ agents can be written as

$\dot{\bar{x}}(t)=(I_{N}\otimes\bar{A}+L\otimes\bar{B}\bar{C})\bar{x}(t).$
(20)

So, this is the delay-free system obtained after our transformation for MAS
with unknown communication delays via partial state coupling.

The synchronization of (20) is equivalent to the asymptotic stability of the
following $N-1$ subsystems,

$\dot{\bar{\eta}}_{i}=(\bar{A}+\ell_{ii}\bar{B}\bar{C})\bar{\eta}_{i},\,\
i=2,\ldots,N.$ (21)

Similar to Lemma 1, we obtain the following lemma for partial state coupling.

######  Lemma 3.3.

The MAS (20) achieves state synchronization if and only if the system (21) is
globally asymptotically stable for $i=2,3,\ldots,N$. The synchronized
trajectory converges to the trajectory of the root agent of (20).

Meanwhile, we have

$\dot{\bar{\eta}}_{1}(t)=A\bar{\eta}_{1}(t),\qquad\bar{\eta}_{1}(0)=\bar{x}_{1}(0),$
(22)

Therefore, the synchronized trajectory given by (22) yields that the
synchronized trajectory is given by

$x_{s}(t)=(I\quad 0)e^{\bar{A}t}\bar{x}_{1}(0)=e^{At}{x}_{1}(0).$ (23)

#### Protocol design:

For partial-state coupling, we design a parameterized dynamic protocol of the
form:

$\left\\{\;\begin{array}[]{ccl}\dot{\chi}_{i}(t)&=&(A+KC)\chi_{i}(t)-K\zeta_{i}(t),\\\
u_{i}(t)&=&-{\beta}^{-1}B^{\mbox{\tiny
T}}{P}_{\delta}\chi_{i}(t),\end{array}\right.$ (24)

where $K$ is a matrix such that $A+KC$ is Hurwitz stable, ${P}_{\delta}>0$ is
the unique solution of the continuous-time algebraic Riccati equation,

$A^{\mbox{\tiny T}}{P}_{\delta}+{P}_{\delta}A-{P}_{\delta}BB^{\mbox{\tiny
T}}{P}_{\delta}+\delta I=0.$ (25)

with $\ell_{ii}\geqslant\beta>0$.

######  Theorem 3.4.

Consider a MAS described by (2) and (3). Let any $\alpha>\beta>0$ be given,
and consider the set of network graphs $\mathbb{G}_{\alpha,\beta}^{N}$ with
$\alpha\geqslant\ell_{ii}\geqslant\beta$.

If $(A,B)$ is stabilizable and $(A,C)$ is observable, then the state
synchronization problem stated in Problem 2 with
$\mathbf{G}=\mathbb{G}_{\alpha,\beta}^{N}$ is solvable. In particular, there
exists a $\delta^{*}>0$ such that for any $\delta\in(0,\delta^{*}]$, the
dynamic protocol (24) solves the state synchronization problem for any graph
$\mathcal{G}\in\mathbb{G}_{\alpha,\beta}^{N}$ and
$\tau_{ij}\in\mathbb{R}^{+}$. Moreover, the synchronized trajectory is given
by (23).

######  Proof 3.5.

For dynamic protocol (24), we have

$\left\\{\;\begin{array}[]{ccl}\dot{\tilde{\chi}}_{i}(t)&=&(A+KC)\tilde{\chi}_{i}(t)-K\tilde{\zeta}_{i}(t),\\\
\tilde{u}_{i}(t)&=&-{\beta}^{-1}B^{\mbox{\tiny
T}}{P}_{\delta}\tilde{\chi}_{i}(t),\end{array}\right.$

by using our delay-free transformation.

From (2), (3), and protocol (24), it means that we only need to prove that the
system

$\left\\{\;\begin{array}[]{ccl}\dot{\tilde{x}}(t)&=&A\tilde{x}(t)-\ell_{ii}\beta^{-1}BB^{\mbox{\tiny
T}}P_{\delta}\tilde{\chi}(t),\\\
\dot{\tilde{\chi}}(t)&=&(A+KC)\tilde{\chi}(t)-KC\tilde{x}(t),\end{array}\right.$
(26)

is asymptotically stable for $\alpha\geqslant\ell_{ii}\geqslant\beta$.

Define $e(t)=\tilde{x}(t)-\tilde{\chi}(t)$. The system (26) can be rewritten
in terms of $x$ and $e$ as

$\left\\{\;\begin{array}[]{ccl}\dot{\tilde{x}}(t)&=&(A-\ell_{ii}\beta^{-1}BB^{\mbox{\tiny
T}}P_{\delta})\tilde{x}(t)+\ell_{ii}\beta^{-1}BB^{\mbox{\tiny
T}}P_{\delta}e(t)\\\ \dot{e}(t)&=&(A+KC+\ell_{ii}\beta^{-1}BB^{\mbox{\tiny
T}}P_{\delta})e(t)-\ell_{ii}\beta^{-1}BB^{\mbox{\tiny
T}}P_{\delta}\tilde{x}(t).\end{array}\right.$ (27)

Since $\ell_{ii}\geqslant\beta$, we have

$(A-\ell_{ii}\beta^{-1}BB^{\mbox{\tiny T}}P_{\delta})^{\mbox{\tiny
T}}P_{\delta}+P_{\delta}(A-\ell_{ii}\beta^{-1}BB^{\mbox{\tiny
T}}P_{\delta})\\\ \leqslant-\delta I-P_{\delta}BB^{\mbox{\tiny T}}P_{\delta}.$

Define $V_{1}=\tilde{x}^{\mbox{\tiny T}}(t)P_{\delta}\tilde{x}(t)$ and
$v=-B^{\mbox{\tiny T}}P_{\delta}\tilde{x}(t)$. We can derive that

$\dot{V}_{1}\leqslant-\delta\|\tilde{x}(t)\|^{2}-\|v\|^{2}+\theta(\delta)\|e(t)\|\|v\|,$

where

$\theta(\delta)=\ell_{ii}{\beta}^{-1}\left\|B^{\mbox{\tiny
T}}P_{\delta}\right\|.$

Clearly, $\theta(\delta)\to 0$ as $\delta\to 0$.

Let $Q$ be the positive definite solution of the Lyapunov equation,

$(A+KC)^{\mbox{\tiny T}}Q+Q(A+KC)=-2I.$

Since $P_{\delta}\to 0$ and $\ell_{ii}$ is bounded (we have
$\ell_{ii}<\alpha$), there exists a $\delta_{1}$ such that for all
$\delta\in(0,\delta_{1}]$,

$(A+KC+\ell_{ii}\beta^{-1}BB^{\mbox{\tiny T}}P_{\delta})^{\mbox{\tiny
T}}Q+Q(A+KC+\ell_{ii}\beta^{-1}BB^{\mbox{\tiny T}}P_{\delta})\leqslant-I.$

Define $V_{2}=e^{\mbox{\tiny T}}(t)Qe(t)$. We get

$\dot{V}_{2}\leqslant-\|e(t)\|^{2}+M\|e(t)\|\|v\|$

where

$M=2\ell_{ii}{\beta}^{-1}\|QB\|.$

Define $V=4M^{2}V_{1}+2V_{2}$. Then

$\dot{V}\leqslant-4M^{2}\delta\|\tilde{x}(t)\|^{2}-2\|e(t)\|^{2}-4M^{2}\|v\|^{2}\\\
+(4M^{2}\theta(\delta)+2M)\|e(t)\|\|v\|.$

There exists a $\delta^{*}\leqslant\delta_{1}$ such that
$4M^{2}\theta(\delta)\leqslant 2M$ for all $\delta\in(0,\delta^{*}]$. Hence
for a $\delta\in(0,\delta^{*}]$,

$\dot{V}\leqslant-4M^{2}\delta\|\tilde{x}(t)\|^{2}-\|e(t)\|^{2}-(\|e(t)\|-2M\|v\|)^{2}.$

We conclude that the system (26) is asymptotically stable for
$\alpha\geqslant\ell_{ii}\geqslant\beta$. Based on Lemma 3.3, the delayed
state synchronization result can be proved.

##  4 Delayed output synchronization for heterogeneous MAS with communication
delays

In this section, we consider the following heterogeneous MAS,

$\begin{array}[]{cl}\dot{x}_{i}(t)&=A_{i}x_{i}(t)+B_{i}u_{i}(t),\\\
y_{i}(t)&=C_{i}x_{i}(t),\end{array}$ (28)

where $x_{i}(t)\in\mathbb{R}^{n_{i}}$, $u_{i}(t)\in\mathbb{R}^{m_{i}}$ and
$y_{i}(t)\in\mathbb{R}^{p}$ are the state, input and the output, respectively,
of agent $i$ for $i=1,\ldots,N$. Meanwhile, the communication network provides
agent $i$ with form of (3) including time delay $\tau_{ij}$. Similarly, we can
obtain _a delay-free transformation_ for (28) by letting
$\tilde{x}_{i}(t)=x_{i}(t+\bar{\tau}_{i,1})$,
$\tilde{y}_{i}(t)=y_{i}(t+\bar{\tau}_{i,1})$,
$\tilde{u}_{i}(t)=u_{i}(t+\bar{\tau}_{i,1})$, and
$\tilde{\zeta}_{i}(t)=\zeta_{i}(t+\bar{\tau}_{i,1})$. Meanwhile, MAS (28)
satisfies the following assumption.

###### Assumption 2

We assume that

  * •

($A_{i}$, $B_{i}$, $C_{i}$) is stabilizable and detectable.

  * •

All eigenvalues of $A_{i}$ are in the closed left half complex plane.

  * •

($A_{i}$, $B_{i}$, $C_{i}$, 0) is right-invertible.

  * •

($A_{i}$, $B_{i}$, $C_{i}$, 0) has no invariant zeros in the closed right-half
complex plane that coincide with the eigenvalues of $A_{1}$ (the system matrix
of the root agent).

Thus, we can transform (2), (3) and (5) as

$\left\\{\;\begin{array}[]{ll}\dot{\tilde{x}}_{i}(t)=A_{i}{\tilde{x}}_{i}(t)+B_{i}\tilde{u}(t)\\\
\tilde{y}_{i}(t)=C_{i}{\tilde{x}}_{i}(t)\\\
\tilde{u}_{i}(t)=\sum_{j=1}^{N}\ell_{ij}F_{i}\tilde{x}_{j}(t).\end{array}\right.$
(29)

Given the model (29) and graph which is a directed spanning tree, all earlier
approaches can also be applied to a heterogeneous MAS. Here we will give
design scheme to obtain the delayed state synchronization results based on
[6].

Since the graph is equal to a directed spanning tree, it only has a single
root which is Agent $1$. Moreover, $u_{1}=0$. In this section, our goal is
achieve delayed output synchronization

$\lim_{t\to\infty}\,\left[y_{i}(t)-y_{j}(t-\tau_{ij})\right]=0.$ (30)

For the heterogeneous MAS (28), we formulate delayed output synchronization
problem as follows.

######  Problem 4.6.

Consider a MAS described by agents (28) and (3) associated with a directed
graph $\mathcal{G}\in\mathbb{G}_{\beta}^{N}$ is equal to a spanning tree where
$\mathbb{G}_{\beta}^{N}$ is defined in Definition 1. The _delayed output
synchronization problem_ given the set of graph $\mathbb{G}_{\beta}^{N}$ in
the presence of unknown, nonuniform, arbitrarily large communication delays is
to find a distributed dynamic protocol of the type (6), for each agent such
that (30) is satisfied for all $i\in\\{1,\ldots,N\\}$, for any directed graph
$\mathcal{G}\in\mathbb{G}_{\beta}^{N}$ and for any communication delay
$\tau_{ij}\in\mathbb{R}^{+}$.

Then, we let

$e_{i}(t)=\tilde{y}_{i}(t)-\tilde{y}_{1}(t),$

and (29) can be rewritten as

$\begin{array}[]{rl}\begin{pmatrix}\dot{\tilde{x}}_{i}(t)\\\
\dot{\tilde{x}}_{1}(t)\end{pmatrix}&=\begin{pmatrix}A_{i}&0\\\
0&A_{1}\end{pmatrix}\begin{pmatrix}\tilde{x}_{i}(t)\\\
\tilde{x}_{1}(t)\end{pmatrix}+\begin{pmatrix}B_{i}\\\
0\end{pmatrix}\tilde{u}_{i}(t)\\\
e_{i}(t)&=\begin{pmatrix}C_{i}&-C_{1}\end{pmatrix}\begin{pmatrix}\tilde{x}_{i}(t)\\\
\tilde{x}_{1}(t)\end{pmatrix}\end{array}$ (31)

Then, let $O_{i}$ be the observability matrix,

$O_{i}=\begin{pmatrix}C_{i}&-C_{1}\\\ C_{i}A_{i}&-C_{1}A_{1}\\\
\vdots&\vdots\\\
C_{i}A_{i}^{n_{i}+n_{1}-1}&-C_{1}A_{1}^{n_{i}+n_{1}-1}\end{pmatrix}.$

Let $q_{i}$ denote the dimension of the null space of $O_{i}$, and define
$k_{i}=n_{1}-q_{i}$. Next, define $\Lambda_{i}^{u}\in\mathbb{R}^{n_{i}\times
q_{i}}$ and $\Phi_{i}^{u}\in\mathbb{R}^{n_{1}\times q_{i}}$ such that

$O_{i}\begin{pmatrix}\Lambda_{i}^{u}\\\
\Phi_{i}^{u}\end{pmatrix}=0,\qquad\operatorname{rank}\begin{pmatrix}\Lambda_{i}^{u}\\\
\Phi_{i}^{u}\end{pmatrix}=q_{i}.$

Because ($A_{i}$, $C_{i}$) is observable, $\Lambda_{i}^{u}$ and $\Phi_{i}^{u}$
have full column rank. Next, we define $\Lambda_{i}^{c}$ and $\Phi_{i}^{c}$
such that $\Lambda_{i}=[\Lambda_{i}^{u}\
\Lambda_{i}^{c}]\in\mathbb{R}^{n_{i}\times n_{i}}$ and
$\Phi_{i}=[\Phi_{i}^{u}\ \Phi_{i}^{c}]\in\mathbb{R}^{n_{1}\times n_{1}}$ are
nonsingular. Thus, we define a new state
$\bar{x}_{i}(t)\in\mathbb{R}^{n_{i}+k_{i}}$ as

$\bar{x}_{i}(t)=\begin{pmatrix}\tilde{x}_{i}(t)-\Lambda_{i}M_{i}\Phi_{i}^{-1}\tilde{x}_{1}(t)\\\
-N_{i}\Phi_{i}^{-1}\tilde{x}_{1}(t)\end{pmatrix},$

where

$M_{i}=\begin{pmatrix}I_{q_{i}}&0\\\
0&0\end{pmatrix}\in\mathbb{R}^{n_{i}\times n_{1}},\text{ and
}N_{i}=\begin{pmatrix}0&I_{k_{i}}\end{pmatrix}\in\mathbb{R}^{k_{i}\times
n_{1}}.$

Based on this new state variable $\bar{x}_{i}(t)$, we can transform (31) as

$\begin{array}[]{rl}\dot{\bar{x}}_{i}(t)&=\bar{A}_{i}\bar{x}_{i}(t)+\bar{B}_{i}\tilde{u}_{i}(t)=\begin{pmatrix}A_{i}&\bar{A}_{12}^{i}\\\
0&\bar{A}_{22}^{i}\end{pmatrix}\bar{x}_{i}(t)+\begin{pmatrix}B_{i}\\\
0\end{pmatrix}\tilde{u}_{i}(t)\\\
e_{i}(t)&=\bar{C}_{i}\bar{x}_{i}(t)=\begin{pmatrix}C_{i}&-\bar{C}_{2}^{i}\end{pmatrix}\bar{x}_{i}(t)\end{array}$
(32)

Further, we define ${\phi}_{i}(t)=\Xi_{i}\bar{x}_{i}(t)$ with

$\Xi_{i}=\begin{pmatrix}\bar{C}_{i}\\\ \bar{C}_{i}\bar{A}_{i}\\\ \vdots\\\
\bar{C}_{i}\bar{A}_{i}^{\bar{n}-1}\end{pmatrix},$

where $\bar{n}\geqslant n_{i}+n_{1}$. Note that $\Xi_{i}$ is not necessarily a
square matrix; however, due to observability of ($\bar{A}_{i}$,
$\bar{C}_{i}$), $\Xi_{i}$ is injective, which implies that
$\Xi_{i}^{\mbox{\tiny T}}\Xi_{i}$ is nonsingular. Meanwhile, we obtain a new
expression of (32),

$\begin{array}[]{l}\dot{\phi}_{i}(t)=(A_{o}+K_{o}^{i})\phi_{i}(t)+B_{o}\tilde{u}_{i}(t)\\\
e_{i}(t)=C_{o}\phi_{i}(t)\end{array}$ (33)

where

$A_{o}=\begin{pmatrix}0&I_{p(\bar{n}-1)}\\\
0&0\end{pmatrix}\in\mathbb{R}^{p\bar{n}\times p\bar{n}},\
C_{o}=\begin{pmatrix}I_{p}&0\end{pmatrix}\in\mathbb{R}^{p\times p\bar{n}},$

$B_{o}=\Xi_{i}\bar{B}_{i}$, and $K_{o}^{i}=\begin{pmatrix}0\\\
G_{i}\end{pmatrix}$ with
$G_{i}=\bar{C}_{i}\bar{A}_{i}^{\bar{n}}(\Xi^{\mbox{\tiny
T}}\Xi)^{-1}\Xi^{\mbox{\tiny T}}$.

Thus, we will design a dynamic protocol of the form

$\begin{array}[]{l}\dot{\hat{\phi}}_{i}(t)=(A_{o}+K_{o}^{i})\hat{\phi}_{i}(t)+B_{o}\tilde{u}_{i}(t)+H_{\varepsilon}Q_{\varepsilon}C_{o}^{\mbox{\tiny
T}}(\tilde{\zeta}(t)\\\ \hskip
133.72786pt-\sum_{j=1}^{N}\ell_{ij}C_{o}\hat{\phi}_{j}(t))\\\
\hat{\bar{x}}_{i}(t)=(\Xi^{\mbox{\tiny T}}\Xi)^{-1}\Xi^{\mbox{\tiny
T}}\hat{\phi}_{i}(t)\\\ \tilde{u}_{i}(t)=F_{i}\hat{\bar{x}}_{i}(t)\end{array}$
(34)

where

$F_{i}=\begin{pmatrix}K_{i}&\Gamma_{i}-K_{i}\Pi_{i}\end{pmatrix}$

$K_{i}$ is chosen such that $A_{i}+B_{i}K_{i}$ is Hurwitz stable. $\Gamma_{i}$
and $\Pi_{i}$ satisfy the following regulator equations,

$\begin{array}[]{l}\Pi_{i}\bar{A}_{22}^{i}=A_{i}\Pi_{i}+\bar{A}_{12}^{i}+B_{i}\Gamma_{i}\\\
C_{i}\Pi_{i}=\bar{C}_{2}^{i}\end{array}$ (35)

$H_{\varepsilon}=\operatorname{diag}(I_{p}\varepsilon^{-1},I_{p}\varepsilon^{-2},\ldots,I_{p}\varepsilon^{-\bar{n}})\in\mathbb{R}^{p\bar{n}\times
p\bar{n}}$, $Q_{\varepsilon}>0$ is the unique solution of the following
algebraic Riccati equation

$(A_{o}+K_{\varepsilon})Q_{\varepsilon}+Q_{\varepsilon}(A_{o}+K_{\varepsilon})^{\mbox{\tiny
T}}-2\beta Q_{\varepsilon}C_{o}^{\mbox{\tiny
T}}C_{o}Q_{\varepsilon}+I_{p\bar{n}}=0$ (36)

with $0<\beta\leqslant\ell_{ii}$ for $i=2,\ldots N$,

$K_{\varepsilon}=\begin{pmatrix}0\\\
\varepsilon^{\bar{n}+1}KH_{\varepsilon}\end{pmatrix}\in\mathbb{R}^{p\bar{n}\times
p\bar{n}},$

$K\in\mathbb{R}^{p\times p\bar{n}}$ is chosen matrix. And
($A_{o}+K_{\varepsilon}$, $C_{o}$) is always observable. Meanwhile,
$Q_{\varepsilon}\to 0$ when $\varepsilon\to 0$.

By designing the dynamic protocol (34), the following theorem can be obtained.

######  Theorem 4.7.

Consider a MAS described by agents (28) and (3) satisfying Assumption 2.

The delayed output synchronization problem stated in Problem 4.6 is solvable
for the set of graphs $\mathbb{G}_{\beta}^{N}$. In particular, protocol (34)
solves the state synchronization problem for any graph
$\mathcal{G}\in\mathbb{G}_{\beta}^{N}$ and any $\tau_{ij}\in\mathbb{R}^{+}$.

######  Proof 4.8.

To achieve output regulation our design includes two steps: observer design
and state feedback design.

Step 1) Observer design.

Let $L_{Q}$ be the matrix obtained by removing the first row and column of $L$
as already used in (11). Clearly, $L_{Q}$ is a lower triangular matrix and all
its eigenvalues are greater than $\beta$, i.e., $\ell_{ii}\geqslant\beta$.

Let $\tilde{\phi}_{i}(t)=\phi_{i}(t)-\hat{\phi}_{i}(t)$ for $i=2,\ldots,N$,
then we have

$\displaystyle\dot{\tilde{\phi}}_{i}(t)$
$\displaystyle=(A_{o}+K_{o}^{i})\tilde{\phi}_{i}(t)-H_{\varepsilon}Q_{\varepsilon}C_{o}^{\mbox{\tiny
T}}(\tilde{\zeta}(t)-\sum_{j=2}^{N}\ell_{ij}C_{o}\hat{\phi}_{j}(t))$
$\displaystyle=(A_{o}+K_{o})\tilde{\phi}_{i}(t)-(K_{o}-K_{o}^{i})\tilde{\phi}_{i}(t)$
$\displaystyle\hskip
113.81102pt-\sum_{j=2}^{N}\ell_{ij}H_{\varepsilon}Q_{\varepsilon}C_{o}^{\mbox{\tiny
T}}C_{o}\tilde{\phi}_{j}(t)$

with $K_{o}=[0,K^{\mbox{\tiny T}}]^{\mbox{\tiny T}}$.

Then, we set
$\xi_{i}(t)=\varepsilon^{-1}H_{\varepsilon}^{-1}\tilde{\phi}_{i}(t)$, we have

$\varepsilon\dot{\xi}_{i}(t)=(A_{o}+K_{\varepsilon})\xi_{i}(t)-K_{\varepsilon}^{i}\xi_{i}(t)-\sum_{j=2}^{N}\ell_{ij}Q_{\varepsilon}C_{o}^{\mbox{\tiny
T}}C_{o}\xi_{j}(t)$ (37)

where

$K_{\varepsilon}^{i}=\begin{pmatrix}0\\\
\varepsilon^{\bar{n}+1}(K_{o}-K_{o}^{i})H_{\varepsilon}\end{pmatrix}.$

Since $\bar{L}$ is a positive lower triangular matrix and the graph is
diverge, we have

$\displaystyle\varepsilon\dot{\xi}_{2}(t)$
$\displaystyle=(A_{o}+K_{\varepsilon}-\ell_{22}Q_{\varepsilon}C_{o}^{\mbox{\tiny
T}}C_{o})\xi_{2}(t)-K_{\varepsilon}^{2}\xi_{2}(t)$
$\displaystyle\varepsilon\dot{\xi}_{j}(t)$
$\displaystyle=(A_{o}+K_{\varepsilon}-\ell_{jj}Q_{\varepsilon}C_{o}^{\mbox{\tiny
T}}C_{o})\xi_{j}(t)-K_{\varepsilon}^{j}\xi_{j}(t)$ $\displaystyle\hskip
113.81102pt-\ell_{ij}Q_{\varepsilon}C_{o}^{\mbox{\tiny T}}C_{o}\xi_{i}(t)$

with $i<j$. That means we just need to prove
$A_{o}+K_{\varepsilon}-\ell_{ii}Q_{\varepsilon}C_{o}^{\mbox{\tiny
T}}C_{o}-K_{\varepsilon}^{i}$ is asymptotically stable.

Thus, based on (36) and $\ell_{ii}\geqslant\beta$, we have

$\displaystyle Q_{\varepsilon}^{-1}(A_{o}$
$\displaystyle+K_{\varepsilon}-\ell_{ii}Q_{\varepsilon}C_{o}^{\mbox{\tiny
T}}C_{o}-K_{\varepsilon}^{i})$
$\displaystyle+(A_{o}+K_{\varepsilon}-\ell_{ii}Q_{\varepsilon}C_{o}^{\mbox{\tiny
T}}C_{o}-K_{\varepsilon}^{i})^{\mbox{\tiny T}}Q_{\varepsilon}^{-1}$
$\displaystyle=$ $\displaystyle
Q_{\varepsilon}^{-1}\left[-I_{p\bar{n}}-2(\ell_{ii}-\beta)Q_{\varepsilon}C_{o}^{\mbox{\tiny
T}}C_{o}Q_{\varepsilon}\right.$
$\displaystyle\qquad\qquad\qquad\left.-K_{\varepsilon}^{i}Q_{\varepsilon}-Q_{\varepsilon}(K_{\varepsilon}^{i})^{\mbox{\tiny
T}}\right]Q_{\varepsilon}^{-1}$ $\displaystyle\leqslant$ $\displaystyle
Q_{\varepsilon}^{-1}\left[-I_{p\bar{n}}-K_{\varepsilon}^{i}Q_{\varepsilon}-Q_{\varepsilon}(K_{\varepsilon}^{i})^{\mbox{\tiny
T}}\right]Q_{\varepsilon}^{-1}.$ (38)

Since $K_{\varepsilon}^{i}$ is a high-order term of $\varepsilon$, there exist
a $\varepsilon^{*}$ such that

$2\|K_{\varepsilon}^{i}Q_{\varepsilon}\|\leqslant 1.$

It means we can obtain
$A_{o}+K_{\varepsilon}-\ell_{ii}Q_{\varepsilon}C_{o}^{\mbox{\tiny
T}}C_{o}-K_{\varepsilon}^{i}$ is asymptotically stable for any
$\varepsilon<\varepsilon^{*}$.

Thus, we obtain $\xi_{i}(t)\to 0$ as $t\to\infty$. That is to say, we have
$\phi_{i}(t)\to\hat{\phi}_{i}(t)$. Finally, we have

$\hat{\bar{x}}_{i}(t)\to\bar{x}_{i}(t),\text{ as }t\to\infty.$

Step 2). State feedback designing to solve output regulation.

The observer designing implies $\tilde{u}_{i}(t)=F_{i}{\bar{x}}_{i}(t)$ for
$t\to\infty$, i.e., control signal converses to a state feedback protocol.

Meanwhile, we need to achieve $\lim_{t\to\infty}e_{i}(t)=0$, it can be
considered as an output regulation problem. From (32), we define
$\bar{x}(t)=[(\bar{x}_{1}^{i}(t))^{\mbox{\tiny
T}},(\bar{x}_{2}^{i}(t))^{\mbox{\tiny T}}]^{\mbox{\tiny T}}$. Thus, we have
exosystem

$\dot{\bar{x}}_{2}^{i}(t)=\bar{A}_{22}^{i}\bar{x}_{2}^{i}(t)$

and the regulation system

$\dot{\bar{x}}_{1}^{i}(t)=A_{i}\bar{x}_{1}^{i}(t)+\bar{A}_{12}^{i}\bar{x}_{2}^{i}(t)+B_{i}\tilde{u}_{i}(t).$

Since ($A_{i}$, $B_{i}$) is stabilizable and the eigenvalues of
$\bar{A}_{22}^{i}$ are in closed right-half plane, we know protocol
$\tilde{u}_{i}(t)=F_{i}{\bar{x}}_{i}$ can solve this regulation problem based
on [19, Theorem 2.3.1], if the regulator equation (35) is solvable. From [19,
Corollary 2.5.1], the regulator equations are solvable if, for each $\lambda$
that is an eigenvalue of $\bar{A}_{22}^{i}$, the rank of Rosenbrock system
matrix

$\operatorname{rank}\begin{pmatrix}A_{i}-\lambda&B_{i}\\\
C_{i}&0\end{pmatrix}=n_{i}+p.$

The Rosenbrock system matrix has normal rank $n_{i}+p$ due to
rightinvertibility of the quadruple ($A_{i}$, $B_{i}$, $C_{i}$, 0) (see [18,
Property 3.1.6]). Since this quadruple has no invariant zeros coinciding with
eigenvalues of $A_{1}$ and the eigenvalues of $\bar{A}_{22}^{i}$ are a subset
of the eigenvalues of $A_{1}$, it follows that the rank of the Rosenbrock
system matrix is equal to the normal rank for each $\lambda$ that is an
eigenvalue of $\bar{A}_{22}^{i}$.

Thus, we can achieve the delayed output synchronization by
$\lim_{t\to\infty}e_{i}(t)=0$, i.e., achieve (30).

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