# Decentralized and Compositional Interconnection Topology Synthesis for
Linear Networked Systems

Shirantha Welikala, Hai Lin and Panos J. Antsaklis The support of the National
Science Foundation (Grant No. CNS-1830335, IIS-2007949) is gratefully
acknowledged.The authors are with the Department of Electrical Engineering,
College of Engineering, University of Notre Dame, IN 46556,
<EMAIL_ADDRESS>
###### Abstract

In this paper, we consider networked systems comprised of interconnected sets
of linear subsystems and propose a decentralized and compositional approach to
stabilize or dissipativate such linear networked systems via optimally
modifying some existing interconnections and/or creating entirely new
interconnections. We also extend this interconnection topology synthesis
approach to ensure the ability to stabilize or dissipativate such linear
networked systems under distributed (local) feedback control. To the best of
the authors’ knowledge, this is the first work that attempts to address the
optimal interconnection topology synthesis problem for linear networked
systems. The proposed approach in this paper only involves solving a sequence
of linear matrix inequality problems (one at each subsystem). Thus, using
standard convex optimization toolboxes, it can be implemented efficiently and
scalably in a decentralized and compositional manner. Apart from many generic
linear networked systems applications (e.g., power grid control), a unique
application for the proposed interconnection topology synthesis approach is in
generating random stable (or dissipative, stabilizable, dissipativate-able)
linear networked systems for simulation purposes. We also include an
interesting case study where the proposed interconnection topology synthesis
approach is compared with an alternative approach that only uses dissipativity
information of the involved subsystems.

##  I Introduction

In recent years, attention towards analysis, controller synthesis, topology
synthesis as well as optimization of large-scale networked systems (comprised
of dynamically coupled subsystems) has been renewed due to their various
emerging applications (e.g., in critical infrastructure networks like supply
chains [1], power grids [2], etc.) and confronting unique challenges (e.g.,
resilience [3], security [4], etc.).

For such networked systems, a large number of distributed control solutions
have been proposed in the literature that can not only stabilize but also
optimize some performance metrics of interest [5] during their operation.
However, almost all such distributed control solutions are synthesized by a
centralized design process which raises concerns related to their security,
scalability, and compositionality [6].

Over the years, there have been several attempts to address this decentralized
controller synthesis problem exploiting weak couplings [7], hierarchical
techniques [8], and decomposition techniques [9]. In particular, the work in
[9] proposes a natural and efficient decomposition technique for analysis and
synthesis of distributed controllers inspired by the Sylvester’s criterion
[10]. Motivated by the attractive qualities of this Sylvester’s criterion
based decomposition approach [9], our recent work in [6] (and its extension
[11]) generalized it so that many fundamental linear control solutions (e.g.,
dissipativity analysis, linear observer design, etc.) can be implemented in a
decentralized as well as compositional manner over large-scale linear
networked systems.

Nevertheless, a major challenge faced by this approach (as well as many other
control solutions proposed for large-scale networked systems) is the
incompatibility between the considered networked system and the proposed
solution. Such an incompatibility may be due to the inherent weaknesses in the
networked system and/or in the proposed solution. For example, a networked
system may not yield a conclusive (and desired) result under a particular
analysis technique. Similarly, a networked system may not be capable of
yielding desired properties under a particular class of controllers.

To address this incompatibility issue, we can either change the networked
system to match the proposed solution (e.g., see [12]), or improve/specialize
the proposed solution so as to handle the considered networked system (e.g.,
see [6]). While in many scenarios it is natural and practical (and even
advisable) to take the latter approach, in some instances, the prior approach
is also a valid and sensible option to take. Most importantly, developing
techniques to systematically change the networked systems can lead to
insightful findings. For example, assuming the proposed control solution
sufficiently rich, we might be able to answer questions like: What kinds of
network topologies are more robust to the disturbances? What are the most
critical interconnections in the networked system? What is the most cost
efficient network topology?

In this paper, we set out to solve the said incompatibility issue faced by the
Sylvester’s criterion based decentralized and compositional approach proposed
in [6] (intended for analysis and distributed controller synthesis of large-
scale linear networked systems). To this end, we propose to change the
networked system so that it matches the approach proposed in [6]. In
particular, in the considered networked system, we treat some inter-subsystem
interconnections (if not all) as design variables and explore the possibility
to: (1) change those variable interconnections from their nominal values, (2)
create entirely new interconnections, and/or (3) remove existing
interconnections, such that the proposed approach in [6] can yield conclusive
as well as desired results. In essence, this can be seen as an effort to
synthesize the interconnection topology for linear networked systems.

In fact, there have been only very few attempts on designing interconnection
topologies for networked systems. For example, the work in [13] considers
designing a network topology to make the communications optimally efficient
for a continuous-time average consensus protocol. The proposed solution in
[13] takes the form of a mixed integer semidefinite program - which does not
scale well. The interconnection matrix synthesis problem is considered limited
to linear and positive networked systems in [14]. Several other
interconnection matrix synthesis techniques such as the ones proposed in [15,
16] and [17] have been reviewed in our recent work [12] (see also its
extension [18]).

In particular, the work in [12] proposes an interconnection matrix synthesis
technique for non-linear networked systems using only the subsystem
dissipativity properties (i.e., without using the complete knowledge of the
non-linear subsystem dynamics). However, in this paper, we limit to linear
networked systems and use the complete knowledge of the linear subsystem
dynamics for interconnection topology synthesis. Nevertheless, as we will show
in this paper (particularly in our case study), there is a clear advantage due
to the use of such additional information regarding the networked system as
compared to [12].

####  I-1 Contributions

Our contributions can be summarized as follows: (1) We take a control
theoretic approach to formulate several interconnection topology synthesis
problems arising related to linear networked systems as LMI problems; (2)
Since the proposed interconnection topology synthesis approach is inspired by
[6], it is inherently decentralized and compositional; (3) Moreover, it can be
used in scenarios where the analysis and controller synthesis approaches
proposed in [6] returns inconclusive; (4) We also provide candidate local
objective functions that can be used to penalize deviations from a nominal set
of specifications (topology). (5) The proposed interconnection topology
synthesis approach can be used to generate random linear networked systems
with certain qualities (e.g., stabilizability) - which is helpful when
designing, testing, and validating control strategies developed for networked
systems. (6) Similar to [6], the proposed approach can be extended to address
a wide range of problems arising related to linear networked systems based on
fundamental linear systems theory (e.g., optimal topology synthesis to ensure
observability). (7) We provide candidate local objective functions that can be
used to penalize deviations from a nominal interconnection topology; (8) We
provide a detailed case study comparing the interconnection topology synthesis
approaches proposed in this paper and in our recent work [12];

####  I-2 Organization

This paper is organized as follows. Section II presents the details of the
considered class of networked systems and motivates the need for
interconnection topology synthesis. Section III summarizes several important
preliminary concepts. Our main theoretical results that address several
different interconnection topology synthesis problems of interest are
presented in Sec. IV along with several important remarks. A case study with
fundamental details, numerical results, discussions, and comparisons are
provided in Sec. V before concluding the paper in Sec. VI.

####  I-3 Notation

The sets of real and natural numbers are denoted by $\mathbb{R}$ and
$\mathbb{N}$, respectively. We define
$\mathbb{N}_{N}\triangleq\\{1,2,\ldots,N\\}$ where $N\in\mathbb{N}$. An
$n\times m$ block matrix $A$ can be represented as
$A=[A_{ij}]_{i\in\mathbb{N}_{n},j\in\mathbb{N}_{m}}$ where $A_{ij}$ is the
$(i,j)$th block of $A$ (for indexing purposes, either subscripts or
superscripts may be used, i.e., $A_{ij}\equiv A^{ij}$). If
$\Psi\triangleq[\Psi^{kl}]_{k,l\in\mathbb{N}_{m}}$ where
$\Psi^{kl}\triangleq[\Psi^{kl}_{ij}]_{i,j\in\mathbb{N}_{n}}$, its block
element-wise form [6] is denoted as
$\mbox{BEW}(\Psi)\triangleq[[\Psi^{kl}_{ij}]_{k,l\in\mathbb{N}_{m}}]_{i,j\in\mathbb{N}_{n}}$.
The transpose of a matrix $A$ is denoted by $A^{\top}$ and
$(A^{\top})^{-1}=A^{-\top}$. The zero and identity matrices are denoted by
$\mathbf{0}$ and $\mathbf{I}$, respectively (dimensions will be clear from the
context). A symmetric positive definite (semi-definite) matrix
$A\in\mathbb{R}^{n\times n}$ is represented as $A=A^{\top}>0$ ($A=A^{\top}\geq
0$). Unless stated otherwise, we assume $A>0\iff A=A^{\top}>0$ (i.e., symmetry
is implied by the positive definiteness). $\mathbf{1}_{\\{\cdot\\}}$ is the
indicator function and
$e_{ij}\triangleq\mathbf{I}\cdot\mathbf{1}_{\\{i=j\\}}$.

##  II Problem Formulation

###  II-A The Networked System

We consider a networked dynamical system $\mathcal{G}_{N}$ comprised of $N$
interconnected subsystems denoted by $\\{\Sigma_{i}:i\in\mathbb{N}_{N}\\}$.
The dynamics of the $i$th subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ are given
by

$\displaystyle\dot{x}_{i}(t)=$
$\displaystyle\sum_{j\in\bar{\mathcal{E}}_{i}}A_{ij}x_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}B_{ij}u_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}E_{ij}w_{j}(t),$
(1) $\displaystyle y_{i}(t)=$
$\displaystyle\sum_{j\in\bar{\mathcal{E}}_{i}}C_{ij}x_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}D_{ij}u_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}F_{ij}w_{j}(t),$

where $x_{i}(t)\in\mathbb{R}^{n_{i}},\ u_{i}(t)\in\mathbb{R}^{p_{i}},\
w_{i}(t)\in\mathbb{R}^{q_{i}}$ and $y_{i}(t)\in\mathbb{R}^{m_{i}}$
respectively represents the state, input, disturbance and output specific to
the subsystems $\Sigma_{i}$ at time $t\in\mathbb{R}_{\geq 0}$. In (1),
$\bar{\mathcal{E}}_{i}\triangleq\mathcal{E}_{i}\cup\\{i\\}$ where
$\mathcal{E}_{i}\subset\mathbb{N}_{N}$ is the set of “in-neighbors” of the
subsystem $\Sigma_{i}$. Formally, any subsystem $\Sigma_{j}$ is an “in-
neighbor” of the subsystem $\Sigma_{i}$ (i.e., $j\in\mathcal{E}_{i}$) iff the
matrices $A_{ij},B_{ij},C_{ij},D_{ij},E_{ij},F_{ij}$ in (1) are not all zero
matrices. Conversely,
$\bar{\mathcal{F}}_{i}\triangleq\mathcal{F}_{i}\cup\\{i\\}$ where
$\mathcal{F}_{i}\triangleq\\{j:j\in\mathbb{N}_{N},\mathcal{E}_{j}\ni i\\}$ is
the set of “out-neighbors” of the subsystem $\Sigma_{i}$. An example networked
system can be seen in Fig. 1.

Figure 1: An example networked dynamical system $\mathcal{G}_{N}$.

By writing (1) for all $i\in\mathbb{N}_{N}$ and concatenating suitably, we can
get the dynamics of the networked system $\mathcal{G}_{N}$ as

$\displaystyle\dot{x}(t)=$ $\displaystyle\ Ax(t)+Bu(t)+Ew(t),$ (2)
$\displaystyle y(t)=$ $\displaystyle\ Cx(t)+Du(t)+Fw(t),$

where $A=[A_{ij}]_{i,j\in\mathbb{N}_{N}}$,
$B=[B_{ij}]_{i,j\in\mathbb{N}_{N}}$, $E=[E_{ij}]_{i,j\in\mathbb{N}_{N}}$,
$C=[C_{ij}]_{i,j\in\mathbb{N}_{N}}$, $D=[D_{ij}]_{i,j\in\mathbb{N}_{N}}$ and
$F=[F_{ij}]_{i,j\in\mathbb{N}_{N}}$ are all $N\times N$ block matrices, and
$x(t)\in\mathbb{R}^{n},\ u(t)\in\mathbb{R}^{p},w(t)\in\mathbb{R}^{q}$ and
$y(t)\in\mathbb{R}^{m}$ (with $n=\sum_{i\in\mathbb{N}_{N}}n_{i}$,
$p=\sum_{i\in\mathbb{N}_{N}}p_{i}$, $q=\sum_{i\in\mathbb{N}_{N}}q_{i}$ and
$m=\sum_{i\in\mathbb{N}_{N}}m_{i}$) are all $N\times 1$ block matrices
respectively representing the networked system’s state, input, disturbance and
output at time $t\in\mathbb{R}_{\geq 0}$.

###  II-B Distributed Controllers

To enforce desired properties (e.g., stability) upon the networked system, a
subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ can use a distributed state feedback
controller:

$u_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}K_{ij}x_{j}(t).$ (3)

By writing (3) for all $i\in\mathbb{N}_{N}$ and concatenating appropriately,
we get the global form of the distributed feedback controller as

$u(t)=Kx(t),$ (4)

where $K=[K_{ij}]_{i,j\in\mathbb{N}_{N}}$.

Note that the unspecified blocks in various block matrices in both (2) and (4)
are zeros matrices (e.g., $A_{ij}=0,\forall j\not\in\mathcal{E}_{i}$).

###  II-C Interconnection Topology Synthesis

Even though state feedback control is a reasonable approach to enforce desired
properties (e.g., stability) upon the networked system, it may be not useful
in two scenarios: (1) when the networked system inherently involves no control
inputs (i.e., when $B=D=\mathbf{0}$ in (2)), or (2) when the networked system
is inherently incapable of achieving the desired properties under state
feedback control (e.g., if (2) is not stabilizable when the desired property
is stability). To address these inherent weaknesses of the networked system,
in this paper, we propose to optimally adjust the interconnection parameters
of the networked system (mainly $A_{ij}$ blocks with $i\neq j$ in (1)). Hence
this approach can be seen as an attempt to synthesize the interconnection
topology of the networked system.

Note also that, for the purposes of analysis and controller synthesis of the
networked system (2), we can use the decentralized and compositional technique
proposed in [6]. However, due to the used assumptions in [6], this
decentralized and compositional technique can return inconclusive (when
analyzing) or infeasible (when synthesizing controllers) [11]. Nevertheless,
as we will show in the sequel, this technical weakness can also be addressed
by optimally adjusting the interconnection parameters of the networked system.

##  III Preliminaries

###  III-A Stability and Dissipativity

Since our main goal is to synthesize the interconnection topology of the
linear networked system (2) so as to enforce properties like stability or
dissipativity (both without or with distributed feedback control (3)), we next
briefly introduce some relevant stability and dissipativity results.

Consider the linear time-invariant (LTI) system

$\displaystyle\dot{x}(t)=Ax(t)+Bu(t),$ (5) $\displaystyle y(t)=Cx(t)+Du(t),$

where $x(t)\in\mathbb{R}^{n},u(t)\in\mathbb{R}^{p}$, and
$y(t)\in\mathbb{R}^{m}$ respectively represent the state, control input, and
output at time $t\in\mathbb{R}_{\geq 0}$.

##### Stability

A well-known necessary and sufficient condition for the stability of (5) is
given in the following lemma as a linear matrix inequality (LMI).

###### Lemma 1

[10] The dynamical system (5) (under $u(t)=\mathbf{0}$) is globally uniformly
(exponentially) stable iff $\exists P>0$ such that

$-A^{\top}P-PA\geq 0\ \ \ \ \ \ \mbox{($-A^{\top}P-PA>0$)}.$ (6)

Note that, henceforth, by ‘stability,’ we simply refer to global exponential
stability.

#####  $(Q,S,R)$-Dissipativity

In general, dissipativity is an important property of dynamical systems that
has many practical uses [19]. In this paper, we consider the quadratic
dissipativity property called $(Q,S,R)$-dissipativity.

###### Definition 1

[20] The dynamical system (5) is $(Q,S,R)$-dissipative from $u(t)$ to $y(t)$,
if there exists a positive definite function
$V(x):\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$ (storage function) such
that for all $t_{1}\geq t_{0}\geq 0,x(t_{0})\in\mathbb{R}^{n}$ and
$u(t)\in\mathbb{R}^{m}$, the inequality
$V(x(t_{1}))-V(x(t_{0}))\leq\int_{t_{0}}^{t_{1}}\begin{bmatrix}y(t)\\\
u(t)\end{bmatrix}^{\top}\begin{bmatrix}Q&S\\\
S^{\top}&R\end{bmatrix}\begin{bmatrix}y(t)\\\ u(t)\end{bmatrix}dt$ holds for
the given $Q\in\mathbb{R}^{m\times m},S\in\mathbb{R}^{m\times p}$ and
$R\in\mathbb{R}^{p\times p}$.

Through appropriate choices of $Q,S$ and $R$ matrices, $(Q,S,R)$-dissipativity
can capture several dynamical properties of interest, as summarized in the
following remark.

###### Remark 1

[20] The dynamical system (5) satisfying Def. 1: (i) is _passive_ iff
$Q=0,S=\frac{1}{2}\mathbf{I},R=0$; (ii) is _strictly passive_ iff
$Q=-\rho\mathbf{I},S=\frac{1}{2}\mathbf{I},R=-\nu\mathbf{I}$ where
$\rho,\nu>0$ ($\nu$, $\rho$ are passivity indices [11]); (iii) is
_$\mathcal{L}_{2}$ -stable_ iff $Q=-\mathbf{I},S=0,R=-\gamma^{2}\mathbf{I}$
where $\gamma\geq 0$ ($\gamma$ is an _$\mathcal{L}_{2}$ -gain_ of the system);
(iv) is _sector bounded_ iff $Q=-\mathbf{I},S=(a+b)\mathbf{I},R=-ab\mathbf{I}$
where $a,b\in\mathbb{R}$ ($a,b$ are sector bound parameters).

A necessary and sufficient condition for $(Q,S,R)$-dissipativity of (5) is
given in the next lemma as an LMI.

###### Lemma 2

[11] The dynamical system (5) is $(Q,S,R)$-dissipative ($-Q>0,R=R^{\top}$)
from $u(t)$ to $y(t)$ iff $\exists P>0$ such that

$\begin{bmatrix}-A^{\top}P-PA&-PB+C^{\top}S&C^{\top}\\\
-B^{\top}P+S^{\top}C&D^{\top}S+S^{\top}D+R&D^{\top}\\\
C&D&-Q^{-1}\end{bmatrix}\geq 0.$ (7)

Note that LMIs in (6) and (7) are “linear” as they contain linear terms in the
corresponding design variable $P$. As shown in [21], LMIs can be solved
efficiently and scalably using standard convex optimization algorithms.

###  III-B Interconnection Topology Synthesis

Due to the similarity between (5) and (2), LMIs (6) and (7) can respectively
be used for stability and dissipativity analysis of the networked system (2).
Note, however, that, in the LMIs (6) and (7), we cannot treat the matrix $A$
(particularly its non-diagonal elements $A_{ij}$ with $i\neq j$) as an
independent design variable separately from $P$. This is because it makes (6)
and (7) bi-linear matrix inequalities - which are non-linear and significantly
harder to solve compared to the corresponding LMIs. Therefore, synthesizing
certain elements of $A$ (i.e., the interconnection parameters of the networked
system) such that stability or dissipativity holds for the networked system
(2) is a non-trivial and challenging problem. Similarly, synthesizing certain
interconnection parameters of the networked system such that stabilizability
or dissipativate-ability holds for the networked system (2) under state
feedback control (3) is also a non-trivial and challenging problem.

We address these challenges by taking a decentralized and compositional
approach. In particular, to analyze or enforce (via state feedback control)
desired properties like stability or dissipativity of the networked system
(2), compared to solving large centralized LMIs like (6) and (7), we propose
to solve their small decentralized and compositional versions proposed in [6].
This approach allows us to sequentially synthesize the interconnection
parameters of the networked system (2) (i.e., step-by-step). In particular, at
each step, we add a new subsystem to the current network and solve a small LMI
problem where some interconnection parameters related to the new subsystem are
treated as design variables while all other interconnection parameters are
treated as fixed.

Before providing more details about this approach, we first need to outline
the decentralized and compositional approach proposed in [6] that can be used
to analyze/enforce centralized LMIs exploiting a concept named “network
matrices.”

###  III-C Network Matrices

Consider a directed network $\mathcal{G}_{n}=(\mathcal{V},\mathcal{E})$ where
$\mathcal{V}\triangleq\\{\Sigma_{i}:i\in\mathbb{N}_{n}\\}$ is the set of
subsystems (nodes), $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ is the
set of inter-subsystem interconnections (edges) and $n\in\mathbb{N}$. We next
recall a class of matrices named “network matrices” introduced in [6]
corresponding to such a network $\mathcal{G}_{n}$.

###### Definition 2

[6] Given a network $\mathcal{G}_{n}=(\mathcal{V},\mathcal{E})$, any $n\times
n$ block matrix
$\Theta=\begin{bmatrix}\Theta_{ij}\end{bmatrix}_{i,j\in\mathbb{N}_{n}}$ is a
corresponding _network matrix_ if: (1) $\Theta_{ij}$ contains information
specific only to the subsystems $\Sigma_{i}$ and $\Sigma_{j}$, and (2)
$(\Sigma_{i},\Sigma_{j})\not\in\mathcal{E}$ and
$(\Sigma_{j},\Sigma_{i})\not\in\mathcal{E}$ implies
$\Theta_{ij}=\Theta_{ji}=\mathbf{0}$, for all $i,j\in\mathbb{N}_{n}$.

According to this definition, any $n\times n$ block matrix
$\Theta=[\Theta_{ij}]_{i,j\in\mathbb{N}_{n}}$ is a network matrix of
$\mathcal{G}_{n}$ if $\Theta_{ij}$ is a coupling weight matrix corresponding
to the edge $(\Sigma_{i},\Sigma_{j})\in\mathcal{V}$. Moreover, any $n\times n$
block diagonal matrix $\Theta=\text{diag}(\Theta_{ii}:i\in\mathbb{N}_{n})$
where $\Theta_{ii}$ contains information specific only to the subsystem
$\Sigma_{i}$, is a network matrix of any network with $n\in\mathbb{N}$
subsystems. The following lemmas provide several useful properties of such
network matrices established in [6].

###### Lemma 3

[6] Given a network $\mathcal{G}_{n}$, a few corresponding block network
matrices $\Theta,\Phi,\\{\Psi^{kl}:k,l\in\mathbb{N}_{m}\\}$, and some
arbitrary block-block matrix
$\Psi\triangleq[\Psi^{kl}]_{k,l\in\mathbb{N}_{m}}$:

  1. 1.

$\Theta^{\top}$, $\alpha\Theta+\beta\Phi$ are network matrices for any
$\alpha,\beta\in\mathbb{R}$.

  2. 2.

$\Phi\Theta$, $\Theta\Phi$ are network matrices whenever $\Phi$ is a block
diagonal network matrix.

  3. 3.

$\mbox{BEW}(\Psi)\triangleq[[\Psi^{kl}_{ij}]_{k,l\in\mathbb{N}_{m}}]_{i,j\in\mathbb{N}_{n}}$
is a network matrix.

The above lemma enables claiming custom block matrices as “network matrices”
by enforcing additional conditions. For example, if $A$ and $P$ are two block
network matrices and $P$ is block diagonal, then: (1) $A^{\top}P,PA$ and
$A^{\top}P+PA$ are all network matrices, and (2) if
$\scriptsize\Psi\triangleq\begin{bmatrix}P&A^{\top}P\\\ PA&P\end{bmatrix}$ is
some block-block matrix, its _block element-wise_ (BEW) form
$\mbox{BEW}(\Psi)\triangleq$
$\scriptsize\begin{bmatrix}\begin{bmatrix}P_{ii}e_{ij}&M_{ji}^{\top}P_{jj}\\\
P_{ii}M_{ij}&P_{ii}e_{ij}\end{bmatrix}\end{bmatrix}_{i,j\in\mathbb{N}_{N}}$ is
a network matrix.

###### Lemma 4

[6] Let $\Psi=[\Psi^{kl}]_{k,l\in\mathbb{N}_{m}}$ be an $m\times m$ block-
block matrix where $\Psi^{kl},\forall k,l\in\mathbb{N}_{m}$ are $n\times n$
block matrices. Then,
$\Psi>0\iff\text{BEW}(\Psi)\triangleq[[\Psi^{kl}_{ij}]_{k,l\in\mathbb{N}_{m}}]_{i,j\in\mathbb{N}_{n}}>0$.

Inspired by Sylvester’s criterion [10], the following lemma provides a
decentralized and compositional testing criterion to evaluate the positive
definiteness of an $N\times N$ block matrix
$W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}$ (for more details, see [11]).

###### Lemma 5

[6] A symmetric $N\times N$ block matrix $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$
if and only if

$\tilde{W}_{ii}\triangleq
W_{ii}-\tilde{W}_{i}\mathcal{D}_{i}\tilde{W}_{i}^{\top}>0,\ \ \ \ \forall
i\in\mathbb{N}_{N},$ (8)

where

$\displaystyle\tilde{W}_{i}\triangleq$ $\displaystyle\
[\tilde{W}_{ij}]_{j\in\mathbb{N}_{i-1}}\triangleq
W_{i}(\mathcal{D}_{i}\mathcal{A}_{i}^{\top})^{-1},$ (9) $\displaystyle
W_{i}\triangleq$ $\displaystyle\ [W_{ij}]_{j\in\mathbb{N}_{i-1}},\ \ \
\mathcal{D}_{i}\triangleq\text{diag}(\tilde{W}_{jj}^{-1}:j\in\mathbb{N}_{i-1}),$
$\displaystyle\mathcal{A}_{i}\triangleq$ $\displaystyle\
\begin{bmatrix}\tilde{W}_{11}&\mathbf{0}&\cdots&\mathbf{0}\\\
\tilde{W}_{21}&\tilde{W}_{22}&\cdots&\mathbf{0}\\\
\vdots&\vdots&\vdots&\vdots\\\
\tilde{W}_{i-1,1}&\tilde{W}_{i-1,2}&\cdots&\tilde{W}_{i-1,i-1}\end{bmatrix}.$

The above lemma shows that testing positive definiteness of an $N\times N$
block matrix $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}$ can be broken down to $N$
separate smaller sequence of tests (iterations). In a network setting where
$W$ is a block network matrix corresponding to a network $\mathcal{G}_{N}$, at
the $i$th iteration (i.e., at the subsystem $\Sigma_{i}$), we now only need to
test whether $\tilde{W}_{ii}>0$, where $\tilde{W}_{ii}$ can be computed using:
(1) $\\{W_{ij}:j\in\mathbb{N}_{i}\\}$ (related blocks to the subsystem
$\Sigma_{i}$ extracted from $W$), (2)
$\\{\tilde{W}_{ij}:j\in\mathbb{N}_{i-1}\\}$ (computed using (9) at subsystem
$\Sigma_{i}$), and (3)
$\\{\\{\tilde{W}_{jk}:k\in\mathbb{N}_{j}\\}:j\in\mathbb{N}_{i-1}\\}$ (blocks
computed using (9) at previous iteratins/subsystems
$\\{\Sigma_{j}:j\in\mathbb{N}_{i-1}\\}$). Note also that, using Schur
complement theory, the matrix inequality $\tilde{W}_{ii}>0$ (8) can be
transformed into a form that is linear in $[W_{ij}]_{j\in\mathbb{N}_{i-1}}$.

Therefore it is clear that Lm. 5 can be used to efficiently test/enforce the
positive definiteness of a network matrix in a decentralized manner. In fact,
as shown in [11], for some network topologies, this process is fully
distributed (i.e., no communications are required between non-neighboring
subsystems). Moreover, the compositionally of this process (i.e., the
resilience to subsystem removals/additions from/to the network) has also been
established in [6]. This decentralized and compositional approach to
test/enforce the positive-definiteness of a network matrix $W$ is outlined in
Alg. 1.

##  IV Main Results

Algorithm 1 Testing/Enforcing $W>0$ in a Network Setting.

1:Input:  $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}$

2:At each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ execute:

3: if $i=1$ then

4: Test/Enforce: $W_{11}>0$

5: Store: $\tilde{W}_{1}\triangleq[W_{11}]$ $\triangleright$ To be sent to
others.

6: else

7: From each subsystem $\Sigma_{j},j\in\mathbb{N}_{i-1}$:

8: Receive:
$\tilde{W}_{j}\triangleq[\tilde{W}_{j1},\tilde{W}_{j2},\ldots,\tilde{W}_{jj}]$

9: Receive: Required info. to compute $W_{ij}$

10: End receiving

11: Construct: $\mathcal{A}_{i},\mathcal{D}_{i}$ and $W_{i}$. $\triangleright$
Using: (9).

12: Compute: $\tilde{W}_{i}\triangleq
W_{i}(\mathcal{D}_{i}\mathcal{A}_{i}^{\top})^{-1}$ $\triangleright$ From (9).

13: Compute: $\tilde{W}_{ii}\triangleq
W_{ii}-\tilde{W}_{i}\mathcal{D}_{i}\tilde{W}_{i}^{\top}$ $\triangleright$ From
(8).

14: Test/Enforce: $\tilde{W}_{ii}>0$

15: Store: $\tilde{W}_{i}\triangleq[\tilde{W}_{i},\tilde{W}_{ii}]$
$\triangleright$ To be sent to others

16: end if

17:End execution

In this section, we present our main theoretical results on decentralized and
compositional synthesis of interconnection topology in linear networked
systems. Note that this synthesis process is designed to enforce stability or
dissipativity both without or with the help of distributed state feedback
control, i.e., we are interested in enforcing: (1) stability, (2)
stabilizability (under feedback control), (3) dissipativity and (4)
dissipativate-ability (under feedback control), via synthesizing an
appropriate interconnection topology.

Based on the subsystem dynamics (1), it is clear that the interconnection
topology of the networked system (2) is determined by the block structures of
the block network matrices $A,B,E,C,D,F$ and $K$ in (2) and (4) (see also Def.
2). Regarding these block network matrices, we make the following two
technical assumptions.

###### Assumption 1

[6] The block network matrices $C,D$, and $F$ (in (2)) are block diagonal
network matrices.

###### Assumption 2

Any block network matrix $M=[M_{ij}]_{i,j\in\mathbb{N}_{N}}$ in the set
$\\{A,B,E,K\\}$ (from (2) and (4)) unless stated otherwise, satisfies the
following statement: Corresponding to a subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$, the intrinsic matrix block $M_{ii}$ and the
interconnection matrix blocks $\\{M_{ij}:j\in\mathcal{E}_{i}\\}$ and
$\\{M_{ji}:j\in\mathcal{F}_{i}\\}$ are known and fixed while the remaining
interconnection matrix blocks $\\{M_{ij}:j\not\in\mathcal{E}_{i}\\}$ and
$\\{M_{ji}:j\not\in\mathcal{F}_{i}\\}$ are free to be designed.

Note that the above assumption relaxes a hard constraint used in [6]. For
example, in [6], $A_{ij}=\mathbf{0},\forall j\not\in\mathcal{E}_{i}$ and
$A_{ji}=\mathbf{0},\forall j\not\in\mathcal{F}_{i}$ was assumed. In contrast,
here we allow new interconnections when necessary via treating, e.g.,
$\\{A_{ij}:j\not\in\mathcal{E}_{i}\\}$ and
$\\{A_{ji}:j\not\in\mathcal{F}_{i}\\}$ as free to be designed.

Note also that we execute this design/synthesis task in a decentralized and
compositional manner. Therefore, in its $i$th step (executed at the subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$, according to Alg. 1), we only need to
synthesize a subset of interconnection matrix blocks, E.g.,
$\\{A_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\}$ and
$\\{A_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$.

###  IV-A Enforcing Stability and Stabilizability

##### Stability

The following theorem considers an un-actuated networked system and provides
how new interconnections can be created via designing the interconnection
matrix blocks $\\{A_{ij}:j\not\in\mathcal{E}_{i}\\}$ and
$\\{A_{ji}:j\not\in\mathcal{F}_{i}\\}$.

###### Theorem 1

(Stability via Topology Synthesis) The networked system (2), under Assumption
2, $u(t)=\mathbf{0}$ and $w(t)=\mathbf{0}$, is stable if at each subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$, the LMI problem

$\displaystyle\mathbb{P}_{1}:\text{Find}$ $\displaystyle\ \
P_{ii},\\{Q_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\},\\{A_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$
(10) such that $\displaystyle\ \ P_{ii}>0,\ \ \tilde{W}_{ii}>0,$

is feasible, where $\tilde{W}_{ii}$ is computed from Alg. 1 (Steps: 3-16) when
analyzing $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ with

$W_{ij}=-P_{ii}A_{ij}\mathbf{1}_{\\{j\in\bar{\mathcal{E}}_{i}\\}}-A_{ji}^{\top}P_{jj}\mathbf{1}_{\\{j\in\bar{\mathcal{F}}_{i}\\}}-Q_{ij}\mathbf{1}_{\\{j\not\in\mathcal{E}_{i}\\}},$
(11)

and the new interconnections are
$\\{A_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$ and
$\\{A_{ij}\triangleq
P_{ii}^{-1}Q_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\}$.

###### Proof:

Let us define $P\triangleq\text{diag}(P_{ii}:i\in\mathbb{N}_{N})$ and
$W\triangleq-A^{\top}P-PA$ where now $A=[A_{ij}]_{i,j\in\mathbb{N}_{N}}$
includes variable interconnection blocks
$\\{A_{ij}:j\not\in\mathcal{E}_{i}\\}$ and
$\\{A_{ji}:j\not\in\mathcal{F}_{i}\\}$ in its every $i$th column and row,
respectively, $i\in\mathbb{N}_{N}$ (replacing the fixed $\mathbf{0}$ blocks
that were there as per the original definition of $A$ given in (2)).

According to Lm. 1, this new networked system is stable if we can find $P>0$
such that $W>0$. Based on the above definition of
$W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}$, it is a symmetric network matrix (see
Def. 2) where

$W_{ij}=-P_{ii}A_{ij}-A_{ji}^{\top}P_{jj}.$ (12)

Thus, we can use Alg. 1 to test $W>0$ in a decentralized and compositional
manner via testing $\tilde{W}_{ii}>0$ at each subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$ sequentially (see Lm. 5 and (8)).

However, according to (9), testing $\tilde{W}_{ii}>0$ will be an LMI problem
only if the terms $\\{W_{ij}:j\in\mathbb{N}_{i-1}\\}$ are linear in the
program variables:
$P_{ii},\\{A_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\}$ and
$\\{A_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$. Based on (12), the
term $-P_{ii}A_{ij}$ in $W_{ij}$ become bi-linear whenever
$j\in\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}$. This calls for a change of
variables:

$Q_{ij}\triangleq P_{ii}A_{ij},\ \
j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1},$ (13)

which transforms $W_{ij}$ in (12) into the form (11) - which is linear in
terms of the new program variables:
$P_{ii},\\{Q_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\}$ and
$\\{A_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$. Consequently,
testing $\tilde{W}_{ii}>0$ takes the form of an LMI problem (10).

If all the local LMI problems (10) are feasible, $\exists P_{ii}>0$ such that
$\tilde{W}_{ii}>0,\forall i\in\mathbb{N}_{N}$ \- which implies that $\exists
P>0$ such that $W>0$. This leads to the conclusion that the new networked
system with new interconnections
$\\{A_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$ and
$\\{A_{ij}\triangleq
P_{ii}^{-1}Q_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\}$ is stable. ∎

###### Remark 2

(Objective Functions) As the objective function of the decentralized LMI
problem (10) proposed in Th. 1, we can use:

$J=\sum_{j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}}c_{ij}\|Q_{ij}-P_{ii}\bar{A}_{ij}\|+\beta_{ii}\sum_{j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}}c_{ji}\|A_{ji}-\bar{A}_{ji}\|,$
(14)

where (1) $\bar{A}_{ij}$ and $\bar{A}_{ji}$ matrices represent
desired/prescribed candidates for $A_{ij}$ and $A_{ji}$, respectively, (2)
$c_{ij}$ and $c_{ji}$ scalars represent the cost of the interconnections
$(\Sigma_{j},\Sigma_{i})$ and $(\Sigma_{i},\Sigma_{j})$, respectively, and (3)
$\beta_{ii}$ is a normalizing constant. It is easy to see that selecting
$\beta_{ii}=\|P_{ii}\|$ equally weights the two components in the above
objective function (14). However, such a choice is not practical as it makes
the objective function non-convex. Therefore, a reasonable alternative would
be to use $\beta_{ii}=\|P_{jj}\|$ \- which is a known constant when evaluating
(10). Note that an intuitive objective function of this form can also be used
with the decentralized LMI problems proposed in the sequel in Theorems 2-4
(with a few minor modifications).

###### Remark 3

(Refining Existing Interconnections) The proposed interconnection topology
synthesis approach can also be used to refine existing interconnections. To
show this, assume we are interested in refining the interconnection $A_{ij}$
for some $j\in\mathcal{E}_{i}$. First, we need to modify the set of in-
neighbors of the subsystem $\Sigma_{i}$ such that
$\mathcal{E}_{i}\rightarrow\mathcal{E}_{i}\backslash\\{j\\}$. Next, the
current value of $A_{ij}$ should be stored as the prescribed value
$\bar{A}_{ij}$ in the LMI objective (14) and then consider $A_{ij}$ as a
design variable to be synthesized. Moreover, if we are interested in removing
the interconnection $A_{ij}$ entirely, we need to set the cost coefficient
$c_{ij}$ in the LMI objective (14) arbitrarily high and set
$\bar{A}_{ij}=\mathbf{0}$. Finally, via solving the LMI problem (10) we can
obtain the refined interconnection topology (with a refined $A_{ij}$ value).

##### Stabilizability

The next theoretical result is on enforcing stabilizability under distributed
state feedback control via interconnection topology synthesis.

###### Theorem 2

(Stabilizability via Topology Synthesis) The networked system (2) (where $B$
is block diagonal), under Assumption 2, $w(t)=\mathbf{0}$ and local state
feedback control (3), is stable if at each subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$, the LMI problem

$\displaystyle\mathbb{P}_{2}:\text{Find}$ $\displaystyle\ \
M_{ii},L_{ii},\\{L_{ij}:j\in\mathbb{N}_{i-1}\\},\\{L_{ji}:j\in\mathbb{N}_{i-1}\\},$
(15) $\displaystyle\ \
\\{A_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\},\\{Q_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$
such that $\displaystyle\ \ M_{ii}>0,\ \ \tilde{W}_{ii}>0,$

is feasible, where $\tilde{W}_{ii}$ is computed from Alg. 1 (Steps: 3-16) when
enforcing $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ with

$\displaystyle W_{ij}=$ $\displaystyle\
-A_{ij}M_{jj}-M_{ii}A_{ji}^{\top}\mathbf{1}_{\\{j\in\bar{\mathcal{F}}_{i}\\}}-Q_{ji}^{\top}\mathbf{1}_{j\not\in\mathcal{F}_{i}}$
(16) $\displaystyle\ -L_{ji}^{\top}B_{jj}^{\top}-B_{ii}L_{ij}.$

The local state feedback controller gains (which includes the new feedback
interconnections $\\{K_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\}$
and $\\{K_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$) are:

$K_{ij}=L_{ij}M_{jj}^{-1}\ \ \mbox{ and }\ \ K_{ji}=L_{ji}M_{ii}^{-1},\ \
\forall j\in\mathbb{N}_{i}$ (17)

and the new system interconnections are:
$\\{A_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\}$ and
$\\{A_{ji}\triangleq
Q_{ji}M_{ii}^{-1}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$.

###### Proof:

Let us define $M\triangleq\text{diag}(M_{ii}:i\in\mathbb{N}_{N})$,
$W\triangleq-MA^{\top}-AM-L^{\top}B^{\top}-BL$ and $K=LM^{-1}$, where now
$A=[A_{ij}]_{i,j\in\mathbb{N}_{N}}$ and $K=[K_{ij}]_{i,j\in\mathbb{N}_{N}}$
includes variable interconnection blocks
$\\{A_{ij}:j\not\in\mathcal{E}_{i}\\}$, $\\{A_{ji}:j\not\in\mathcal{F}_{i}\\}$
and $\\{K_{ij}:j\not\in\mathcal{E}_{i}\\}$,
$\\{K_{ji}:j\not\in\mathcal{F}_{i}\\}$, respectively (replacing the fixed
$\mathbf{0}$ blocks that were there in $A$ (2) and $K$ (4)).

Starting from Lm. 1, it is easy to show that if there exists $M>0$ and $L$
such that $W>0$, the feedback controller gains given by $K=LM^{-1}$ guarantees
the closed-loop stability of the networked system (with new interconnections).
Based on the above definition of $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}$, it is a
symmetric network matrix (see Def. 2) where

$W_{ij}=-M_{ii}A_{ji}^{\top}-A_{ij}M_{jj}-L_{ji}^{\top}B_{jj}^{\top}-B_{ii}L_{ij}.$
(18)

Therefore, we can use Alg. 1 to enforce $W>0$ in a decentralized and
compositional manner via enforcing $\tilde{W}_{ii}>0$ at each subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$ sequentially.

As in the proof of Th. 1, to make this local enforcement $\tilde{W}_{ii}>0$ an
LMI problem, we need to replace any bi-linear term in $W_{ij}$ (18) using a
change of variables. In this case, the term $-M_{ii}A_{ji}^{\top}$ in $W_{ij}$
(18) is bi-linear, and thus we introduce:

$Q_{ji}^{\top}\triangleq M_{ii}A_{ji}^{\top}\iff Q_{ji}\triangleq
A_{ji}M_{ii},\ \ \forall j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1},$ (19)

which transforms $W_{ij}$ in (18) into the form (16). Consequently, testing
$\tilde{W}_{ii}>0$ takes the form of an LMI problem (10). The remainder of the
proof is similar to Th. 1, and is therefore omitted. ∎

###  IV-B Enforcing Dissipativity and Dissipativate-ability

Next, we provide decentralized and compositional interconnection topology
synthesis techniques to ensure dissipativity and dissipativate-ability. In
particular, we consider the $(Q,S,R)$-dissipativity property introduced in
Def. 1, and regarding the specification matrices $Q,S$ and $R$, we assume: (1)
they are network matrices, (2) $Q$ is a block diagonal, (3) $-Q>0$ (see Rm.
1), and (3) $R=R^{\top}$.

###### Theorem 3

(Dissipativity via Topology Synthesis) The networked system (2) (where $C,D$
are block diagonal) under $w(t)=\mathbf{0}$ is $(Q,S,R)$-dissipative from
$u(t)$ to $y(t)$ if at each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the LMI
problem

$\displaystyle\mathbb{P}_{3}:\text{Find}$ $\displaystyle\ \
P_{ii},\\{G_{ij}:j\not\in\mathcal{E}_{i}\\},\\{A_{ji}:j\not\in\mathcal{F}_{i}\\},$
(20) $\displaystyle\ \
\\{H_{ij}:j\not\in\mathcal{E}_{i}\\},\\{B_{ji}:j\not\in\mathcal{F}_{i}\\}$
such that $\displaystyle\ \ P_{ii}>0,\ \ \tilde{W}_{ii}>0,$

is feasible, where $\tilde{W}_{ii}$ is computed from Alg. 1 (Steps: 3-16) when
analyzing $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ with

$W_{ij}=\begin{bmatrix}W_{ij}^{11}&W_{ij}^{12}&C_{ii}^{\top}e_{ij}\\\
W_{ij}^{21}&D_{ii}^{\top}S_{ij}+S_{ji}^{\top}D_{jj}+R_{ij}&D_{ii}^{\top}e_{ij}\\\
C_{jj}e_{ij}&D_{jj}e_{ij}&-Q_{ii}^{-1}e_{ij}\end{bmatrix}.$ (21)

where

$\displaystyle W_{ij}^{11}=$ $\displaystyle\
-P_{ii}A_{ij}\mathbf{1}_{\\{j\in\bar{\mathcal{E}}_{i}\\}}-G_{ij}\mathbf{1}_{\\{j\not\in\mathcal{E}_{i}\\}}-A_{ji}^{\top}P_{jj},$
$\displaystyle W_{ij}^{12}=$ $\displaystyle\
-P_{ii}B_{ij}\mathbf{1}_{\\{j\in\bar{\mathcal{E}}_{i}\\}}-H_{ij}\mathbf{1}_{\\{j\not\in\mathcal{E}_{i}\\}}+C_{ii}^{\top}S_{ij},$
$\displaystyle W_{ij}^{21}=$ $\displaystyle\
-B_{ji}^{\top}P_{jj}+S_{ji}^{\top}C_{jj}.$

The new system interconnections are: $\\{A_{ij}\triangleq
P_{ii}^{-1}G_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\}$ and
$\\{A_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$, and the new input
interconnections are: $\\{B_{ij}\triangleq
P_{ii}^{-1}H_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\}$ and
$\\{B_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$.

###### Proof:

The proof starts by defining
$P\triangleq\text{diag}(P_{ii}:i\in\mathbb{N}_{N})$ and

$W\triangleq
BEW\big{(}\scriptsize\begin{bmatrix}-A^{\top}P-PA&-PB+C^{\top}S&C^{\top}\\\
-B^{\top}P+S^{\top}C&D^{\top}S+S^{\top}D+R&D^{\top}\\\
C&D&-Q^{-1}\end{bmatrix}\big{)}$

(inspired by (7) and including the interested variable interconnection
blocks), and proceeds using Prop. 2 in a similar manner to the proof of Th. 1
(note that here Lm. 4 is needed to deduce $W=\text{BEW}(\Psi)>0\iff\Psi>0$).
Therefore, explicit details of the proof are omitted here. ∎

###### Theorem 4

(Dissipativate-ablity via Topology Synthesis) The networked system (2) (where
$B,C,F$ are block diagonal) under $D=\mathbf{0}$ and local state feedback
control (3) is $(Q,S,R)$-dissipative from $w(t)$ to $y(t)$ if at each
subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the LMI problem

$\displaystyle\mathbb{P}_{4}:\text{Find}$ $\displaystyle\ \
M_{ii},L_{ii},\\{L_{ij}:j\in\mathbb{N}_{i-1}\\},\\{L_{ji}:j\in\mathbb{N}_{i-1}\\},$
(22) $\displaystyle\ \
\\{A_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\},\\{G_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\},$
$\displaystyle\ \
\\{E_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\},\\{E_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\},$
such that $\displaystyle\ \ M_{ii}>0,\ \ \tilde{W}_{ii}>0,$

is feasible, where $\tilde{W}_{ii}$ is computed from Alg. 1 (Steps: 3-16) when
enforcing $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ with $W_{ij}$ given in (23).
The local state feedback controller gains (which include new feedback
interconnections $\\{K_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\}$
and $\\{K_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$) are given by
(17), the new system interconnections are:
$\\{A_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\}$ and
$\\{A_{ji}\triangleq
G_{ji}M_{ii}^{-1}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$, and the new
input interconnections are:
$\\{E_{ij}:j\not\in\mathcal{E}_{i}\cap\mathbb{N}_{i-1}\\}$ and
$\\{E_{ji}:j\not\in\mathcal{F}_{i}\cap\mathbb{N}_{i-1}\\}$.

###### Proof:

The proof is similar to that of Th. 3. ∎

###### Remark 4

(Designing Intrinsic Parameters) When using the decentralized LMI problems
proposed in Th. 3 and Th. 4, if the application allows, we can also treat some
subsystem intrinsic parameters (in addition to interconnection parameters) as
design variables - without compromising the LMI format of the problem. For
example, in the LMI problem (20), we can include $C_{ii}$ and/or $D_{ii}$ as
design variables as $W_{ij}$ (21) is linear in both $C_{ii}$ and $D_{ii}$.

$W_{ij}=\begin{bmatrix}-A_{ij}M_{jj}-M_{ii}A_{ji}^{\top}\mathbf{1}_{\\{j\in\bar{\mathcal{F}}_{i}\\}}-G_{ji}^{\top}\mathbf{1}_{\\{j\not\in\mathcal{F}_{i}\\}}-B_{ii}L_{ij}-L_{ji}^{\top}B_{jj}&-E_{ij}+M_{ii}C_{ii}^{\top}S_{ij}&M_{ii}C^{\top}_{ii}e_{ij}\\\
-E_{ji}^{\top}+S_{ji}^{\top}C_{jj}M_{jj}&F_{ii}^{\top}S_{ij}+S_{ji}F_{jj}+R_{ij}&F_{ii}^{\top}e_{ij}\\\
C_{jj}M_{jj}e_{ij}&F_{jj}e_{ij}&-Q_{ii}^{-1}e_{ij}\end{bmatrix}$ (23)

##  V Case Study

In this section, we compare: (1) the decentralization-based topology synthesis
(DeTS) approach proposed for linear networked systems in this paper with (2)
the dissipativity-based topology synthesis (DiTS) approach proposed for non-
linear networked systems in our recent work [12]. In particular, we use two
variants of the DiTS approach based on the accuracy of the used dissipativity
information of the subsystems. They are denoted by the acronyms W-DiTS and
S-DiTS, representing scenarios where the available dissipativity information
of the subsystems is weak (less precise) and strong (more precise),
respectively. Note that, due to space constraints and for simplicity, we limit
this case study to scenarios where topology synthesis is required to ensure
the stability of a certain randomly generated networked system.

###  V-A Considered Networked System

In this case study, we consider a randomly generated networked system of the
form (2) with $N=5,n_{i}=3,\forall i\in\mathbb{N}_{N}$ and $B=E=\mathbf{0}$.
The initial values of a subset of system matrices
$\\{A_{ij}:j\in\bar{\mathcal{E}}_{i},i\in\mathbb{N}_{N}\\}$ are given in (24).
The corresponding initial interconnection topology is shown in the graph in
Fig. 2 (left). It is worth noting that this initial networked system is
unstable and the decentralized stability analysis proposed in [6] returns
inconclusive.

$\displaystyle\scriptsize
A_{22}=\begin{bmatrix}-2.06&1.43&-1.83\cr-1.79&-2.01&1.78\cr
1.50&-2.08&-2.00\end{bmatrix},\ A_{23}\begin{bmatrix}-1.35&-1.62&-8.97\cr
1.51&-1.40&-13.70\cr 8.99&13.68&-1.43\end{bmatrix},\
A_{24}=\begin{bmatrix}-2.96&0.12&2.47\cr 1.94&-4.44&0.11\cr
1.53&1.94&-2.98\end{bmatrix},$ (24) $\displaystyle\scriptsize
A_{55}=\begin{bmatrix}-2.86&-0.56&-1.64\cr-0.56&-2.20&-1.06\cr-1.64&-1.06&-4.93\end{bmatrix},\
A_{52}=\begin{bmatrix}-2.54&1.13&-0.13\cr-0.02&-1.56&1.40\cr
1.14&0.83&-1.78\end{bmatrix},\ A_{54}=\begin{bmatrix}-1.42&0.09&-0.61\cr
0.09&-0.30&-0.18\cr-0.61&-0.18&-0.86\end{bmatrix}.\normalsize$
Figure 2: Numerical Example : Initial Topology

Two interconnection cost matrices inspired by this initial interconnection
topology are shown in the same figure (i.e., $C_{f}$ and $C_{d}$). Note that
$C_{f}$ has fixed cost levels while $C_{d}$ has graphical distance-inspired
cost levels for different interconnections. Elements of these cost matrices
are used in the topology synthesis processes (e.g., as
$[c_{ij}]_{i,j\in\mathbb{N}_{N}}$ in (14)) to penalize deviations from the
initial topology.

Since the proposed topology synthesis approach in this paper (DeTS) is
decentralized, for comparison purposes, we use the following two centralized
cost functions:

$\displaystyle J_{Dev}\triangleq$
$\displaystyle\sum_{i,j\in\mathbb{N}_{N},i\neq
j}c_{ij}\|A_{ij}^{*}-\bar{A}_{ij}\|,$ (25) $\displaystyle J_{Nom}\triangleq$
$\displaystyle\sum_{i,j\in\mathbb{N}_{N},i\neq j}\|A_{ij}^{*}\|,$

where $c_{ij}$ is the interconnection cost coefficient (taken from either
$C_{f}$ or $C_{d}$), $\bar{A}_{ij}$ is the initial (given) system matrix
block, and $A_{ij}^{*}$ is the optimally synthesized system matrix block -
corresponding to the interconnection $(\Sigma_{j},\Sigma_{i})$. Note that, in
(25), $J_{Dev}$ represents the weighted deviation from the initial topology
while $J_{Nom}$ represents the nominal coupling strength of the synthesized
topology.

###  V-B Dissipativity Based Topology Synthesis (DiTS)

The work in [12] considers networked systems comprised of non-linear
subsystems $\tilde{\Sigma}_{i}:u_{i}\rightarrow y_{i},i\in\mathbb{N}_{N}$
interconnected through a static interconnection matrix $M$ (e.g., see Fig. 3).
A key advantage of the solution proposed in [12] is that it only requires the
knowledge of $(Q,S,R)$-dissipativity properties of the subsystems:
$\\{(Q_{i},S_{i},R_{i}):i\in\mathbb{N}_{N}\\}$ (in lieu of exact dynamic
models of the subsystems). Even though subsystem dissipativity information is
limited, it provides a simple, robust, and energy-based representation for the
subsystems. In particular, the work in [12] uses such subsystem dissipativity
information to formulate an LMI problem so as to synthesize the optimal
interconnection matrix $M$ (and hence, the interconnection topology) for the
non-linear networked system under some minor assumptions. To make this paper
self-contained, we have summarized this dissipativity-based topology synthesis
approach in the following proposition.

###### Proposition 1

[12, Prop. 5] Under $R_{i}<0,\forall i\in\mathbb{N}_{N}$, a stabilizing
interconnection matrix $M$ for the non-linear networked system shown in Fig. 3
can be found via solving the LMI problem (centralized):

$\displaystyle\mathbb{P}_{5}:\text{Find}$ $\displaystyle\ \
L,\\{p_{i}\in\mathbb{R}:i\in\mathbb{N}_{N}\\}$ (26) such that $\displaystyle\
\ p_{i}>0,\forall i\in\mathbb{N}_{N},$ $\displaystyle\ \
\begin{bmatrix}\textbf{R}_{p}&L\\\
L^{\top}&-(L^{\top}\textbf{X}+\textbf{X}^{\top}L+\textbf{Q}_{p})\end{bmatrix}$

as $M\triangleq\textbf{R}_{p}^{-1}L$, where
$\textbf{R}_{p}\triangleq\text{diag}(p_{i}R_{i}\mathbf{I}:i\in\mathbb{N}_{N})$,
$\textbf{Q}_{p}\triangleq\text{diag}(p_{i}Q_{i}\mathbf{I}:i\in\mathbb{N}_{N})$,
and $\textbf{X}\triangleq\text{diag}(R_{i}^{-1}S_{i}:i\in\mathbb{N}_{N})$.

Figure 3: A non-linear networked dynamical system configuration considered in
[12] (compare with Fig. 1).

To apply Prop. 1 for the considered linear networked system, we first need to
identify the corresponding construction of a nonlinear subsystem
$\tilde{\Sigma}_{i},i\in\mathbb{N}_{N}$ and the interconnection matrix
$M=[M_{ij}]_{i,j\in\mathbb{N}_{N}}$ (shown in Fig. 3). This can be achieved by
re-arranging the dynamics of a considered linear subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$ as:

$\Sigma_{i}:\Big{\\{}\dot{x}_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}A_{ij}x_{j}(t)=A_{ii}x_{i}(t)+\sum_{j\in\mathcal{E}_{i}}A_{ij}x_{j}(t).$

Now, the dynamics of a corresponding non-linear (in name only) subsystem
$\tilde{\Sigma}_{i},i\in\mathbb{N}_{N}$ can be written as:

$\tilde{\Sigma}_{i}:\begin{cases}\dot{x}_{i}(t)=&A_{ii}x_{i}(t)+u_{i}(t),\\\
y_{i}(t)=&x_{i}(t),\end{cases}$ (27)

where
$u_{i}(t)\triangleq\sum_{j\in\mathcal{E}_{i}}A_{ij}y_{j}(t)\equiv\sum_{j\in\mathbb{N}_{N}}M_{ij}y_{j}(t)$.
Therefore, $M_{ij}\triangleq A_{ij}\mathbf{1}_{\\{i,j\in\mathbb{N}_{N},i\neq
j\\}}$, and thus, using Prop. 1, we can synthesize the system matrices
$\\{A_{ij}:i,j\in\mathbb{N}_{N},i\neq j\\}$ which implies the optimal
interconnection topology for the considered networked system.

Recall that, per our As. 2, the system matrices
$\\{A_{ii},i\in\mathbb{N}_{N}\\}$ and the non-linear subsystems
$\\{\tilde{\Sigma}_{i}:i\in\mathbb{N}_{N}\\}$ (27) are prespecified. However,
to apply Prop. 1, we only require the subsystem dissipativity properties:
$\\{(Q_{i},S_{i},R_{i}):i\in\mathbb{N}_{N}\\}$. Here we assume each subsystem
$\tilde{\Sigma}_{i},i\in\mathbb{N}_{N}$ to have input and output passivity
indices as $\nu_{i}$ and $\rho_{i}$, respectively. In other words, subsystem
$\tilde{\Sigma}_{i}$ (27) is assumed to be
$(-\rho_{i}\mathbf{I},0.5\mathbf{I},-\nu_{i}\mathbf{I})$-dissipative (see Def.
1). Candidate values for such passivity indices were estimated by applying Lm.
2 under: (1) a trial and error approach (which led to weak/less precise
passivity indices), and (2) an LMI-based optimization approach [12] (which
lead to strong/precise passivity indices). It is worth noting that there are
several other alternative approaches to estimate such passivity indices (e.g.,
see [22, 23, 24]). Note that the said two types of passivity indices estimates
led to the two dissipativity-based topology synthesis approaches: (1) W-DiTS
and (2) S-DiTS mentioned earlier.

Note also that, inspired by Rm. 2, to penalize deviations from the initial
interconnection topology, when solving the LMI problem (26) in Prop. 1, we use
the objective function

$J=\left\|[c_{ij}(L_{ij}-p_{i}R_{i}\bar{A}_{ij})]_{i,j\in\mathbb{N}_{N}}\right\|.$
(28)

###  V-C Decentralization Based Topology Synthesis (DeTS)

For the considered networked system in this case study, the application of the
proposed DeTS approach is straightforward as it only involves solving the
sequence of LMI problems given in Th. 1. In the implementation, as the LMI
objective function, we used (14) (with $\beta_{i}i=\|P_{j}j\|$) as proposed in
Rm. 2. Note also that, as suggested in Rm. 3, we considered the possibility of
refining all the existing interconnections. Consequently, in numerical
results, we observed that it is possible (and even preferred) to get optimal
interconnection topologies where some initial interconnections have been
removed completely to preserve stability while minimizing deviations from the
initial topology.

###  V-D Observations and Discussion

Figure 4 illustrates the synthesized optimal interconnection topologies under
the aforementioned topology synthesis techniques: W-DiTS (Figs. 4ab), S-DiTS
(Figs. 4cd), and DeTS (Figs. 4ef), and under the said interconnection cost
matrices $C_{f}$ (left) and $C_{d}$ (right). Moreover, the observed deviation
and nominal cost values proposed in (25) are given in the titles of the
subfigures in Fig. 4.

(a) Under W-DiTS and $C_{f}$.

(b) Under W-DiTS and $C_{d}$.

(c) Under S-DiTS and $C_{f}$.

(d) Under S-DiTS and $C_{d}$.

(e) Under DeTS and $C_{f}$.

(f) Under DeTS and $C_{d}$.

Figure 4: Obtained optimal interconnection topologies for the considered
linear networked system under different topology synthesis methods: W-DiTS,
S-DiTS, and DeTS, under different interconnection cost matrics: $C_{f}$ and
$C_{d}$. The blue, green, and red edges in the sub-figures represent the
initial, newly added, and entirely removed interconnections, respectively. The
titles of the sub-figures indicate the deviation and nominal cost values as
defined in (25)

Based on the observed newly added (green-colored) and entirely removed (red-
colored) edges with respect to the initial topology (blue-colored) in each
scenario, we can clearly see the practical advantage of the proposed DeTS
approach in this paper compared to both W-DiTS and S-DiTS methods adapted from
[12]. In essence, the proposed DeTS approach has mainly resorted to removing a
single edge from the initial topology to ensure the stability of the
considered networked system. In contrast, the dissipativity-based approaches
W-DiTS and S-DiTS have mainly resorted to creating several new
interconnections to achieve the same goal. Note also that both such approaches
may have also refined some existing interconnections (this is also implied by
the change in the nominal cost observed in Fig. 4f compared to that in Fig.
4e).

Another interesting observation is that when using the interconnection cost
matrix $C_{d}$ as opposed to $C_{f}$, the number of newly added edges
decreases (particularly the lengthy ones, e.g., compare Figs. 4ac with Figs.
4bd). However, during the same process, the observed nominal cost increases
owing to the internal changes required to stabilize the considered networked
system without using additional new edges.

Note also that a similar reduction in the number of newly added edges occurs
when we have more precise/stronger passivity information regarding the
subsystems (e.g., compare Figs. 4ab with Figs. 4cd). This is because
dissipativity-based topology synthesis [12] becomes more flexible (as opposed
to becoming more constrained/conservative) when underlying subsystems are
strongly passive. This can also be understood by the fact that strongly
passive systems not only can easily be stabilized but also can tolerate other
connected weakly passive subsystems (due to the compositionality of
passivity). Nevertheless, similar to before, an increment in the nominal cost
can be seen due to the internal changes required to achieve stability without
using additional new edges.

In terms of the deviation cost (25), when using $C_{f}$, the S-DiTS approach
has provided the best performance. Note, however, that, in this case, the
proposed DeTS approach performs closely to the S-DiTS approach while also
having a better nominal cost. Moreover, when using $C_{d}$, the proposed DeTS
approach performs the best (which is also the overall best deviation cost
value observed in this case study).

We conclude this paper by acknowledging some unique attributes of the S-DiTS
approach adapted from [12] as opposed to the DeTS approach proposed in this
paper. Let us consider the amount of information required to synthesize
topologies under each approach. First, note that both these approaches use the
initial interconnection system matrices $\\{A_{ij}:i,j\in\mathbb{N}_{N},i\neq
j\\}$ as a reference to penalize deviations from them. However, the intrinsic
system matrices $\\{A_{ii}\in\mathbb{R}^{n_{i}\times
n_{i}}:i\in\mathbb{N}_{N}\\}$ are only used in the proposed DeTS approach in
this paper. In contrast, only two scalar passivity indices per each subsystem:
$\\{(\nu_{i},\rho_{i})\in\mathbb{R}^{2}:i\in\mathbb{N}_{N}\\}$ are being used
in the S-DiTS approach. In real-world scenarios, such scalar passivity indices
can be estimated conveniently and accurately - compared to having to estimate
the entire intrinsic system matrices. Moreover, as detailed in [12], the
S-DITS approach is applicable to a variety of networked systems comprised of
non-linear subsystems.

##  VI Conclusion

This paper considered networked systems comprised of interconnected linear
subsystems and proposed a decentralized and compositional approach to
stabilize or dissipativate such linear networked systems via synthesizing an
optimal set of interconnections (topology) for the subsystems. The proposed
topology synthesis approach was then extended to ensure the ability to
stabilize or dissipativate linear networked systems using distributed feedback
control. We gain this ability to optimally synthesize topologies by improving
an existing decentralized and compositional approach for various analysis and
controller synthesis tasks related to linear networked systems. The proposed
topology synthesis approach only involves solving a sequence of linear matrix
inequality problems - which can be implemented efficiently and scalably using
standard convex optimization toolboxes. The presented case study showed that
the proposed topology synthesis approach provides simplistic and high-
performing solutions compared to an existing topology synthesis approach
proposed for more general non-linear networked systems with limited
information about the subsystems. Future work aims to closely study several
critical real-world applications (e.g., supply chain networks, micro grids,
vehicular platoons, and multi-robot systems) where the proposed topology
synthesis approach can be directly applied to optimize existing networks in
such applications.

## References

  * [1] D. Ivanov, S. Sethi, A. Dolgui, and B. Sokolov, “A Survey on Control Theory Applications to Operational Systems, Supply Chain Management, and Industry 4.0,” _Annual Reviews in Control_ , vol. 46, pp. 134–147, 2018.
  * [2] W. Tang and P. Daoutidis, “Dissipativity Learning Control (DLC): Theoretical Foundations of Input–Output Data-Driven Model-Free Control,” _Systems & Control Letters_, vol. 147, p. 104831, 2021.
  * [3] R. D. Tordecilla, A. A. Juan, J. R. Montoya-Torres, C. L. Quintero-Araujo, and J. Panadero, “Simulation-Optimization Methods for Designing and Assessing Resilient Supply Chain Networks Under Uncertainty Scenarios: A Review.” _Simulation Modelling Practice and Theory_ , vol. 106, p. 102166, 2021.
  * [4] T. Samad and A. M. Annaswamy, “Controls for Smart Grids: Architectures and Applications,” _Proc. of the IEEE_ , vol. 105, no. 11, pp. 2244–2261, 2017\.
  * [5] G. Antonelli, “Interconnected Dynamic Systems: An Overview on Distributed Control,” _IEEE Control Systems Magazine_ , vol. 33, no. 1, pp. 76–88, 2013\.
  * [6] S. Welikala, H. Lin, and P. J. Antsaklis, “A Generalized Distributed Analysis and Control Synthesis Approach for Networked Systems with Arbitrary Interconnections,” in _Proc. of 30th Mediterranean Conf. on Control and Automation_ , 2022, pp. 803–808.
  * [7] A. N. Michel, “On the Status of Stability of Interconnected Systems,” _IEEE Trans. on Automatic Control_ , vol. 28, no. 6, pp. 639–653, 1983.
  * [8] T. Ishizaki, H. Sasahara, M. Inoue, T. Kawaguchi, and J.-i. Imura, “Modularity in Design of Dynamical Network Systems: Retrofit Control Approach,” _IEEE Trans. on Automatic Control_ , vol. 66, no. 11, pp. 5205–5220, 2021.
  * [9] E. Agarwal, S. Sivaranjani, V. Gupta, and P. J. Antsaklis, “Distributed Synthesis of Local Controllers for Networked Systems with Arbitrary Interconnection Topologies,” _IEEE Trans. on Automatic Control_ , vol. 66, no. 2, pp. 683–698, 2021.
  * [10] P. J. Antsaklis and A. N. Michel, _Linear Systems_. Birkhauser, 2006.
  * [11] S. Welikala, H. Lin, and P. J. Antsaklis, “A Generalized Distributed Analysis and Control Synthesis Approach for Networked Systems with Arbitrary Interconnections,” _arXiv e-prints_ , p. 2204.09756, 2022. [Online]. Available: http://arxiv.org/abs/2204.09756
  * [12] ——, “Non-Linear Networked Systems Analysis and Synthesis using Dissipativity Theory,” in _Proc. of American Control Conf. (accepted)_ , 2023.
  * [13] M. Rafiee and A. M. Bayen, “Optimal Network Topology Design in Multi-Agent Systems for Efficient Average Consensus,” in _49th IEEE Conf. on Decision and Control_ , 2010, pp. 3877–3883.
  * [14] Y. Ebihara, D. Peaucelle, and D. Arzelier, “Analysis and Synthesis of Interconnected Positive Systems,” _IEEE Trans. on Automatic Control_ , vol. 62, no. 2, pp. 652–667, 2017.
  * [15] A. S. Rufino Ferreira, C. Meissen, M. Arcak, and A. Packard, “Symmetry Reduction for Performance Certification of Interconnected Systems,” _IEEE Trans. on Control of Network Systems_ , vol. 5, no. 1, pp. 525–535, 2018.
  * [16] D. Xue, A. Gusrialdi, and S. Hirche, “Robust Distributed Control Design for Interconnected Systems Under Topology Uncertainty,” _Proc. of American Control Conf._ , pp. 6541–6546, 2013.
  * [17] L. B. Cremean and R. M. Murray, “Stability Analysis of Interconnected Nonlinear Systems under Matrix Feedback,” in _Proc. of 42nd IEEE Conf. on Decision and Control_ , vol. 3, 2003, pp. 3078–3083.
  * [18] S. Welikala, H. Lin, and P. J. Antsaklis, “Centralized and Decentralized Techniques for Analysis and Synthesis of Non-Linear Networked Systems,” _arXiv e-prints_ , p. 2209.14552, 2022. [Online]. Available: http://arxiv.org/abs/2209.14552
  * [19] J. C. Willems, “Dissipative Dynamical Systems Part I: General Theory,” _Archive for Rational Mechanics and Analysis_ , vol. 45, no. 5, pp. 321–351, 1972.
  * [20] N. Kottenstette, M. J. McCourt, M. Xia, V. Gupta, and P. J. Antsaklis, “On Relationships Among Passivity, Positive Realness, and Dissipativity in Linear Systems,” _Automatica_ , vol. 50, no. 4, pp. 1003–1016, 2014.
  * [21] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, _Linear Matrix Inequalities in System and Control Theory_. SIAM, 1994.
  * [22] S. Welikala, H. Lin, and P. J. Antsaklis, “On-line Estimation of Stability and Passivity Metrics,” in _Proc. of 61st IEEE Conf. on Decision and Control_ , 2022.
  * [23] A. Koch, J. M. Montenbruck, and F. Allgower, “Sampling Strategies for Data-Driven Inference of Input–Output System Properties,” _IEEE Trans. on Automatic Control_ , vol. 66, no. 3, pp. 1144–1159, 2021.
  * [24] M. Tanemura and S. Azuma, “Efficient Data-Driven Estimation of Passivity Properties,” _IEEE Control Systems Letters_ , vol. 3, no. 2, pp. 398–403, 2019.