# Distributed Design of Glocal Controllers
via Hierarchical Model Decomposition

Hampei Sasahara  Member, IEEE, Takayuki Ishizaki  Member, IEEE, Jun-ichi Imura
Senior Member, IEEE, Henrik Sandberg  Member, IEEE, and Karl Henrik Johansson
Fellow, IEEE H. Sasahara, H. Sandberg, and K. H. Johansson are with the
Division of Decision and Control Systems, KTH Royal Institute of Technology,
Stockholm, SE-100 44 Sweden e-mail: {hampei, hsan<EMAIL_ADDRESS>Ishizaki
and J. Imura are with the Graduate School of Engineering, Tokyo Institute of
Technology, Tokyo, 152-8552 Japan e-mail: {ishizaki,
<EMAIL_ADDRESS>work was supported by JST MIRAI Grant Number
18077648, Japan, JSPS KAKENHI, Japan Grant Number 18K13774, the Swedish
Research Council (grant 2016-00861), the Swedish Foundation for Strategic
Research (project CLAS), the Swedish Civil Contingencies Agency (project
CERCES).Manuscript received Xxx xx, 20xx; revised Xxx xx, 20xx.

###### Abstract

This paper proposes a distributed design method of controllers having a glocal
(global/local) information structure for large-scale network systems.
Distributed design, independent design of all subcontrollers that constitute a
structured controller, facilitates scalable controller synthesis. While
existing distributed design methods confine attention to the decentralized or
distributed information structures, this study addresses distributed design of
glocal-structured controllers. Glocal control exploits the nature that network
system’s behavior can typically be represented as a superposition of spatially
local fluctuations and global interarea oscillations by incorporating a global
coordinating subcontroller with local decentralized subcontrollers. The key
idea to distributed design of glocal controllers is to represent the original
network system as a hierarchical cascaded system composed of reduced-order
models representing the global and local dynamics, referred to as hierarchical
model decomposition. Distributed design is achieved by independently designing
and implementing subcontrollers for the reduced-order models while preserving
the cascade structure. This paper provides a condition for existence of the
hierarchical model decomposition, a specific representation of the
hierarchical system, a clustering method appropriate for the proposed
approach, and a robust extension. Numerical examples of a power grid evidence
the practical relevance of the proposed method.

Distributed design, glocal control, large-scale systems, model reduction,
network systems.

##  1 Introduction

The recent development of cyber and physical technologies facilitates large-
scale dynamical systems, but increases also the complexity of the network
systems to be controlled [1, 2]. For large systems, instead of placing a
centralized controller operating the entire system, it is preferred to deploy
subcontrollers each of which monitors and actuates a small network unit while
communicating with the others and thereby reducing the control structure
complexity. A representative information structure is the decentralized
structure, where each subcontroller measures output signals within the
assigned local network and transmits its control signals to local actuators.
Another common structure is the distributed structure, which allows
communication among the subcontrollers with their neighborhood for
accomplishing more complicated tasks through coordination and cooperation. By
virtue of their sparse communication topology, implementation of such
structured controllers is scalable, which leads to their broad applicability
to large-scale systems [3, 4, 5].

Nevertheless, for most conventional methods in the literature, design of such
structured controllers is _not_ necessarily scalable owing to its implicit
philosophy of _centralized design_ , where a unique authority designs the
entire controller for a fixed network system. In practical large-scale
systems, there are often multiple subcontroller designers, each of whom
designs and implements a subcontroller according to her control policy
independently of the others. For example, a power grid is governed by multiple
companies each of whom is responsible for managing a subgrid. Accordingly,
each controller for frequency regulation is independently designed and
operated by each company [6]. In consequence, even if a controller is
optimally designed for the grid at some time instant, the control performance
is no longer guaranteed once one of the companies changes its control policy.

As the opposite of centralized design, the notion of _distributed design_ ,
where each subcontroller is designed independently of the others, has been
introduced [7]. Despite its practical importance, few studies on distributed
design are available in the literature owing to the technical difficulty that
each subcontroller must be designed to be capable of handling variations of
the other subcontrollers. To overcome this obstacle, several sophisticated
distributed design methods have been proposed over the last decade. As a
distributed design method of decentralized controllers, retrofit control has
been proposed [8, 9, 10]. Distributed design methods of distributed
controllers having a general communication topology have also been proposed[7,
11, 12, 13, 14, 15, 16, 17, 18].

With the aforementioned background, this study addresses the distributed
design problem of controllers having a specific _glocal (global/local)_
structure. Glocal control, originally proposed in [19], employs a structured
controller inspired by the fact that network system’s behaviors can typically
be represented as a superposition of spatially global and local behaviors. For
instance, the behavior of a power grid can be decomposed into global interarea
oscillations and local fluctuations [20]. To utilize this nature, a global
coordinating subcontroller is combined with local decentralized subcontrollers
in the glocal control framework.

The objective of this paper is to develop a distributed design method for
glocal controllers. To this end, we introduce _hierarchical model
decomposition_ , a hierarchical cascaded representation whose upstream and
downstream parts stand for local and global reduced-order models,
respectively. Hierarchical model decomposition is an alternative
representation of the original network system to be controlled. Our
fundamental idea is to design and implement subcontrollers for the reduced-
order models while preserving the cascade structure of hierarchical model
decomposition. Owing to the cascade structure, the stability of the entire
closed-loop system can be guaranteed as long as every reduced-order model is
stabilized by its corresponding subcontroller. In this paper, we resolve
technical issues related to this idea.

The main contributions of this paper are outlined as follows: First, we
propose a systematic method of distributed design of glocal controllers based
on hierarchical model decomposition. Specifically, we provide a necessary and
sufficient geometric condition on the existence of a hierarchical model
decomposition. We further derive an implicit representation of all
hierarchical model decompositions through linear matrix equations and show how
the designed control policy can be implemented based on a functional observer.
Second, we develop a clustering algorithm that produces clusters appropriate
for the proposed method based on a greedy approach. Third, we extend the
framework to the case where no exact hierarchical model decompositions exist.
To handle this situation, we introduce a robust hierarchical model
decomposition with approximation error, where also the error dynamics is
decomposed into a hierarchical form. Fourth and finally, we illustrate the
potential impact of our theoretical findings through practical examples. In
particular, we design a glocal controller via the proposed approach for the
48-machine NPCC (Northeast Power Coordinating Council) system [21], a model of
the power grid in New York and neighboring areas. Preliminary versions of this
work have been presented in [22, 23], but they did not include the clustering
algorithm, the robust extension, and the power grid example.

### Related work

A few studies on distributed design of structured controllers can be found.
Before the notion of distributed design in [7], similar problems have been
discussed in [12, 13]. The underlying idea is to reduce the norm of the input-
output maps relevant to interactions among the subsystems and to guarantee the
overall stability based on the small-gain theorem. Distributed design of
distributed controllers guaranteeing bounded-input bounded-output stability
has been proposed in [14] on the premise that interaction signals are bounded.
Different from the small-gain approaches, retrofit control has been proposed
for distributed design of decentralized controllers [8, 9, 10]. Regarding
distributed information structure, deadbeat control [7, 11], integral
quadratic constraint approach [15], passivity-based approach [16], and system-
level synthesis [17, 18] approaches have been proposed.

Glocal control has been introduced in [19] based on [24, 25]. One of its
features is hierarchical structure with spatial multiple-resolutions. The idea
of glocal control is to implement multi-resolved subcontrollers. Especially in
power system control, considerable efforts have been devoted to designing
controllers having hierarchical structure [26]. In the classical approach,
referred to as multi-level control [27, 28, 29], the control signal is
decomposed for each machine into two components generated by subcontrollers at
global and local levels. More recently, wide-area control [30, 31] has
attracted attention along with the advancement of wide-area measurement system
technology with sophisticated phasor measurement units. Accordingly, several
applications based on wide-area control have been proposed, e.g., [32, 33, 34,
35]. However, distributed design of the hierarchical structured controllers
has not been discussed so far in the power systems literature.

Finally, we emphasize that hierarchical model decomposition, which provides an
equivalent system representation composed of reduced-order models, cannot be
obtained with standard model reduction techniques, such as the projection-
based model reduction [36] or the singular perturbation method through
coordinate transformation [37]. Indeed, our idea relies on modification of the
state space, which can make the dimension of the proposed representation
larger than that of the original system, although the models used for
designing each subcontroller are decomposed.

### Organization and Notation

In Sec. 2, we present an illustration of our proposed method using a simple
second-order network system example, and subsequently, the problem is
formulated. Sec. 3, where the proposed distributed design via hierarchical
model decomposition is presented, gives the main technical results related to
our approach. Based on these findings, Sec. 4 develops a clustering algorithm
that produces clusters appropriate for the proposed design framework. In Sec.
5, we propose a robust extension of the proposed method. Sec. 6 verifies the
theoretical findings and demonstrates its practical effectiveness through
numerical examples of power grids. Finally, Sec. 7 draws conclusion.

We denote the set of real numbers by $\mathbb{R}$, the $n$-dimensional
identity matrix by $I_{n}$, the $n\times m$ zero matrix by $0_{n\times m}$,
the $n$-dimensional all-ones vector by $\mathds{1}_{n}$, the vector where
$x_{i}$ for $i\in\mathcal{I}$ are concatenated vertically by ${\rm
col}(x_{i})_{i\in\mathcal{I}}$, and the block diagonal matrix whose diagonal
blocks are composed of matrices $M_{i}$ for $i\in\mathcal{I}$ by ${\rm
D}({M_{i}})_{i\in\mathcal{I}}$. The subscript for the variables is omitted
when the dimension is clear from the context. Moreover, we denote the
Kronecker product by $\otimes$, the transpose and a pseudoinverse of a matrix
$M$ by $M^{\sf T}$ and $M^{\dagger}$, respectively, the direct sum and the sum
space of linear subspaces $\mathcal{X}$ and $\mathcal{Y}$ by
$\mathcal{X}\oplus\mathcal{Y}$ and $\mathcal{X}+\mathcal{Y}$, respectively,
the image space of a matrix $M$ by ${\rm im}\,{M}$, the set
$\\{y=Mx:x\in\mathcal{X}\\}$ for a matrix $M$ and a set $\mathcal{X}$ by
$M\mathcal{X}$, and the controllable subspace with respect to the pair $(A,B)$
by $\mathcal{R}(A,B)$. Appendix contains the proofs.

##  2 Problem Formulation

###  2.1 Motivating Example

This subsection provides an example that motivates us to introduce the glocal
control structure to be presented. We here consider the network system
illustrated in Fig. 1 representing a power grid [38] where each component’s
dynamics is given as a second-order system. For $k=1,\ldots,9$, each component
$\Sigma_{[k]}$ is given by

$\Sigma_{[k]}:m_{[k]}\ddot{\theta}_{[k]}+d_{[k]}\dot{\theta}_{[k]}+v_{[k]}+u_{[k]}=0,\quad
y_{[k]}=\omega_{[k]}$ (1)

where $\theta_{[k]}\in\mathbb{R}$ and
$\omega_{[k]}:=\dot{\theta}_{[k]}\in\mathbb{R}$ are the state,
$v_{[k]}\in\mathbb{R}$ is an interaction signal given by

$\textstyle{v_{[k]}=\sum_{l\in\mathcal{N}_{[k]}}\alpha_{[k,l]}(\theta_{[k]}-\theta_{[l]}),}$
(2)

$u_{[k]}\in\mathbb{R}$ and $y_{[k]}\in\mathbb{R}$ are the control input and
the measurement output, respectively, and $\mathcal{N}_{[k]}$ represents the
index set corresponding to the components connected to $\Sigma_{[k]}$. Let the
strength of the interaction among the components be given by
$\alpha_{[k,l]}=1,$ for any $l,k=1,\ldots,9.$ Note that the underlying graph
has a certain symmetry. For instance, the components $\Sigma_{1},\Sigma_{2},$
and $\Sigma_{3}$ are equivalent from the viewpoint of the other components.

Figure 1: Example: a network system where each component’s dynamics is given
as a second-order system. The parameters of the components represented by the
same shape are assumed to be identical.

The glocal control framework addresses network systems whose components are
partially synchronized. Noticing the symmetry of the graph structure, we
impose this property by assuming that symmetric components are partially
homogeneous. Specifically, the parameters of the components represented by the
same shape in Fig. 1 are assumed to be identical and given as

$(m_{[k]},d_{[k]})=\left\\{\begin{array}[]{cl}(3,0.4),&k=1,2,3,\\\
(2,0.3),&k=4,5,\\\ (1,0.2),&k=6,7,8,9.\end{array}\right.$ (3)

Then we can naturally arrange clusters as

$\mathcal{I}_{1}=\\{1,2,3\\},\quad\mathcal{I}_{2}=\\{4,5\\},\quad\mathcal{I}_{3}=\\{6,7,8,9\\},$
(4)

each of which contains homogeneous components. Taking a single cluster, we can
expect that its components’ responses to external signals injected into other
clusters are the same owing to the partial homogeneity and network symmetry.
Fig. 2 depicts the responses to a disturbance occurring inside
$\mathcal{I}_{1}$ at the initial time. As expected, the behaviors of
$\Sigma_{[4]}$ and $\Sigma_{[5]}$, which belong to $\mathcal{I}_{2}$, are
synchronized with each other. Similarly, the components in $\mathcal{I}_{3}$
are synchronized. This behavior can be interpreted as a global interarea
oscillation among the clusters. Thus, the entire state behavior in response to
the local disturbance in $\mathcal{I}_{1}$ can be regarded as a superposition
of global and local oscillations. Further, because the system is linear, the
same interpretation can apply for any disturbance by separating the
disturbance to the sum of local disturbances.

Figure 2: Free response of the second-order system in Fig. 1 in response to an
initial disturbance occurring inside $\mathcal{I}_{1}$. The top, middle, and
bottom correspond to $\mathcal{I}_{1},$ $\mathcal{I}_{2},$ and
$\mathcal{I}_{3}$, respectively.

This observation leads us to the glocal control structure. Let the control
input and the measurement output for the $i$th cluster be given by

$u_{i}:={\rm col}(u_{[k]})_{k\in\mathcal{I}_{i}},\quad y_{i}:={\rm
col}(y_{[k]})_{k\in\mathcal{I}_{i}},\quad i=1,2,3.$

Consider constructing the control input as a composition of global and local
inputs where the global input maintains the states in each cluster
synchronized. Because the control inputs in each cluster affect the
corresponding component’s state in the same manner, we take broadcast-type
global control inputs as the global input. Then the whole control input is
given by

$\left[\begin{array}[]{c}u_{1}\\\ u_{2}\\\
u_{3}\end{array}\right]=\hat{\mathbf{u}}_{0}+\left[\begin{array}[]{c}\hat{u}_{1}\\\
\hat{u}_{2}\\\
\hat{u}_{3}\end{array}\right],\quad\hat{\mathbf{u}}_{0}:=\left[\begin{array}[]{c}\mathds{1}_{3}\hat{u}_{0,1}\\\
\mathds{1}_{2}\hat{u}_{0,2}\\\ \mathds{1}_{4}\hat{u}_{0,3}\end{array}\right].$

Similarly, the global measurement signal reads as $y_{0}:=[\mathds{1}_{3}^{\sf
T}y_{1}\ \mathds{1}_{2}^{\sf T}y_{2}\ \mathds{1}_{4}^{\sf T}y_{3}]^{\sf T}$.
The global and local control inputs are basically used for suppressing global
and local oscillations, respectively. We design a glocal controller that
contains global and local subcontrollers associated with these signals.

###  2.2 System Description

Let us now introduce the general system description. Consider a linear time-
invariant interconnected system containing $N_{0}$ components

$\Sigma_{[k]}:\left\\{\begin{array}[]{cl}\dot{x}_{[k]}&=A_{[k]}x_{[k]}+L_{[k]}\sum_{l\in\mathcal{N}_{[k]}}v_{[l]}+B_{[k]}u_{[k]}\\\
y_{[k]}&=C_{[k]}x_{[k]}\end{array}\right.$ (5)

for $k=1,\ldots,N_{0}$ with the interaction given by

$\left[\begin{array}[]{c}v_{[1]}\\\ \vdots\\\
v_{[N_{0}]}\end{array}\right]=\left[\begin{array}[]{ccc}M_{[1,1]}&\cdots&M_{[1,N_{0}]}\\\
\vdots&&\vdots\\\
M_{[N_{0},1]}&\cdots&M_{[N_{0},N_{0}]}\end{array}\right]\left[\begin{array}[]{c}x_{[1]}\\\
\vdots\\\ x_{[N_{0}]}\end{array}\right]$ (6)

with matrices $M_{[k,l]}$ for $k,l=1,\ldots,N_{0}$. The signals
$x_{[k]},v_{[k]},u_{[k]},y_{[k]}$ denote the state, the interaction signal,
the control input, and the measurement signal, respectively, and the set
$\mathcal{N}_{[k]}$ denotes the neighborhood of $\Sigma_{[k]}$. We assume that
the dimensions of all the signals are identical with respect to $k$ and also
that $u_{[k]}$ and $y_{[k]}$ are one-dimensional to avoid notational burden.

Next we introduce the glocal control structure for the given interconnected
system. First, for grouping the components, we arrange $N$ clusters
$\mathcal{I}_{i}\subset\\{1,\ldots,N_{0}\\}$ for $i=1,\ldots,N$ that satisfy
$\mathcal{I}_{i}\cap\mathcal{I}_{j}=\emptyset$ for $i\neq j$ and
$\bigcup_{i=1}^{N}\mathcal{I}_{i}=\\{1,\ldots,N_{0}\\}$. The clusters should
be chosen such that the components in each cluster are synchronized in
response to disturbance occurring outside. The clustering problem will be
addressed in Sec. 4.

Given clusters, the subsystem regarding the $i$th cluster can be written as

$\Sigma_{i}:\left\\{\begin{array}[]{cl}\dot{x}_{i}&=A_{i}x_{i}+L_{i}\sum_{j\in\mathcal{N}_{i}}v_{j}+B_{i}u_{i}\\\
y_{i}&=C_{i}x_{i},\end{array}\right.$

where the state is defined by $x_{i}:={\rm
col}(x_{[k]})_{k\in\mathcal{I}_{i}}$ and the other signals are defined in a
similar manner, for $i=1,\ldots,N$. The interaction among them is given by

$\left[\begin{array}[]{c}v_{1}\\\ \vdots\\\
v_{N}\end{array}\right]=\underbrace{\left[\begin{array}[]{ccc}M_{1,1}&\cdots&M_{1,N}\\\
\vdots&&\vdots\\\
M_{N,1}&\cdots&M_{N,N}\end{array}\right]}_{=:M}\left[\begin{array}[]{c}x_{1}\\\
\vdots\\\ x_{N}\end{array}\right].$

We denote the dimension of the state and the input in $\mathcal{I}_{i}$ by
$n_{i}$ and $r_{i}$, respectively.

As in the motivating example, we let the control input be composed of global
and local control inputs. The following assumption is made.

###### Assumption 1

The input and output matrices in each cluster are identical, that is,

$B_{[k]}=B_{[l]},\quad C_{[k]}=C_{[l]},\quad\forall k,l\in\mathcal{I}_{i}$

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

Accordingly, we form the control input as

${\rm col}(u_{i})_{i=1}^{N}=\hat{\mathbf{u}}_{0}+{\rm
col}(\hat{u}_{i})_{i=1}^{N},\quad\hat{\mathbf{u}}_{0}:=E_{0}\hat{u}_{0}$

with $E_{0}:={\rm D}({\mathds{1}_{r_{i}}})_{i=1}^{N}$ where $\hat{u}_{0}$ and
$\hat{u}_{i}$ for $i=1,\ldots,N$ represent global and local control inputs,
respectively. Similarly, the global measurement signal is defined by
$y_{0}:=E_{0}^{\sf T}y.$ Note that when Assumption 1 does not hold, it
suffices to modify $E_{0}$ appropriately such that only the synchronized
behavior in each cluster is excited by the modified global control input
$E_{0}\hat{u}_{0}$.

The dynamics of _the clustered interconnected system_ is given by

$\left\\{\begin{array}[]{cl}\dot{x}&=Ax+P_{0}B_{0}\hat{u}_{0}+\sum_{i=1}^{N}P_{i}B_{i}\hat{u}_{i}\\\
y_{0}&=C_{0}P_{0}^{\sf T}x\\\ y_{i}&=C_{i}P_{i}^{\sf T}x\end{array}\right.$
(7)

where $x:={\rm col}(x_{i})_{i=1}^{N}$, $A:={\rm D}({A_{i}})_{i=1}^{N}+{\rm
D}({L_{i}})_{i=1}^{N}M$, and the matrices $P_{0}$ and $P_{i}$, which are
consistent in broadcasting and embedding matrices, are defined by

$P_{0}:=\left[\begin{array}[]{ccc}\mathds{1}_{r_{1}}\otimes
I_{n_{0,1}}&\cdots&0_{n_{1}\times n_{0,N}}\\\ \vdots&\ddots&\vdots\\\
0_{n_{N}\times n_{0,1}}&\cdots&\mathds{1}_{r_{N}}\otimes
I_{n_{0,N}}\end{array}\right],\,P_{i}:=\left[\begin{array}[]{c}0_{n_{1}\times
n_{i}}\\\ \vdots\\\ I_{n_{i}}\\\ \vdots\\\ 0_{n_{N}\times
n_{i}}\end{array}\right],$ (8)

and $n_{0,i}:=n_{i}/r_{i}$ is the dimension of the state of a subsystem in
$\mathcal{I}_{i}$. The matrices $B_{0}$ and $C_{0}$ are chosen such that

$P_{0}B_{0}={\rm D}({B_{i}})E_{0},\quad C_{0}P_{0}^{\sf T}=E_{0}^{\sf T}{\rm
D}({C_{i}}).$

Note that there always exist such $B_{0}$ and $C_{0}$ owing to Assumption 1.
Finally, let $K_{i}$ be a subcontroller to be designed corresponding to
$\hat{u}_{i}$ and $y_{i}$ for $i=0,1,\ldots,N$. The specific information
structure of the entire controller is discussed in Sec. 3.

###  2.3 Problem Formulation

Based on the system description, we consider distributed design of a glocal
controller, where each subcontroller can be designed independently of the
others. Let us introduce subcontroller sets $\mathcal{K}_{i}$ such that the
entire closed-loop system is internally stable for any combination of
subcontrollers in $\mathcal{K}_{i}$. Distributed design of the subcontrollers
is achieved by designing the $i$th subcontroller $K_{i}$ to be an element of
$\mathcal{K}_{i}$ in the sense that $K_{i}$ can be chosen independently of the
other subcontrollers.

###### Problem 1

Design a collection of subcontroller sets $\\{\mathcal{K}_{i}\\}_{i=0}^{N}$
such that the clustered interconnected system (7) with the subcontrollers
$K_{0},K_{1},\ldots,K_{N}$ is internally stable for any choice of a tuple

$(K_{0},K_{1},\ldots,K_{N})\in\mathcal{K}_{0}\times\mathcal{K}_{1}\times\cdots\times\mathcal{K}_{N}.$

In Problem 1, the subcontroller sets $\mathcal{K}_{i}$ are designed instead of
subcontrollers $K_{i}$ themselves. Note that a trivial solution can be given
as singletons $\mathcal{K}_{i}=\\{K_{i}\\}$ where the unique controller
stabilizes the entire system. However, this choice is unfavorable for
distributed design since the designed controller has no flexibility. In this
study, we seek for larger subcontroller sets.

Figure 3:  Hierarchical model decomposition of the second-order network system
of the motivating example. Top: block diagrams of the original network system
and the hierarchical system. Bottom: the responses of $\omega_{i}$ in
$\Sigma_{i}$ and its representation as a superposition of $\hat{\omega}_{i}$
in $\Xi_{i}$ and $\hat{\omega}_{0,i}$ in $\Xi_{0}$ for $i=1,2,3$.

In Sec. 3, we solve Problem 1, when the system and clusters are given in
advance. However, since the performance of the distributed design method
depends on the choice of clusters, clustering should be included in the design
process. Thus we also address the following problem in Sec. 4 based on the
results in Sec. 3.

###### Problem 2

Develop a clustering method that produces proper clusters for the proposed
distributed design method.

##  3 Distributed Design via Hierarchical Model Decomposition

###  3.1 Motivating Example Revisited

This subsection describes our basic idea for distributed design of a glocal
controller through the motivating example in Sec. 2.1. As observed, any local
disturbance occurring in a cluster excites only the synchronized behavior in
the other clusters. Thus behaviors of the network system in response to any
disturbance can be interpreted as a superposition of global and local
behaviors excited by the corresponding local disturbances. The key idea is to
derive a _hierarchical_ representation that explicitly describes these
behaviors.

Consider describing the state variables as a superposition of global and local
states, e.g.,

$\underbrace{\left[\begin{array}[]{c}\omega_{[1]}\\\ \omega_{[2]}\\\
\omega_{[3]}\end{array}\right]}_{=:\omega_{1}}=\underbrace{\left[\begin{array}[]{c}\hat{\omega}_{[1]}\\\
\hat{\omega}_{[2]}\\\
\hat{\omega}_{[3]}\end{array}\right]}_{=:\hat{\omega}_{1}}+\left[\begin{array}[]{c}1\\\
1\\\ 1\end{array}\right]\hat{\omega}_{0,1}$ (9)

where $\hat{\omega}_{[k]}$ for $k=1,2,3$ and $\hat{\omega}_{0,1}$ represent
the local and global behaviors of $\omega_{[k]}$ in the first cluster,
respectively. Similarly, we consider $\hat{\omega}_{i}$ and
$\hat{\omega}_{0,i}$ for the second and third clusters. The global variable
with respect to all clusters is denoted by $\hat{\omega}_{0}:={\rm
col}(\hat{\omega}_{0,i})_{i=1,2,3}$. The variables with respect to
$\theta_{i}$, denoted by $\hat{\theta}_{0}$ and $\hat{\theta}_{i}$, are also
prepared in a similar manner.

Now we consider the dynamics that should be followed by the global and local
variables in compliance with the superposition representation in (9). Because
such a dynamics is not necessarily unique, we can possibly choose a
representation that has a desirable property for our controller design.
Indeed, we impose a _hierarchical_ structure into the dynamics. We can show
that there exists a hierarchical system

$\left\\{\begin{array}[]{lcl}\Xi_{i}:&\left[\begin{array}[]{c}\dot{\hat{\theta}}_{i}\\\
\dot{\hat{\omega}}_{i}\end{array}\right]&=\hat{A}_{i}\left[\begin{array}[]{c}\hat{\theta}_{i}\\\
\hat{\omega}_{i}\end{array}\right]+B_{i}\hat{u}_{i},\quad i=1,2,3\\\
\Xi_{0}:&\left[\begin{array}[]{c}\dot{\hat{\theta}}_{0}\\\
\dot{\hat{\omega}}_{0}\end{array}\right]&=\hat{A}_{0}\left[\begin{array}[]{c}\hat{\theta}_{0}\\\
\hat{\omega}_{0}\end{array}\right]\displaystyle{+\sum_{i=1}^{N}\hat{R}_{i}}\left[\begin{array}[]{c}\hat{\theta}_{i}\\\
\hat{\omega}_{i}\end{array}\right]+B_{0}\hat{u}_{0}\end{array}\right.$

with certain system matrices such that the original states $\theta_{i}$ and
$\omega_{i}$ can be reproduced as a superposition of the states for any
control inputs as long as the initial condition is consistent. Although the
specific description of the matrices is omitted here, its general derivation
is discussed in Sec. 3.2. Block diagrams of the original network system and
the hierarchical system are illustrated at the top of Fig. 3.

The hierarchical representation can be interpreted as dynamics in response to
multiple local disturbances. The dynamics $\Xi_{0}$ stands for a reduced-order
global model of the original network system, while the dynamics of
$\Xi_{1},\Xi_{2},\Xi_{3}$ represent local models. In this sense, the
hierarchical representation decomposes the entire model into global and local
models. At the bottom of Fig. 3, the state trajectories without control in
response to a disturbance occurring in the first two clusters are depicted.
Accordingly, the local state $\hat{\omega}_{3}$ in the third cluster is not
excited at all, which is induced from the hierarchical structure. Moreover,
the original state trajectories can be represented as a superposition of the
states in the hierarchical system as shown at the bottom of this figure. In a
similar manner, this hierarchical decomposition can reproduce the original
state as a superposition of global and local states for any disturbance and
any control inputs.

Consider utilizing this decomposition for our problem. A block diagram of
$\Xi$ with control inputs is illustrated in Fig. 4a. From the hierarchical
structure, there are no feedback paths around the reduced-order models. Thus
we can guarantee stability of the whole system by attaching subcontrollers as
in Fig. 4b provided that each subcontroller stabilizes the corresponding
reduced-order system. Following this procedure, we can perform distributed
design of a glocal controller, where $\mathcal{K}_{i}$ is given as a set
containing stabilizing controllers for $\Xi_{i}$.

(a)

(b)

Figure 4: Block diagrams of the hierarchical model decomposition with control
signals.

In the rest of this section, we adopt the scheme explained above to the
general systems and then present some important properties for control design
and implementation.

###  3.2 Definition of Hierarchical Model Decomposition

The key idea of the proposed method is to represent the entire network system
as a hierarchical system consisting of reduced-order models. A definition of
the hierarchical model, referred to as _hierarchical model decomposition,_ is
given as follows.

######  Definition 1 (Hierarchical Model Decomposition)

Consider the hierarchical system $\Xi$ composed of

$\left\\{\begin{array}[]{lcl}\Xi_{i}:&\dot{\xi}_{i}&=\hat{A}_{i}\xi_{i}+B_{i}\hat{u}_{i},\quad
i=1,\ldots,N\\\
\Xi_{0}:&\dot{\xi}_{0}&=\hat{A}_{0}\xi_{0}+\sum_{i=1}^{N}\hat{R}_{i}\xi_{i}+B_{0}\hat{u}_{0}.\end{array}\right.$
(10)

The system $\Xi$ is said to be a hierarchical model decomposition of the
clustered interconnected system in (7) if

$\textstyle{x(t)=\sum_{i=1}^{N}P_{i}\xi_{i}(t)+P_{0}\xi_{0}(t),\quad\forall
t\geq 0}$ (11)

holds for arbitrary initial conditions and control inputs provided that
$x(0)=\sum_{i=1}^{N}P_{i}\xi_{i}(0)+P_{0}\xi_{0}(0)$.

If we can obtain a hierarchical model decomposition (10), then distributed
design can be achieved. The technical questions related to the decomposition
are as follows:

  1. 1.

Does there exist a hierarchical model decomposition for the given system and
clusters?

  2. 2.

How to obtain a specific representation of a hierarchical model decomposition
if it exists?

  3. 3.

How to implement the designed controller preserving the cascade structure?

Those questions are addressed in the remainder of this section.

###  3.3 Existence Condition and Implicit Representation

We first give a necessary and sufficient condition on existence of
hierarchical model decomposition from the viewpoint of controllable subspaces
in terms of given clusters.

######  Theorem 1 (Existence Condition)

Under Assumption 1, a hierarchical model decomposition of (7) exists if and
only if the condition

$\displaystyle\mathcal{R}(A,P_{i})\subset{\rm im}\,{P_{i}}+{\rm
im}\,{P_{0}},\quad i=1,\ldots,N$ (12a)
$\displaystyle\mathcal{R}(A,P_{0})\subset{\rm im}\,{P_{0}}$ (12b)

holds.

The condition in Theorem 1 can be interpreted as follows. Consider possible
effects of a disturbance and a control input injected into the $i$th cluster.
Condition (12a) indicates that state trajectories excited by the inputs are
restricted in the space of the right hand side of (12a). Consequently, the
behavior in response to the disturbance and control input can be represented
as a sum of global and local behaviors. Condition (12b) implies that ${\rm
im}\,{P_{0}}$ is an invariant subspace of $A$. Then the global control input
$\hat{u}_{0}$ in (7) can excite only global interarea behaviors restricted to
${\rm im}\,{P_{0}}$. The relationship between the controllable subspaces and
the image spaces of the matrices $\\{P_{i}\\}_{i=0}^{N}$ induces (10), which
has the special structure illustrated in Fig. 4a.

Based on Theorem 1, we derive a necessary and sufficient condition that $\Xi$
must satisfy to be a hierarchical model decomposition.

######  Theorem 2 (Implicit Representation)

Under Assumption 1, the system $\Xi$ in (10) is a hierarchical model
decomposition of (7) if and only if the condition

$\left\\{\begin{array}[]{l}AP_{i}-P_{0}\hat{R}_{i}-P_{i}\hat{A}_{i}=0,\quad
i=1,\ldots,N\\\ AP_{0}-P_{0}\hat{A}_{0}=0\end{array}\right.$ (13)

holds.

Theorem 2 gives an implicit representation of all hierarchical model
decompositions through the linear matrix equations (13), which can readily be
solved.

_Remark:_ _Retrofit control_ [8, 9] proposes a distributed design method of
decentralized controllers through the following hierarchical system
representation

$\left\\{\begin{array}[]{cl}\dot{\xi}_{i}&=A_{i}\xi_{i}+B_{i}\hat{u}_{i},\quad
i=1,\ldots,N\\\
\dot{\xi}_{0}&=A\xi_{0}+\sum_{i=1}^{N}(AP_{i}-P_{i}A_{i})\xi_{i}.\end{array}\right.$
(14)

The representation (14) can be obtained by choosing

$N=N_{0},\quad\mathcal{I}_{i}=\\{i\\},\quad\hat{A}_{i}=A_{i},\quad\hat{u}_{0}=0$

in the framework of this paper. Under this choice, the existence condition in
Theorem 1 always holds since ${\rm im}\,{P_{0}}$ contains the entire space.
Thus, any network systems can be represented in the form (14), through which
distributed design of local subcontrollers can be carried out. In this sense,
hierarchical model decomposition proposed in this paper can be interpreted as
an extension of hierarchical state-space expansion in [8].

###  3.4 Controller Implementation

We consider implementation of subcontrollers designed based on a hierarchical
model decomposition. Let $\hat{K}_{0},\ldots,\hat{K}_{N}$ be stabilizing
controllers for the subsystems in $\Xi$ obtained through Theorem 2. Then the
closed-loop systems

$\left\\{\begin{array}[]{cl}\dot{\xi}_{i}&=\hat{A}_{i}\xi_{i}+B_{i}\hat{u}_{i}\\\
\hat{u}_{i}&=\hat{K}_{i}(C_{i}\xi_{i})\end{array},\quad i=0,\ldots,N\right.$
(15)

are internally stable. Consequently, the entire network system with those
control inputs is also stabilized from the cascade structure of $\Xi$ and the
identity (11). However, since the virtual variable $\xi_{i}$ is inaccessible,
the control inputs in (15) cannot be created straightforwardly. The aim of
this subsection is to develop an implementation method to circumvent this
problem.

First, notice that the global measurement signal $y_{0}$ can be used instead
of $C_{0}\xi_{0}$ in (15) for stabilizing $\Xi_{0}$, the downstream part of
$\Xi$. Suppose that $\hat{K}_{0}$ is a linear controller and $y_{0}$ is used
as a feedback signal to $\hat{K}_{0}$. Then the control input can be
represented by

$\textstyle{\hat{u}_{0}=\hat{K}_{0}(y_{0})=\hat{K}_{0}(C_{0}\xi_{0})+\hat{K}_{0}\left(C_{0}P_{0}^{\sf
T}\sum_{i=1}^{N}P_{i}\xi_{i}\right).}$

From the perspective of the closed-loop system composed of $\Xi_{0}$ and
$\hat{K}_{0}$, the second term can be regarded as an external input signal
from the upstream parts. Because there is no feedback path from $\Xi_{0}$ to
$\Xi_{i}$ for $i=1,\ldots,N$, the stability of the downstream part can be
guaranteed even if the original global measurement signal $y_{0}$ is used as
an alternative to $C_{0}\xi_{0}$. For this reason, we henceforth confine
attention only to the upstream parts associated with $\xi_{1},\ldots,\xi_{N}$.

The idea for implementation is to estimate $C_{i}\xi_{i}$ for $i=1,\ldots,N$
using functional observers [39, 40]. A functional observer is a state observer
producing a signal that tracks partial states of interest. For designing a
functional observer, we consider the particular information structure given by

$\hat{u}_{0}=K_{0}(y_{0}),\quad\hat{u}_{i}=K_{i}(y_{i},\hat{u}_{0}),\quad
i=1,\ldots,N.$

This structure is illustrated in Fig. 5, where the whole controller has a star
topology. The global subcontroller transmits its control input to all local
subcontrollers, while the local subcontrollers do not directly communicate
with each other.

Figure 5: The information structure of the glocal controller to be designed,
where local subcontrollers $K_{1},K_{2},K_{3}$ receive the global control
input from the global subcontroller $K_{0}$ and all local subcontrollers do
not directly communicate with each other.

The following theorem claims that the entire network system can be stabilized
by combining functional observers of $C_{1}\xi_{1},\ldots,C_{N}\xi_{N}$ with
the subcontrollers (15).

######  Theorem 3 (Stabilization through Functional Observers)

Assume that $\Xi$ is a hierarchical model decomposition of (7) and that the
dynamical systems

$\Phi_{i}:\left\\{\begin{array}[]{cl}\dot{\phi}_{i}&=\mathbf{A}_{i}\phi_{i}+\mathbf{B}_{i}\hat{u}_{i}+\mathbf{D}_{i}\hat{u}_{0}+\mathbf{E}_{i}y_{i}\\\
\psi_{i}&=\mathbf{C}_{i}\phi_{i}+\mathbf{F}_{i}y_{i}\end{array}\right.$ (16)

for $i=1,\ldots,N$ are functional observers of $C_{i}\xi_{i}$, i.e.,

$\textstyle{\lim_{t\rightarrow+\infty}\left(C_{i}\xi_{i}(t)-\psi_{i}(t)\right)=0}$

holds for any initial condition and inputs. Then the glocal controller
composed of

$\hat{u}_{0}=\hat{K}_{0}y_{0},\quad\hat{u}_{i}=\hat{K}_{i}\psi_{i},\quad
i=1,\ldots,N$ (17)

with (16) stabilizes the clustered interconnected system (7) for any
$\hat{K}_{0},\ldots,\hat{K}_{N}$ such that the closed-loop systems (15) are
internally stable.

Theorem 3 implies that the stability of the original system can be guaranteed
using the estimated signal $\psi_{i}$ instead of $C_{i}\xi_{i}$ itself.

We next provide a specific functional observer for (16). For simplicity, we
consider the case where the interaction signal $v_{i}$ can be measured by the
$i$th local subcontroller in addition to the local measurement signal $y_{i}$
for $i=1,\ldots,N$.

###### Proposition 1

Assume that $A_{i}$ and $\hat{A}_{i}$ are stable. Then

$\Phi_{i}:\left\\{\begin{array}[]{cl}\dot{\phi}_{i}&=\hat{A}_{i}\phi_{i}+(A_{i}-\hat{A}_{i})\hat{x}_{i}+L_{i}v_{i}+P_{i}^{\sf
T}P_{0}B_{0}\hat{u}_{0}\\\
\dot{\hat{x}}_{i}&=A_{i}\hat{x}_{i}+B_{i}\hat{u}_{i}+L_{i}v_{i}+P_{i}^{\sf
T}P_{0}B_{0}\hat{u}_{0}\\\ \psi_{i}&=-C_{i}\phi_{i}+y_{i}\end{array}\right.$
(18)

is a functional observer of $C_{i}\xi_{i}$.

The idea to construct the observer in Proposition 1 is as follows. Notice that
$y_{i}=C_{i}\xi_{i}+C_{i}P_{i}^{\sf T}P_{0}\xi_{0}$ implies that if we use
$y_{i}$ as a feedback signal then the cascade structure of $\Xi$ is collapsed
because of the arising feedback path from $\Xi_{0}$ to $\Xi_{i}$. To extract
$C_{i}\xi_{i}$ alone, consider estimating $P_{i}^{\sf T}P_{0}\xi_{0}$ using
$v_{i},\hat{u}_{0},$ and an estimated local state $\hat{x}_{i}$ generated
through the second differential equation in (18). Because the dynamics of
$P_{i}^{\sf T}P_{0}\xi_{0}$ can be represented by the first differential
equation in (18), we use $\phi_{i}$ as a replacement of $P_{i}^{\sf
T}P_{0}\xi_{0}$, which induces $\psi_{i}=y_{i}-C_{i}\phi_{i}$ as an estimation
of $C_{i}\xi_{i}=y_{i}-C_{i}P_{i}^{\sf T}P_{0}\xi_{0}$. We stress that the
requirement for each functional observer is independent of the other
subsystems, subcontrollers, and functional observers. Hence, design of
functional observers does not hinder distributed design of the glocal
controller.

Our solution to Problem 1 can be summarized by the following procedure.

  1. 1.

Identify existence of a hierarchical model decomposition for the given
clustered interconnected system based on Theorem 1.

  2. 2.

Construct a hierarchical model decomposition based on Theorem 2.

  3. 3.

Design $\mathcal{K}_{i}$ to be the set whose elements are the controllers
composed of the functional observer (18) and an internal controller that
stabilizes (15) for $i=0,\ldots,N$.

In the above procedure, it is assumed that the clusters are given in advance
and satisfy the existence condition in Theorem 1. In Sec. 4, we develop a
clustering method that provides a solution to Problem 2 based on the results
in this section. Moreover, an extension of the proposed approach to the case
where the existence condition is _not_ satisfied is discussed in Sec. 5.

##  4 Clustering Method

###  4.1 Clustering Algorithm

This section addresses Problem 2. For the given interconnected system composed
of (5) and the interaction (6), we find a cluster set for which there exists a
hierarchical model decomposition of the clustered interconnected system (7).
Before proceeding, we state an alternative assumption to Assumption 1.
Instead, the following assumption is made.

###### Assumption 2

The input and output matrices of all components are identical, that is,
$B_{[k]}=B_{[l]}$ and $C_{[k]}=C_{[l]}$ for any $k,l=1,\ldots,N_{0}$.

It is clear that if Assumption 2 holds then Assumption 1 is satisfied for any
clusters. When Assumption 2 does not hold, it suffices to modify the global
signals as explained in Sec. 2.2 after the clustering.

The desirable clusters are characterized through (12a) and (12b). Since
cluster sets that fulfill the conditions are not unique, we have to make a
criterion to choose. A trivial cluster set is given by
$\mathcal{I}_{i}=\\{i\\}$ for $i=1,\ldots,N_{0}$. As mentioned in the remark
in Sec. 3.3, this choice does not reduce the complexity of designing a global
subcontroller at all. Noticing that the number of the clusters is maximized by
choosing the trivial cluster set, we aim at minimizing the number of clusters.
An important observation is that the condition (12a) becomes a sufficient
condition of (12b) under a mild condition.

###### Proposition 2

Assume that $(A,\\{P_{i}\\}_{i=0}^{N})$ satisfies

${\rm im}\,{P_{0}^{(i)}}\subset\mathcal{R}(A,[P_{1}\ \cdots P_{i-1}\ P_{i+1}\
\cdots\ P_{N}]),\quad i=1,\ldots,N$ (19)

where $P_{0}^{(i)}:=\pi_{{\rm im}\,{P_{i}}}P_{0}$ with $\pi_{{\rm
im}\,{P_{i}}}:=P_{i}P_{i}^{\sf T}$, a projection matrix onto ${\rm
im}\,{P_{i}}$. Then, if (12a) holds, (12b) holds as well.

Note that $P_{0}^{(i)}$ satisfies the relationship

${\rm im}\,{P_{0}^{(i)}}={\rm im}\,{P_{i}}\cap{\rm im}\,{P_{0}},$

which suggests that ${\rm im}\,{P_{0}^{(i)}}$ contains synchronized states in
the $i$th cluster. In Proposition 2, the condition (19) means that the
subspace ${\rm im}\,{P_{0}^{(i)}}$ is reachable from one of the other
clusters. Since this condition is not very restrictive, we first develop an
algorithm producing clusters that satisfy (12a) disregarding (12b), and
subsequently we extend the algorithm to satisfy (19) as well.

We exemplify our proposed algorithm through the motivating example in Sec.
2.1, where the desired clusters are given by (4). The proposed algorithm is
based on a greedy approach. We begin with an initial cluster set, partition
one of the clusters such that (12a) is satisfied, and repeat this process
until all partitioned clusters satisfy (12a). Suppose that the current step is
the $\tau$th step at which the temporary cluster set is given as the one in
Fig. 6a. We check if the clusters satisfy (12a). Observe that

$P_{2}^{(\tau)}=\left[\begin{array}[]{c}0_{10\times 8}\\\
I_{8}\end{array}\right],\quad
P_{0}^{(\tau)}=\left[\begin{array}[]{cc}\mathds{1}_{5}\otimes I_{2}&0\\\
0&\mathds{1}_{4}\otimes I_{2}\end{array}\right]$

and the controllable subspace from the second cluster

$\mathcal{R}(A,P_{2}^{(\tau)})={\rm im}\,{P^{(\tau)}_{2}}\oplus{\rm
im}\,{\left[\begin{array}[]{c}\mathds{1}_{3}\otimes I_{2}\\\ 0_{4\times 2}\\\
0_{8\times 2}\end{array}\right]}\oplus{\rm
im}\,{\left[\begin{array}[]{c}0_{6\times 2}\\\ \mathds{1}_{2}\otimes I_{2}\\\
0_{8\times 2}\end{array}\right]}.$ (20)

Thus (12a) does not hold. Accordingly, we choose $\mathcal{I}^{(\tau)}_{1}$ to
be partitioned into multiple clusters at the next step. From (20), the
components $\Sigma_{[k]}$ for $k=1,2,3$ can be lumped together, and
$\Sigma_{[4]}$ and $\Sigma_{[5]}$ can be lumped together as well. This
procedure can systematically be performed by comparing rows of the
controllability matrix. This partition results in the clusters illustrated by
Fig. 6b at the $(\tau+1)$th step. Then we can terminate the algorithm by
confirming that the condition (12a) is satisfied for $i=1,2,3$. Note that (19)
is satisfied in this case and hence (12b) also holds.

(a)

(b)

Figure 6: An illustrative example of the clusters during the algorithm.

The detailed description of the clustering algorithm is shown in Algorithm 1,
where $f_{P_{i}}(\mathcal{I}_{i})$ generates the matrix $P_{i}$ according to
(8), $f_{\mathbf{R}}(A,P_{i})$ generates the controllability matrix
$\mathbf{R}_{i}$ with respect to the pair $(A,P_{i})$, ${\rm
par}(\mathbf{R}_{i},P_{i})$ generates the minimal clusters that satisfy the
condition (12a) for $P_{i}$ based on $\mathbf{R}_{i}$ through the optimization
problem

$\displaystyle{\min_{\\{\mathcal{I}_{j}\\}_{j=1}^{N}}}N\ {\rm s.t.}\ \pi_{{\rm
im}\,{P_{i}}^{\perp}}\,\mathcal{R}(A,P_{i})\subset\pi_{{\rm
im}\,{P_{i}}^{\perp}}\,{\rm
im}\,{f_{P_{0}}(\\{\mathcal{I}_{j}\\}_{j=1}^{N})},$ (21)

with $\pi_{{\rm im}\,{P_{i}}^{\perp}}:=I-P_{i}P_{i}^{\sf T}$, a projection
matrix onto the orthogonal subspace of ${\rm im}\,{P_{i}}$, with which the
state of the components in the $i$th cluster is disregarded, and
$f_{P_{0}}(\\{\mathcal{I}_{i}\\})$ generates the matrix $P_{0}$ according to
(8). The subproblem, which can be regarded as a greedy part, can be solved by
comparing rows of $\pi_{{\rm im}\,{P_{i}}^{\perp}}\,\mathbf{R}_{i}$ as
mentioned above.

Clearly, Algorithm 1 produces a cluster set that satisfies (12a). It should be
remarked that if we use the single largest cluster
$\mathcal{I}^{(0)}_{1}=\\{1,\ldots,N_{0}\\}$ as the initial cluster set then
the condition (12a) holds for any systems. Thus Algorithm 1 is immediately
terminated with the initial cluster set, not producing any beneficial cluster
set. For this reason, the initial cluster set must be set to have at least two
clusters.

Algorithm 1 Clustering Algorithm

0: $A,\\{\mathcal{I}_{i}^{(0)}\\}_{i=1}^{N^{(0)}}$

0: $\\{\mathcal{I}_{i}\\}_{i=1}^{N}$

1: $\tau\leftarrow 0$

2: repeat

3: $\tau\leftarrow\tau+1$

4: for $i=1,\ldots,N^{(\tau-1)}$ do

5: $P_{i}^{(\tau-1)}\leftarrow f_{P_{i}}(\mathcal{I}^{(\tau-1)}_{i})$

6: $\mathbf{R}^{(\tau-1)}_{i}\leftarrow f_{\mathbf{R}}(A,P_{i}^{(\tau-1)})$

7: if the condition (12a) is not satisfied for $i$ then

8: $\\{\mathcal{I}\\}_{i=1}^{N^{(\tau)}}\leftarrow{\rm
par}(\mathbf{R}_{i}^{(\tau-1)},P_{i}^{(\tau-1)})$

9: break

10: end if

11: end for

12: until
$\\{\mathcal{I}^{(\tau)}_{i}\\}_{i=1}^{N^{(\tau)}}=\\{\mathcal{I}_{i}^{(\tau-1)}\\}_{i=1}^{N^{(\tau-1)}}$

13:
$\\{\mathcal{I}_{i}\\}_{i=1}^{N}=\\{\mathcal{I}_{i}^{(\tau)}\\}_{i=1}^{N^{(\tau)}}$

###  4.2 Algorithm Property

We show the optimality of Algorithm 1. The following notion is needed.

######  Definition 2 (Partition of Clusters)

A cluster set $\\{\mathcal{I}_{i}\\}_{i=1}^{N}$ is said to be a _partition_ of
$\\{\mathcal{I}_{i^{\prime}}\\}_{i^{\prime}=1}^{N^{\prime}}$ when for any
$i\in\\{1,\ldots,N\\}$ there exists $i^{\prime}\in\\{1,\ldots,N^{\prime}\\}$
such that $I_{i}\subset I_{i^{\prime}}$.

Let $\mathfrak{F}(\\{\mathcal{I}_{i}\\}_{i=1}^{N})$ be the family of all
cluster sets that are partitions of $\\{\mathcal{I}_{i}\\}_{i=1}^{N}$.
Intuitively, $\mathfrak{F}(\\{\mathcal{I}_{i}\\}_{i=1}^{N})$ contains all
cluster sets that can be produced from the original cluster set
$\\{\mathcal{I}_{i}\\}_{i=1}^{N}$ by partitioning some of the clusters.
Moreover, let
$\mathfrak{G}(\\{\mathcal{I}_{i}\\}_{i=1}^{N})\subset\mathfrak{F}(\\{\mathcal{I}_{i}\\}_{i=1}^{N})$
denote the family of all cluster sets in
$\mathfrak{F}(\\{\mathcal{I}_{i}\\}_{i=1}^{N})$ such that (12a) is satisfied.
The following theorem shows minimality of the resulting cluster set.

######  Theorem 4 (Algorithm Property)

The cluster set produced by Algorithm 1 is the minimum cluster set in
$\mathfrak{G}(\\{\mathcal{I}^{(0)}_{i}\\}_{i=1}^{N^{(0)}})$.

Essentially, Theorem 4 is a direct consequence of the following lemma.

###### Lemma 1

The relation

$\mathfrak{G}(\\{\mathcal{I}^{(0)}_{i}\\}_{i=1}^{N^{(0)}})=\mathfrak{G}(\\{\mathcal{I}^{(\tau)}_{i}\\}_{i=1}^{N^{(\tau)}})$

holds for any $\tau\geq 0$.

Lemma 1 implies that Algorithm 1 preserves the admissible clusters. Therefore,
we can guarantee minimality of the resulting cluster set in
$\mathfrak{G}(\\{\mathcal{I}^{(0)}_{i}\\}_{i=1}^{N^{(0)}})$.

###  4.3 Extended Clustering Algorithm

We next extend Algorithm 1 to the case when the resulting cluster set does not
satisfy (12b). In this case, (19) is neither satisfied. The idea for the
extension is to apply Algorithm 1 only to a subset of clusters that do not
satisfy (19) inspired by the following proposition.

###### Proposition 3

Let $\\{\mathcal{I}_{i}\\}_{i=1}^{N}$ be a cluster set such that (12a) is
satisfied for any $i\in\\{1,\ldots,N\\}$ and (19) is not satisfied for
$j\in\mathcal{J}\subset\\{1,\ldots,N\\}$. Then for any cluster set
$\\{\mathcal{I}^{\prime}_{j^{\prime}}\\}_{j^{\prime}\in\mathcal{J}^{\prime}}$
in $\mathfrak{F}(\\{\mathcal{I}_{j}\\}_{j\in\mathcal{J}})$, the following
cluster set

$\\{\mathcal{I}_{i}\\}_{i\notin\mathcal{J}}\cup\\{\mathcal{I}_{j^{\prime}}\\}_{j^{\prime}\in\mathcal{J}^{\prime}}$

which is obtained by partitioning $\\{\mathcal{I}_{j}\\}_{j\in\mathcal{J}}$
into
$\\{\mathcal{I}^{\prime}_{j^{\prime}}\\}_{j^{\prime}\in\mathcal{J}^{\prime}}$,
satisfies (12a) as well as (19) for $i\notin\mathcal{J}$.

Proposition 3 implies that, once (12a) and (19) are satisfied for some
clusters, this property is preserved even under partition of the other
clusters. Thanks to Proposition 3, we can reduce the clustering problem into a
subproblem for the subclusters that do not satisfy (19).

The proposed clustering algorithm is described in Algorithm 2. As mentioned
above, the idea is to apply Algorithm 1 repeatedly with an initial partition
of the clusters such that (19) is not satisfied. The resulting clusters
obviously satisfy the conditions in Theorem 1 from Propositions 2 and 3.

###### Proposition 4

Consider the clusters produced by Algorithm 2. Under Assumption 2, there
exists a hierarchical model decomposition of the resulting clustered
interconnected system.

Algorithm 2 Extended Clustering Algorithm

0: $A,\\{\mathcal{I}_{i}\\}^{(0)}$

0: $\\{\mathcal{I}_{i}\\}_{i=1}^{N}$

1: $\tau^{\prime}\leftarrow 0$

2:
$\\{\mathcal{I}_{i}\\}^{(\tau^{\prime})}\leftarrow\\{\mathcal{I}_{i}\\}^{(0)}$

3: repeat

4: $\tau^{\prime}\leftarrow\tau^{\prime}+1$

5: $\\{\mathcal{I}_{i}\\}^{(\tau^{\prime})}\leftarrow$ Algorithm 1 with
$\\{\mathcal{I}_{i}\\}^{(\tau^{\prime}-1)}$

6: if (19) is not satisfied for $j\in\mathcal{J}\subset\\{1,\ldots,N\\}$ then

7: provide
$\\{\mathcal{I^{\prime}}_{j^{\prime}}\\}_{j^{\prime}\in\mathcal{J}^{\prime}}$
in $\mathfrak{F}(\\{\mathcal{I}_{j}\\}_{j\in\mathcal{J}})$

8:
$\\{\mathcal{I}_{i}\\}^{(\tau^{\prime})}\leftarrow\\{\mathcal{I}_{i}\\}_{i\notin\mathcal{J}}\cup\\{\mathcal{I^{\prime}}_{j^{\prime}}\\}_{j^{\prime}\in\mathcal{J}^{\prime}}$

9: break

10: end if

11: until the condition (19) is satisfied for all clusters

12: $\\{\mathcal{I}_{i}\\}_{i=1}^{N}=\\{\mathcal{I}_{i}\\}^{(\tau^{\prime})}$

##  5 Extension to Indecomposable Systems

In the framework proposed in Sec. 3, the state $x$ is needed to be perfectly
represented as the superposition of $\xi_{0},\xi_{1},\ldots,\xi_{N}$ without
error as in (11). This section extends the proposed method to systems for
which no exact hierarchical model decompositions exist.

###  5.1 Motivating Example Revisited

Consider the motivating example again.. We assume that the model parameters
are perturbed away from their nominal values. Specifically, let
$m_{[k]}=(1+\delta_{m,[k]})\overline{m}_{[k]}$ and
$d_{[k]}=(1+\delta_{d,[k]})\overline{d}_{[k]}$ where
$(\overline{m}_{[k]},\overline{d}_{[k]})$ are the nominal values given in (3)
and $(\delta_{m,[k]},\delta_{d,[k]})$ are independently and randomly generated
scalars following the uniform distribution with the interval $[-0.2,0.2]$. For
this system, there exist no hierarchical model decompositions. However, as
depicted by Fig. 7, it seems like the system behavior is similar to that in
the preceding case of Fig. 3. The bottom of Fig. 7 suggests that the state in
$\mathcal{I}_{3}$ can be approximated by the global behavior with a small
error. This example motivates us to extend the proposed method by admitting
approximation error in the hierarchical model decomposition.

Figure 7: Free response of the second-order system with slightly perturbed
model parameters.

###  5.2 Robust Hierarchical Model Decomposition

To handle approximation error, we consider the following system

$\left\\{\begin{array}[]{cl}\dot{\xi}_{i}&=\hat{A}_{i}\xi_{i}+\hat{E}_{i}e+B_{i}\hat{u}_{i},\quad
i=1,\ldots,N\\\
\dot{\xi}_{0}&=\hat{A}_{0}\xi_{0}+\sum_{i=1}^{N}\hat{R}_{i}\xi_{i}+\hat{E}_{0}e+B_{0}\hat{u}_{0}\\\
\dot{e}&=\hat{A}_{e}e+\hat{F}_{0}\xi_{0}+\sum_{i=1}^{N}\hat{F}_{i}\xi_{i},\end{array}\right.$
(22)

which is obtained by introducing the error signal $e$ and interaction between
$e$ and $\xi_{i}$ for $i=0,1,\ldots,N$ into $\Xi$ in (10). From simple
calculation, it turns out that

$\textstyle{x(t)=\sum_{i=1}^{N}P_{i}\xi_{i}(t)+P_{0}\xi_{0}(t)+e(t),\quad\forall
t\geq 0}$

holds for any control inputs and initial states provided that
$x(0)=\sum_{i=1}^{N}P_{i}\xi_{i}(0)+P_{0}\xi_{0}(0)+e(0)$ if and only if

$\begin{array}[]{l}\hat{A}_{e}=A-P_{0}\hat{E}_{0}-\sum_{i=1}^{N}P_{i}\hat{E}_{i},\quad\hat{F}_{0}=AP_{0}-P_{0}\hat{A}_{0},\\\
\hat{F}_{i}=AP_{i}-P_{i}\hat{A}_{i}-P_{0}\hat{R}_{i},\quad
i=1,\ldots,N\end{array}$ (23)

where $\hat{A}_{0},\hat{A}_{i},\hat{R}_{i},\hat{E}_{0},\hat{E}_{i}$ are free
parameters. The block diagram of the system (22) for the case $N=3$ is
depicted in the left of Fig. 8 where $\Xi_{e}$ represents the dynamics of $e$.
Because there exist feedback paths from $\Xi_{e}$ to $\Xi_{0}$ and $\Xi_{i}$
for $i=1,\ldots,N$ as shown in this figure, the system (22) no longer has a
cascade structure. Hence, the entire stability cannot be guaranteed even if we
attach subcontrollers each of which stabilizes the corresponding subloop.

Figure 8: Block diagrams of the hierarchical model decomposition with error
dynamics in (22) when $N=3$. Left: the general case. Right: the case under the
choice (24) where the system has a hierarchical structure including the error
dynamics.

An important observation from (22) is that the hierarchical cascade structure
can be recovered by choosing the free parameters appropriately. In particular,
when the condition

$\hat{E}_{i}=0,\quad\forall i=1,\ldots,N$ (24)

or

$\hat{E}_{0}=0\ {\rm and}\ \hat{F}_{0}=0$

is satisfied, the cascade structure is preserved. In the former case, there
are no signals flowing from the error dynamics $\Xi_{e}$ to the upstream parts
$\Xi_{1},\ldots,\Xi_{N}$, and the error dynamics forms a feedback loop only
with the downstream part $\Xi_{0}$, a block diagram of which is illustrated in
the right of Fig. 8. In the latter case, the error dynamics has no direct
interaction with $\Xi_{0}$. However, there exists a free parameter
$\hat{A}_{0}$ that satisfies $\hat{F}_{0}=0$ only when ${\rm im}\,{P_{0}}$ is
$A$-invariant, which is a restrictive requirement. Therefore, we hereinafter
consider only the former case.

When (24) is satisfied, the other parameters should be chosen so as to reduce
the norm of the transfer matrix from $\xi_{0},\xi_{1},\ldots,\xi_{N}$ to $e$
given by

$G_{e}:=(sI-\hat{A}_{e})^{-1}[\hat{F}_{0}\ \hat{F}_{1}\ \cdots\ \hat{F}_{N}]$

while satisfying (23). A reasonable policy to determine the parameters is to
reduce the norm of the input matrices through

$\begin{array}[]{cl}\hat{A}_{0}&\in\displaystyle{\operatorname*{arg\,min}_{X}\|\hat{F}_{0}(X)\|,}\\\
(\hat{A}_{i},\hat{R}_{i})&\in\displaystyle{\operatorname*{arg\,min}_{(X,Y)}\|\hat{F}_{i}(X,Y)\|,\quad
i=1,\ldots,N}\\\ \end{array}$ (25)

with an appropriate norm $\|\cdot\|$ where

$\hat{F}_{0}(X):=AP_{0}-P_{0}X,\quad\hat{F}_{i}(X,Y):=AP_{i}-P_{i}X-P_{0}Y.$

Note that this choice yields the exact hierarchical model decomposition with
(13) in Theorem 2 if the conditions (12a) and (12b) in Theorem 1 hold. In this
sense, the system (22) with the parameters (24) and (25) can be regarded as a
generalization of hierarchical model decomposition. Accordingly, we define
_robust hierarchical model decomposition._

######  Definition 3 (Robust Hierarchical Model Decomposition)

The system in (22) with the parameters (23), (24), and (25) is said to be a
robust hierarchical model decomposition of (7).

There always exists a robust hierarchical model decomposition for any system
and choice of clusters as claimed by the following proposition.

###### Proposition 5

There always exists a robust hierarchical model decomposition of (7) for any
cluster set.

Based on the result, we subsequently discuss distributed design with a robust
hierarchical model decomposition.

###  5.3 Distributed Design for Indecomposable Systems

Through the obtained robust hierarchical model decomposition, distributed
design can be achieved by designing subcontrollers for each subsystem as long
as the global subcontroller, which corresponds to the downstream part, can
cope with the error signal. This fact is described by the following theorem.

######  Theorem 5 (Stabilization under Approximation Error)

Consider a robust hierarchical model decomposition (22). Let
$\hat{K}_{1},\ldots,\hat{K}_{N}$ be controllers such that the closed-loop
systems (15) are internally stable. Moreover, let $\hat{K}_{0}$ be a
controller such that the closed-loop system

$\left\\{\begin{array}[]{cl}\dot{\xi}_{0}&=\hat{A}_{0}\xi_{0}+\hat{E}_{0}e+B_{0}\hat{u}_{0}\\\
\dot{e}&=\hat{A}_{e}e+\hat{F}_{0}\xi_{0}\\\
\hat{u}_{0}&=\hat{K}_{0}(C_{0}\xi_{0}+C_{0}P_{0}^{\sf T}e)\end{array}\right.$

is internally stable. Then the controller composed of (17) and functional
observers (16) stabilizes the clustered interconnected system (7).

Theorem 5 implies that the local subcontrollers can be designed without any
concern about approximation error as long as the downstream part is stabilized
by the global subcontroller, for design of which robust control [41] can
apply.

Further, the following proposition shows that the functional observer (18)
still works for robust hierarchical model decomposition.

###### Proposition 6

Consider a robust hierarchical model decomposition (22). Then the system (18)
is a functional observer of $C_{i}\xi_{i}$ for (22) as well.

The idea of the construction is almost the same as that in Proposition 1. The
only difference is that $\phi_{i}$ is an estimation of $P_{i}^{\sf
T}P_{0}\xi_{0}+P_{i}^{\sf T}e$ instead of $P_{i}^{\sf T}P_{0}\xi_{0}$. Thus,
we can estimate $C_{i}\xi_{i}$ using $\phi_{i}$ even when approximation errors
are present.

##  6 Numerical Examples

###  6.1 Motivating Example Revisited

Consider the motivating example. As each local internal controller
$\hat{K}_{i}$ in (17) for $i=1,2,3$, we employ a linear quadratic regulator
(LQR) under the state weight
$Q_{i}=I_{r_{i}}\otimes\mathrm{D}(q_{\theta},q_{\omega})$ with
$(q_{\theta},q_{\omega})=(1,10^{4})$ and the input weight
$R_{i}=10^{2}I_{r_{i}}$ with a state observer whose observer gain is
determined according to the LQR method under the state weight
$10^{3}I_{2r_{i}}$ and the input weight $R_{i}$. Similarly, the global
subcontroller $\hat{K}_{0}$ in (17) is designed based on the LQR method under
$Q_{0}=I_{N}\otimes\mathrm{D}(q_{\theta},q_{\omega})$ and $R_{0}=10^{2}I_{N}$
with the state observer. For implementation of local subcontrollers, the
functional observer (18) is employed.

The responses under the same initial condition as that of Fig. 3 are
illustrated in Fig. 9, where Fig. 9a, Fig. 9b, Fig. 9c, and Fig. 9d correspond
to the cases in which no controllers, only the local subcontrollers, only the
global subcontroller, and the glocal controller is implemented, respectively.
It is observed in Fig. 9b that stationary interarea oscillation remains. In
Fig. 9c, local oscillation inside $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$
cannot be suppressed only with the global subcontroller although interarea
oscillation is removed. In contrast to the above two cases, both of
global/local behaviors can be suppressed by using the glocal controller as
illustrated in Fig. 9d. This result evidences the effectiveness of the glocal
structure.

(a)

(b)

(c)

(d)

Figure 9: Responses of the second-order network system with different control
policies.

Next, we confirm scalability of controller design by comparing computation
times elapsed for designing a glocal controller and a centralized controller.
Consider increasing the number of subsystems inside each cluster in Fig. 1.
Let $n_{0}$ be an index to determine the size of the system and set the number
of components to $N_{0}=9n_{0}$. Consider the three clusters constructed as

$\begin{array}[]{l}\mathcal{I}_{1}=\\{1,\ldots,3n_{0}\\},\quad\mathcal{I}_{2}=\\{3n_{0}+1,\ldots,5n_{0}\\},\\\
\mathcal{I}_{3}=\\{5n_{0}+1,\ldots,9n_{0}\\}\end{array}$ (26)

and let the parameters of the components in each cluster be the same as those
in Sec. 2.1. We consider the centralized controller ${\rm
col}(u_{i})_{i=1}^{N}=K_{\rm c}{\rm col}(y_{i})_{i=1}^{N}$ with a dense
information structure. For the centralized controller design, we employ the
LQR method under the state weight $Q_{\rm
c}=I_{n}\otimes\mathrm{D}(q_{\theta},q_{\omega})$ and the input weight $R_{\rm
c}=10^{2}I_{N}$ with the state observer where $n$ is the dimension of all the
states. We also suppose that the global/local subcontrollers are designed in
accordance with the previous ones. The average computation times elapsed for
designing the controllers with varied $n_{0}$ ranging from 10 to 25 are
depicted in Fig. 10 on a logarithmic scale. It can be observed that the
computation time is significantly reduced through hierarchical model
decomposition, which indicates scalability of the proposed distributed design.
Standard model reduction techniques, e.g., balanced truncation method [36] and
singular perturbation method [37], are ineffective in this case. The Hankel
singular values $\sigma_{i}$ of the transfer matrix from ${\rm col}(u_{i})$ to
${\rm col}(y_{i})$ for $n_{0}=20$ are given by

$\sigma_{i}\in\\{1.3,1.6,1.7,2.4,2.5,+\infty\\},\quad i=1,\ldots,360,$

where the unique singular value $+\infty$ corresponds to the semistable pole
with the eigenvector $\mathds{1}_{N_{0}}\otimes[1\ 0]^{\sf T},$ which
represents the direction of synchronized angles. Because the ratio between the
maximum singular value except for $+\infty$ and the minimum one is not very
large, all states of the entire system are irreducible through the balanced
truncation method even if the semistable pole is disregarded. Further, since
the time scales of the global/local behaviors are not very different as seen
in Fig. 9a, the singular perturbation approach is unprofitable for this
system. Indeed, using only the global subcontroller, which corresponds to the
case where the local dynamics is reduced as a fast dynamics, cannot suppress
the local oscillation as shown in Fig. 9c. Therefore, the computation time for
designing $K_{\rm c}$ cannot efficiently be reduced using the traditional
methods.

Figure 10: Average computation times for designing the local subcontrollers
$K_{1},K_{2},K_{3}$, the global subcontroller $K_{0}$, and the centralized
controller $K_{\rm c}$ for varied $n_{0}$ ranging from $10$ to $25$ on a
logarithmic scale. Figure 11: Average computation times of the clustering
algorithm for varied $n$, the dimension of the entire state space, ranging
from $180$ to $900$ on a logarithmic scale.

We next investigate computational complexity of the proposed clustering
algorithm. Consider two situations:

  1. (i)

The size of the components increases under a fixed number of clusters.

  2. (ii)

The number of clusters increases under a fixed number of components in each
cluster.

Set $N_{0}=9n_{0}$. In case (i), the parameters of the components in each
cluster are set to be identical, which leads to the three clusters in (26). In
case (ii), the parameters of the components are set to give the clusters as

$\begin{array}[]{l}\mathcal{I}_{3i+1}=\\{9i+1,9i+2,9i+3\\},\
\mathcal{I}_{3i+2}=\\{9i+4,9i+5\\},\\\
\mathcal{I}_{3i+3}=\\{9i+6,9i+7,9i+8,9i+9\\},\ i=0,\ldots,n_{0}-1.\end{array}$

The average computation times for the clustering algorithm are illustrated in
Fig. 11 where the number of components $N_{0}$ ranges from $90$ to $450$ and
the dimension of the entire state space $n$ ranges from $180$ to $900$. It is
observed that the computational complexity in case (i) is small compared with
case (ii). The computationally most expensive part in the algorithm is the
calculation of the controllability matrix with respect to the pair
$(A,P_{i})$. The overall computation time in case (ii), where the number of
the clusters is $3n_{0}$, is longer than that in case (i), where the number of
clusters is three.

###  6.2 NPCC system

To illustrate the practical relevance of our proposed control structure, we
consider the 48-machine NPCC system [21], a model of the power grid in New
York and the neighboring areas. The NPCC 140-bus, 48-machine, 233-branch model
can be found in the Power System Toolbox [42]. The interconnection parameters
in (2) are calculated based on its tie-line parameters. Although each
generator is modeled as the second-order system described by (1) as in the
previous example, the parameters are not strictly homogeneous. Hence, there
does not exist an exact hierarchical model decomposition for any non-trivial
cluster set, and hence the proposed clustering algorithm cannot be applied to
this system. Accordingly, as a given cluster set, we employ the nine clusters
depicted by Fig. 3.5 in [21], which is obtained through the coherency-based
aggregation. We apply the robust version of hierarchical model decomposition
proposed in Sec. 5. Every internal subcontroller is designed as an LQR
controller with a state observer.

The frequency deviations of all generators are depicted in Fig. 12, where
Figs. 12a, 12b, 12c, and 12d show the responses without any controller, only
with the local subcontrollers, only with the global subcontroller, and with
the proposed glocal controller, respectively. It can be observed that local
oscillations are efficiently suppressed with local subcontrollers. The average
frequency deviations of all generators in the cases only with the local
subcontrollers and with the glocal controller are depicted in Fig. 13. As
shown in this figure, the excitation of the slow global dynamics, which
remains in Fig. 12b, is suppressed through the global subcontroller. The
result highlights the potential effectiveness of the proposed glocal control
for practical systems.

(a)

(b)

(c)

(d)

Figure 12: Responses of the NPCC testbed with different controllers. Figure
13: Average frequency deviations of all generators with the local
subcontrollers and with the glocal controller.

##  7 Conclusion

In this paper, distributed design of glocal controllers has been proposed for
large-scale network systems. The proposed idea is to transform the original
system into a cascade structured system, called hierarchical model
decomposition. Thanks to this structure, stability of the entire system can be
guaranteed by designing subcontrollers each of which stabilizes the
corresponding subsystem. We have provided a condition for existence of the
hierarchical model decomposition, a specific representation, a clustering
method, and a robust extension.

Future research directions on the proposed framework include development of a
clustering method that can handle approximation errors. The proposed algorithm
can result in a conservative decomposition for this case. Moreover, although
we have confined our attention only to the glocal structure, distributed
design of controllers with other particular information structures is an open
problem.

[Proof of Theorems]

##  Appendix A Proofs of Theorems

###### Proof.

_Proof of Theorem 1:_ Sufficiency is shown in the proof of Theorem 2 by
construction. We here show necessity. Assume that $\Xi$ is a hierarchical
model decomposition. Let $x_{0}\in{\rm im}\,{P_{0}}$ and
$x(0)=x_{0},\xi_{0}(0)=P_{0}^{\dagger}x_{0},\xi_{i}(0)=0$ for $i=1,\ldots,N$.
Then $P_{0}P_{0}^{\dagger}x_{0}=x_{0}$ and
$x(0)=P_{0}\xi_{0}(0)+\sum_{i=1}^{N}P_{i}\xi_{i}(0)$. Because $\Xi$ is a
hierarchical model decomposition, $x(t)=P_{0}\xi_{0}(t)$ holds for any $t\geq
0$ with $\hat{u}_{i}=0$ for any $i$. Thus $x(t)=e^{At}x_{0}\in{\rm
im}\,{P_{0}}$ for any $t\geq 0$. Since ${\rm im}\,{P_{0}}$ is a closed
subspace, $\lim_{t\rightarrow 0}(e^{At}x_{0}-x_{0})/t=Ax_{0}\in{\rm
im}\,{P_{0}}$. Because $x_{0}$ is arbitrary in ${\rm im}\,{P_{0}}$, ${\rm
im}\,{P_{0}}$ is an invariant subspace of $A$. Hence
$\mathcal{R}(A,P_{0})\subset{\rm im}\,{P_{0}}$. Similarly, it can be shown
that $\mathcal{R}(A,P_{i})\subset{\rm im}\,{P_{0}}+{\rm im}\,{P_{i}}$ for
$i=1,\ldots,N$ by taking $x(0)\in{\rm im}\,{P_{i}}$. ∎

###### Proof.

_Proof of Theorem 2:_ We first show sufficiency. As a preparation, we show
that there exist matrices $\hat{A}_{i}$ and $\hat{R}_{i}$ such that (13) holds
when the conditions (12a) and (12b) are satisfied. From (12a), $A\,{\rm
im}\,{P_{i}}\subset{\rm im}\,{P_{i}}+{\rm im}\,{P_{0}}$ for $i=1,\ldots,N$.
Hence there exist $X_{i}$ and $X_{0i}$ such that
$AP_{i}=P_{i}X_{i}+P_{0}X_{0i}.$ Moreover, from (12b), ${\rm im}\,{P_{0}}$ is
$A$-invariant and hence there exists $X_{0}$ such that $AP_{0}=P_{0}X_{0}.$
Thus the condition (13) holds with
$\hat{A}_{i}=X_{i},\hat{R}_{i}=X_{0i},\hat{A}_{0}=X_{0}$.

Let us assume that the condition (13) holds. Define the error signal
$\textstyle{e:=x-P_{0}\xi_{0}-\sum_{i=1}^{N}P_{i}\xi_{i}}$ and then

$\begin{array}[]{cl}\dot{e}&=Ax-
P_{0}(\hat{A}_{0}\xi_{0}+\sum_{i=1}^{N}\hat{R}_{i}\xi_{i})-\sum_{i=1}^{N}P_{i}\hat{A}_{i}\xi_{i}\\\
&=Ax-
P_{0}\hat{A}_{0}\xi_{0}-\sum_{i=1}^{N}(P_{0}\hat{R}_{i}+P_{i}\hat{A}_{i})\xi_{i}\\\
&=Ax-AP_{0}\xi_{0}-\sum_{i=1}^{N}AP_{i}\xi_{i}=Ae.\end{array}$

When $x(0)=\sum_{i=1}^{N}P_{i}\xi_{i}(0)+P_{0}\xi_{0}(0)$ holds, $e(0)=0$ and
hence $e(t)=0$ for any $t\geq 0$ and $\hat{u}_{i}$. Thus $\Xi$ is a
hierarchical model decomposition.

We next show the necessity part. Note that, from Theorem 1
$\mathcal{R}(A,P_{0})\subset{\rm im}\,{P_{0}}$ holds. We first show that
$AP_{0}-P_{0}\hat{A}_{0}=0,$ which is equivalent to
$\hat{A}_{0}=P_{0}^{\dagger}AP_{0}$ under $\mathcal{R}(A,P_{0})\subset{\rm
im}\,{P_{0}}$. When $\hat{u}_{i}=0$ and $\xi_{i}(0)=0$, $\xi_{i}(t)=0$ for any
$t\geq 0$. Then $x(t)=P_{0}\xi_{0}(t)$ for any $t\geq 0$ for arbitrary
$\hat{u}_{0}$ provided that $x(0)=P_{0}\xi_{0}(0)$. Define
$e_{0}:=P_{0}^{\dagger}x-\xi_{0}$ and then
$\dot{e}_{0}=P_{0}^{\dagger}AP_{0}e_{0}+(P_{0}^{\dagger}AP_{0}-\hat{A}_{0})\xi_{0}.$
Since $e_{0}(t)=0$ for any $t\geq 0$, $\dot{e}_{0}(0)=0$ for any $\xi_{0}$.
Therefore the kernel of $P_{0}^{\dagger}AP_{0}-\hat{A}_{0}$ contains the
entire space, which leads to $\hat{A}_{0}=P_{0}^{\dagger}AP_{0}$. Similarly,
it can be shown that $AP_{i}-P_{0}\hat{R}_{i}-P_{i}\hat{A}_{i}=0$ for
$i=1,\ldots,N$. ∎

###### Proof.

_Proof of Theorem 3:_ From the necessary and sufficient condition of
functional observers [40, Lemma 2], $\mathbf{A}_{i}$ is stable and there
exists a matrix $U_{i0}$ such that

$\begin{array}[]{l}U_{i0}\left[\begin{array}[]{cc}\hat{A}_{i}&0\\\
\hat{R}_{i}&\hat{A}_{0}\end{array}\right]-\mathbf{A}_{i}U_{i0}=\mathbf{E}_{i}C_{i}[I\
P_{i}^{\sf T}P_{0}],\\\ \left[\mathbf{B}_{i}\
\mathbf{D}_{i}\right]=U_{i0}\left[\begin{array}[]{cc}B_{i}&0\\\
0&B_{0}\end{array}\right],\ [C_{i}\
0]=\mathbf{C}_{i}U_{i0}+\mathbf{F}_{i}C_{i}[I\ P_{i}^{\sf T}P_{0}]\end{array}$
(27)

for any $i=1,\ldots,N$. Let $U_{i0}=[U_{i}\ U_{0}]$ and define

$\epsilon_{i}:=U_{i}\xi_{i}+U_{0}\xi_{0}-\phi_{i}.$

Then because $\Phi_{i}$ is a functional observer, the dynamics of
$\epsilon_{i}$ can be represented as
$\dot{\epsilon}_{i}=\mathbf{A}_{i}\epsilon_{i}$, which is stable. Moreover, we
have

$\begin{array}[]{cl}\psi_{i}&=\mathbf{C}_{i}\phi_{i}+\mathbf{F}_{i}C_{i}[I\
P_{i}^{\sf T}P_{0}][\xi_{i}^{\sf T}\ \xi_{0}^{\sf T}]^{\sf T}\\\
&=\mathbf{C}_{i}([U_{i}\ U_{0}][\xi_{i}^{\sf T}\ \xi_{0}^{\sf T}]^{\sf
T}-\epsilon_{i})+\mathbf{F}_{i}C_{i}[I\ P_{i}^{\sf T}P_{0}][\xi_{i}^{\sf T}\
\xi_{0}^{\sf T}]^{\sf T}\\\
&=C_{i}\xi_{i}-\mathbf{C_{i}}\epsilon_{i}\end{array}$

in view of the third identity in (27). Thus the entire closed-loop system
composed of $\Xi$ and $\\{K_{i}\\}_{i=0}^{N}$ can be described by

$\left\\{\begin{array}[]{l}\left\\{\begin{array}[]{cl}\dot{\epsilon}_{i}&=\mathbf{A}_{i}\epsilon_{i}\\\
\dot{\xi}_{i}&=\hat{A}_{i}\xi_{i}+B_{i}\hat{u}_{i}\\\
\hat{u}_{i}&=\hat{K}_{i}(C_{i}\xi_{i}-\mathbf{C}_{i}\epsilon_{i})\end{array},\quad
i=1,\ldots,N\right.\\\
\begin{array}[]{cl}\dot{\xi}_{0}&=\hat{A}_{0}\xi_{0}+\sum_{i=1}^{N}\hat{R}_{i}\xi_{i}+B_{0}u_{0}\\\
\hat{u}_{0}&=\hat{K}_{0}C_{0}(\xi_{0}+\sum_{i=1}^{N}P_{0}^{\sf
T}P_{i}\xi_{i}).\end{array}\end{array}\right.$ (28)

From the cascade structure of (28) and the assumption on stability of every
closed-loop system, the entire system is internally stable. Because the
original state can be represented by superposition of the states in (28), the
original system with the controllers is also internally stable. ∎

###### Proof.

_Proof of Proposition 1:_ The whole observer composed of
$\Phi_{1},\ldots,\Phi_{N}$ can be represented by

$\left\\{\begin{array}[]{cl}\dot{\phi}&={\rm D}({A_{i}})\phi+{\rm
D}({A_{i}-\hat{A}_{i}})\hat{x}+{\rm D}({L_{i}})v+P_{0}B_{0}\hat{u}_{0}\\\
\dot{\hat{x}}&={\rm D}({A_{i}})\hat{x}+{\rm D}({B_{i}}){\rm
col}(\hat{u}_{i})+{\rm D}({L_{i}})v+P_{0}B_{0}\hat{u}_{0}\\\ \psi&=-{\rm
D}({C_{i}})\phi+y\end{array}\right.$

with $\phi:={\rm col}(\phi_{i}),\hat{x}:={\rm col}(\hat{x}_{i}),\psi:={\rm
col}(\psi_{i})$ where the measurement signals are represented by

$\left[\begin{array}[]{c}y\\\
v\end{array}\right]=\left[\begin{array}[]{cc}{\rm D}({C_{i}})&{\rm
D}({C_{i}})P_{0}\\\ M&MP_{0}\end{array}\right]\left[\begin{array}[]{c}{\rm
col}(\xi_{i})_{i=1}^{N}\\\ \xi_{0}\end{array}\right].$

It suffices to show that the matrix

$U:=\left[\begin{array}[]{cc}0&P_{0}\\\ I&P_{0}\end{array}\right]$

satisfies the conditions

$\begin{array}[]{l}U\left[\begin{array}[]{cc}{\rm D}({\hat{A}_{i}})&0\\\
\left[\hat{R}_{1}\
\cdots\hat{R}_{N}\right]&\hat{A}_{0}\end{array}\right]-\left[\begin{array}[]{cc}{\rm
D}({\hat{A}_{i}})&{\rm D}({A_{i}-\hat{A}_{i}})\\\ 0&{\rm
D}({A_{i}})\end{array}\right]U\vspace{1mm}\\\ =\left[\begin{array}[]{cc}0&{\rm
D}({L_{i}})\\\ 0&{\rm
D}({L_{i}})\end{array}\right]\left[\begin{array}[]{cc}{\rm D}({C_{i}})&{\rm
D}({C_{i}})P_{0}\\\ M&MP_{0}\end{array}\right],\end{array}$ (29)

and

$\begin{array}[]{l}\left[\begin{array}[]{cc}0&P_{0}B_{0}\\\ {\rm
D}({B_{i}})&P_{0}B_{0}\end{array}\right]=U\left[\begin{array}[]{cc}{\rm
D}({B_{i}})&0\\\ 0&B_{0}\end{array}\right],\vspace{1mm}\\\ [{\rm D}({C_{i}})\
0]=-[{\rm D}({C_{i}})\ 0]U+[{\rm D}({C_{i}})\ {\rm
D}({C_{i}})P_{0}].\end{array}$

The second and third identities obviously hold. Regarding the first condition,
the right-hand side of (29) is described by

$\begin{array}[]{cl}{\rm(RHS)}&=[I\ I]^{\sf T}{\rm D}({L_{i}})M\left[I\
P_{0}\right]\vspace{1mm}\\\ &=[I\ I]^{\sf T}(A-{\rm D}({A_{i}}))\left[I\
P_{0}\right]\end{array}$ (30)

and the left-hand side is described by

$\begin{array}[]{l}{\rm(LHS)}\\\ =\left[\begin{array}[]{c}I\\\
I\end{array}\right]\left[P_{0}\left[\hat{R}_{1}\ \cdots\
\hat{R}_{N}\right]-{\rm D}({A_{i}-\hat{A}_{i}})\ P_{0}\hat{A}_{0}-{\rm
D}({A_{i}})P_{0}\right].\end{array}$

Since $\Xi$ is a hierarchical model decomposition,
$P_{0}\hat{R}_{i}=AP_{i}-P_{i}\hat{A}_{i}$ for $i=1,\ldots,N$ and
$P_{0}\hat{A}_{0}=AP_{0}$. Therefore

${\rm(LHS)}=[I\ I]^{\sf T}\left[A-{\rm D}({A_{i}})\ AP_{0}-{\rm
D}({A_{i}})P_{0}\right],$

which is equal to (30). ∎

###### Proof.

_Proof of Proposition 2:_ We show that $A\,{\rm im}\,{P_{0}^{(1)}}\subset{\rm
im}\,{P_{0}}.$ Noting that $\mathcal{R}(A,[P_{2}\ \cdots\ P_{N}])\subset{\rm
im}\,{P_{0}^{(1)}}+{\rm im}\,{[P_{2}\ \cdots\ P_{N}]}$ from (12a), we have

$\begin{array}[]{cl}A\,{\rm im}\,{P_{0}^{(1)}}&\subset A\,{\rm im}\,{P_{0}}\\\
&\subset A\,\mathcal{R}(A,[P_{2}\ \cdots\ P_{N}])\\\
&\subset\mathcal{R}(A,[P_{2}\ \cdots\ P_{N}])\\\ &\subset{\rm
im}\,{P_{0}^{(1)}}+{\rm im}\,{[P_{2}\ \cdots\ P_{N}]}\end{array}$

because of (19) and $A$-invariance of the controllable subspace. Moreover,
since ${\rm im}\,{P_{0}^{(1)}}\subset{\rm im}\,{P_{1}}$, we have

$\begin{array}[]{cl}A\,{\rm im}\,{P_{0}^{(1)}}&\subset A\,{\rm im}\,{P_{1}}\\\
&\subset\mathcal{R}(A,P_{1})\\\ &\subset{\rm im}\,{P_{1}}+{\rm
im}\,{P_{0}}\end{array}$

from (12a). Therefore, it follows that

$\begin{array}[]{cl}A\,{\rm im}\,{P_{0}^{(1)}}&\subset({\rm
im}\,{P_{0}^{(1)}}+{\rm im}\,{[P_{2}\ \cdots\ P_{N}]})\cap({\rm
im}\,{P_{1}}+{\rm im}\,{P_{0}})\\\ &=({\rm im}\,{P_{0}^{(1)}}\cap{\rm
im}\,{P_{1}})\oplus({\rm im}\,{[P_{2}\ \cdots\ P_{N}]}\cap{\rm
im}\,{P_{0}})\\\ &={\rm im}\,{P_{0}}.\end{array}$

Similarly, we can show the same inclusion property for the other clusters.
Hence, ${\rm im}\,{P_{0}}$ is $A$-invariant and the condition (12b) holds. ∎

###### Proof.

_Proof of Theorem 4:_ Let $\\{\mathcal{I}_{i}\\}$ be the cluster set produced
by Algorithm 1. From Lemma 1, any cluster set in
$\mathfrak{G}(\\{\mathcal{I}^{(0)}_{i}\\}_{i=1}^{N^{(0)}})$ can be generated
from $\\{\mathcal{I}_{i}\\}$. Since the number of clusters increases by
partition, the claim holds. ∎

###### Proof.

_Proof of Lemma 1:_ We prove the claim by induction. It suffices to show
$\mathfrak{G}(\\{\mathcal{I}^{(\tau)}_{i}\\})\subset\mathfrak{G}(\\{\mathcal{I}^{(\tau+1)}_{i}\\})$
for any $\tau$. Take a cluster set $\\{\mathcal{I}_{i}\\}_{i=1}^{N}$ that
belongs to $\mathfrak{G}(\\{\mathcal{I}^{(\tau)}_{i}\\})$. Because the
clusters satisfy (12a) and (12b), it suffices to show that
$\\{\mathcal{I}_{i}\\}$ belongs to
$\mathfrak{F}(\\{\mathcal{I}^{(\tau+1)}_{i}\\})$. Let
$k\in\\{1,\ldots,N^{(\tau)}\\}$ be the minimum index such that

$\mathcal{R}(A,P_{k}^{(\tau)})\not\subset{\rm im}\,{P_{k}^{(\tau)}}+{\rm
im}\,{P_{0}^{(\tau)}}$

where $P_{k}^{(\tau)}:=f_{P_{i}}(\mathcal{I}_{k}^{(\tau)})$ and
$P_{0}^{(\tau)}:=f_{P_{0}}(\\{\mathcal{I}_{i}^{(\tau)}\\})$. Because
$\mathcal{I}_{k}^{(\tau)}$ is not partitioned at the $\tau$th step, there
exists $j\in\\{1,\ldots,N^{(\tau+1)}\\}$ such that
$\mathcal{I}_{j}^{(\tau+1)}=\mathcal{I}_{k}^{(\tau)}.$ Hence

$\mathcal{R}(A,P_{j}^{(\tau+1)})=\mathcal{R}(A,P_{k}^{(\tau)})\subset{\rm
im}\,{P_{k}^{(\tau)}}+{\rm im}\,{P_{0}^{(\tau+1)}}$

since (12a) holds for $j$. Because
$\\{\mathcal{I}_{i}\\}\subset\mathfrak{F}(\\{\mathcal{I}_{i}^{\tau}\\})$,
there exists an index set $\mathcal{L}\subset\\{1,\ldots,N\\}$ such that
$\cup_{l\in\mathcal{L}}\mathcal{I}_{l}=\mathcal{I}_{k}^{(\tau)}=\mathcal{I}_{j}^{(\tau+1)}$.
Because $\\{\mathcal{I}_{i}\\}$ satisfies (12a), we have

$\mathcal{R}(A,P_{\mathcal{L}})=\mathcal{R}(A,P_{k}^{(\tau)})\subset{\rm
im}\,{P_{k}^{(\tau)}}+{\rm im}\,{P_{0}}$

where $P_{\mathcal{L}}$ is composed of $P_{l}:=f_{P_{i}}(\mathcal{I}_{l})$ for
$l\in\mathcal{L}$ and $P_{0}:=f_{P_{0}}(\\{\mathcal{I}_{i}\\})$. By taking the
intersection, we have

$\mathcal{R}(A,P_{k}^{(\tau)})\subset{\rm im}\,{P_{k}^{(\tau)}}+({\rm
im}\,{P_{0}^{(\tau+1)}}\cap{\rm im}\,{P_{0}}).$

Taking the projection onto the orthogonal subspace of ${\rm
im}\,{P_{k}}^{(\tau)}$, we have

$\pi_{{\rm
im}\,{P_{k}}^{(\tau)\perp}}\mathcal{R}(A,P_{k}^{(\tau)})\subset\pi_{{\rm
im}\,{P_{k}}^{(\tau)\perp}}({\rm im}\,{P_{0}^{(\tau+1)}}\cap{\rm
im}\,{P_{0}}).$

The optimality of $\\{\mathcal{I}_{i}^{(\tau+1)}\\}$ in the problem (21)
yields

$\pi_{{\rm im}\,{P_{k}}^{(\tau)\perp}}({\rm im}\,{P_{0}^{(\tau+1)}}\cap{\rm
im}\,{P_{0}})=\pi_{{\rm im}\,{P_{k}}^{(\tau)\perp}}{\rm
im}\,{P_{0}^{(\tau+1)}}.$

This identity and the relation
$\cup_{l\in\mathcal{L}}\mathcal{I}_{l}=\mathcal{I}_{k}^{(\tau)}=\mathcal{I}_{j}^{(\tau+1)}$
imply that {$\mathcal{I}_{i}\\}$ is a partition of
$\\{\mathcal{I}_{i}^{(\tau+1)}\\}$. ∎

###### Proof.

_Proof of Proposition 3:_ Let $P_{i}$ and $P_{0}$ be the corresponding
matrices of the initial cluster set and $P_{i}^{\prime}$ and $P_{0}^{\prime}$
be the ones corresponding to the expanded cluster set. Because ${\rm
im}\,{P_{0}}\subset{\rm im}\,{P_{0}^{\prime}}$, we have

$\mathcal{R}(A,P_{i})\subset{\rm im}\,{P_{i}}+{\rm im}\,{P_{0}}\subset{\rm
im}\,{P_{i}}+{\rm im}\,{P_{0}^{\prime}},\quad i\notin\mathcal{J}.$

Similarly, because ${\rm im}\,{P_{\mathcal{J}}}\subset{\rm
im}\,{P_{\mathcal{J^{\prime}}}},$ where $P_{\mathcal{J}}$ is composed of
$P_{j}$ for $j\in\mathcal{J}$, the condition (19) holds for
$i\notin\mathcal{J}$. ∎

###### Proof.

_Proof of Proposition 4:_ From Assumption 2, the condition of Assumption 1
holds. Thus Theorem 1 leads to the claim. ∎

###### Proof.

_Proof of Proposition 5:_ Because $\hat{F}_{i}$ for $i=0,1,\ldots,N$ are
affine maps and any norm operator is convex, there exist $\hat{A}_{0}$ and
$(\hat{A}_{i},\hat{R}_{i})$ that satisfy (25) for any $i=0,1,\ldots,N$. ∎

###### Proof.

_Proof of Theorem 5:_ As in the proof of Theorem 3, we can obtain an
equivalent system

$\left\\{\begin{array}[]{l}\left\\{\begin{array}[]{cl}\dot{\epsilon}_{i}&=\mathbf{A}_{i}\epsilon_{i}\\\
\dot{\xi}_{i}&=\hat{A}_{i}\xi_{i}+B_{i}\hat{u}_{i}\\\
\hat{u}_{i}&=\hat{K}_{i}(C_{i}\xi_{i}-\mathbf{C}_{i}\epsilon_{i})\end{array},\quad
i=1,\ldots,N\right.\\\
\begin{array}[]{cl}\dot{\xi}_{0}&=\hat{A}_{0}\xi_{0}+\sum_{i=1}^{N}\hat{R}_{i}\xi_{i}+B_{0}u_{0}\\\
\dot{e}&=\hat{A}_{e}e+\hat{F}_{0}\xi_{0}\\\
\hat{u}_{0}&=\hat{K}_{0}C_{0}(\xi_{0}+\sum_{i=1}^{N}P_{0}^{\sf
T}P_{i}\xi_{i}+P_{0}^{\sf T}e).\end{array}\end{array}\right.$

From the cascade structure and the assumption on internal stability, the claim
holds. ∎

###### Proof.

_Proof of Proposition 6:_ By simple algebra, it can be confirmed that the
matrix

$U_{e}:=\left[\begin{array}[]{ccc}0&P_{0}&I\\\ I&P_{0}&I\end{array}\right]$

satisfies the conditions for the functional observers of ${\rm
col}(C_{i}\xi_{i})$. ∎

## References

  * [1] R. Rajkumar, “A cyber-physical future,” _Proc. IEEE_ , vol. 100, no. Special Centennial Issue, pp. 1309–1312, 2012.
  * [2] M. D. Ilić, L. Xie, U. A. Khan, and J. M. F. Moura, “Modeling of future cyber-physical energy systems for distributed sensing and control,” _IEEE Trans. Syst., Man, and Cybern. A, Syst., Humans_ , vol. 40, no. 4, pp. 825–838, 2010.
  * [3] N. Sandell, P. Varaiya, M. Athans, and M. Safonov, “Survey of decentralized control methods for large scale systems,” _IEEE Trans. Autom. Control_ , vol. 23, no. 2, pp. 108–128, 1978.
  * [4] L. Bakule, “Decentralized control: An overview,” _Annual Reviews in Control_ , vol. 32, no. 1, pp. 87–98, 2008.
  * [5] D. D. Šiljak, _Decentralized Control of Complex Systems_. Courier Corporation, 2011.
  * [6] E. V. Larsen and D. A. Swann, “Applying power system stabilizers part I: General concepts,” _IEEE Trans. Power App. Syst._ , vol. PAS-100, no. 6, pp. 3017–3024, 1981.
  * [7] C. Langbort and J. Delvenne, “Distributed design methods for linear quadratic control and their limitations,” _IEEE Trans. Autom. Control_ , vol. 55, no. 9, pp. 2085–2093, 2010.
  * [8] T. Ishizaki, T. Sadamoto, J. Imura, H. Sandberg, and K. H. Johansson, “Retrofit control: Localization of controller design and implementation,” _Automatica_ , vol. 95, pp. 336–346, 2018.
  * [9] T. Ishizaki, H. Sasahara, M. Inoue, T. Kawaguchi, and J. Imura, “Modularity-in-design of dynamical network systems: Retrofit control approach,” _IEEE Trans. Autom. Control_ , 2020, (to appear). [Online]. Available: https://arxiv.org/abs/1902.01625.
  * [10] H. Sasahara, T. Ishizaki, and J. Imura, “Parameterization of all output-rectifying retrofit controllers,” 2020, (submitted). [Online]. Available: https://arxiv.org/abs/2008.11552.
  * [11] F. Farokhi, C. Langbort, and K. H. Johansson, “Optimal structured static state-feedback control design with limited model information for fully-actuated systems,” _Automatica_ , vol. 49, no. 2, pp. 326–337, 2013\.
  * [12] Z.-P. Jiang, I. M. Mareels, and Y. Wang, “A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems,” _Automatica_ , vol. 32, no. 8, pp. 1211–1215, 1996.
  * [13] S. N. Dashkovskiy, B. S. Rüffer, and F. R. Wirth, “Small gain theorems for large scale systems and construction of ISS Lyapunov functions,” _SIAM Journal on Control and Optimization_ , vol. 48, no. 6, pp. 4089–4118, 2010.
  * [14] S. Riverso, F. Boem, G. Ferrari-Trecate, and T. Parisini, “Plug-and-play fault detection and control-reconfiguration for a class of nonlinear large-scale constrained systems,” _IEEE Trans. Autom. Control_ , vol. 61, no. 12, pp. 3963–3978, 2016.
  * [15] R. Pates and G. Vinnicombe, “Scalable design of heterogeneous networks,” _IEEE Trans. Autom. Control_ , vol. 62, no. 5, pp. 2318–2333, 2017.
  * [16] Z. Qu and M. A. Simaan, “Modularized design for cooperative control and plug-and-play operation of networked heterogeneous systems,” _Automatica_ , vol. 50, no. 9, pp. 2405–2414, 2014.
  * [17] Y. Wang, N. Matni, and J. C. Doyle, “Separable and localized system-level synthesis for large-scale systems,” _IEEE Trans. Autom. Control_ , vol. 63, no. 12, pp. 4234–4249, 2018.
  * [18] J. Anderson, J. Doyle, S. Low, and N. Matni, “System level synthesis,” _Annual Reviews in Control_ , vol. 47, pp. 364–393, 2019.
  * [19] S. Hara, J. Imura, K. Tsumura, T. Ishizaki, and T. Sadamoto, “Glocal (global/local) control synthesis for hierarchical networked systems,” in _Proc. Multi-Conference on Systems and Control_ , 2015, pp. 107–112.
  * [20] J. H. Chow and P. V. Kokotović, “Time scale modeling of sparse dynamic networks,” _IEEE Trans. Autom. Control_ , vol. 30, no. 8, pp. 714–722, 1985\.
  * [21] J. H. Chow, “Slow coherency and aggregation,” in _Power System Coherency and Model Reduction_ , J. H. Chow, Ed. Springer, 2013, ch. 3.
  * [22] H. Sasahara, T. Ishizaki, T. Sadamoto, J. Imura, H. Sandberg, and K. H. Johansson, “Glocal control for network systems via hierarchical state-space expansion,” in _Proc. 56th IEEE Conference on Decision and Control (CDC)_ , 2017, pp. 6346–6351.
  * [23] H. Sasahara, T. Ishizaki, J. Imura, H. Sandberg, and K. H. Johansson, “Hierarchical model decomposition for distributed design of glocal controllers,” in _Proc. 58th IEEE Conference on Decision and Control (CDC)_ , 2019, pp. 3540–3545.
  * [24] S. Hara, H. Shimizu, and T. Kim, “Consensus in hierarchical multi-agent dynamical systems with low-rank interconnections: Analysis of stability and convergence rates,” in _Proc. 2009 American Control Conference_ , 2009, pp. 5192–5197.
  * [25] D. Tsubakino and S. Hara, “Eigenvector-based intergroup connection of low rank for hierarchical multi-agent dynamical systems,” _Systems & Control Letters_, vol. 61, no. 2, pp. 354–361, 2012.
  * [26] I. Kamwa, R. Grondin, and Y. Hebert, “Wide-area measurement based stabilizing control of large power systems–a decentralized/hierarchical approach,” _IEEE Trans. Power Syst._ , vol. 16, no. 1, pp. 136–153, 2001\.
  * [27] M. Brucoli, P. Pugliese, F. Torelli, and M. Trovato, “Decentralized two-level excitation control of multimachine power systems,” _International Journal of Electrical Power & Energy Systems_, vol. 6, no. 4, pp. 230–238, 1984\.
  * [28] A. Rubaai, “Transient stability control: A multi-level hierarchical approach,” _IEEE Trans. Power Syst._ , vol. 6, no. 1, pp. 262–268, 1991\.
  * [29] T. L. Huang, T. Y. Hwang, and W. T. Yang, “Two-level optimal output feedback stabilizer design,” _IEEE Trans. Power Syst._ , vol. 6, no. 3, pp. 1042–1048, 1991.
  * [30] A. Chakrabortty, “Wide-area communication and control: A cyber-physical perspective,” in _Smart Grid Control: Overview and Research Opportunities_ , J. Stoustrup, A. Annaswamy, A. Chakrabortty, and Z. Qu, Eds. Springer International Publishing, 2019, pp. 139–164.
  * [31] A. Chakrabortty and P. P. Khargonekar, “Introduction to wide-area control of power systems,” in _2013 American Control Conference (ACC)_ , 2013, pp. 6758–6770.
  * [32] D. Soudbakhsh, A. Chakrabortty, and A. M. Annaswamy, “A delay-aware cyber-physical architecture for wide-area control of power systems,” _Control Engineering Practice_ , vol. 60, pp. 171–182, 2017.
  * [33] I. Zenelis and X. Wang, “Wide-area damping control for interarea oscillations in power grids based on PMU measurements,” _IEEE Contr. Syst. Lett._ , vol. 2, no. 4, pp. 719–724, 2018.
  * [34] N. Xue and A. Chakrabortty, “Control inversion: A clustering-based method for distributed wide-area control of power systems,” _IEEE Control Netw. Syst._ , vol. 6, no. 3, pp. 937–949, 2019.
  * [35] A. Thakallapelli and S. Kamalasadan, “Alternating direction method of multipliers (ADMMs) based distributed approach for wide-area control,” _IEEE Trans. Ind. Appl._ , vol. 55, no. 3, pp. 3215–3227, 2019.
  * [36] A. C. Antoulas, _Approximation of Large-Scale Dynamical Systems_. Society for Industrial and Applied Mathematics, 2005.
  * [37] P. Kokotović, H. K. Khalil, and J. O’Reilly, _Singular Perturbation Methods in Control: Analysis and Design_. Society for Industrial and Applied Mathematics, 1999.
  * [38] P. Kundur, _Power System Stability and Control_. McGraw-Hill Education, 1994.
  * [39] M. Darouach, “Existence and design of functional observers for linear systems,” _IEEE Trans. Autom. Control_ , vol. 45, no. 5, pp. 940–943, 2000\.
  * [40] T. Fortmann and D. Williamson, “Design of low-order observers for linear feedback control laws,” _IEEE Trans. Autom. Control_ , vol. 17, no. 3, pp. 301–308, 1972.
  * [41] K. Zhou, J. C. Doyle, and K. Glover, _Robust and Optimal Control_. Prentice Hall, 1996.
  * [42] P. W. Sauer, M. A. Pai, and J. H. Chow, _Power System Toolbox_. John Wiley & Sons, Ltd., 2017, ch. 11, pp. 305–325.

Hampei Sasahara(M’??) Biography text here.

Takayuki Ishizaki(M’??) Biography text here.

Jun-ichi Imura(M’??) Biography text here.

Henrik Sandberg(M’??) Biography text here.

Karl Henrik Johansson(M’??) Biography text here.