# A Decentralized Analysis and Control Synthesis Approach for Networked
Systems with Arbitrary Interconnections

Shirantha Welikala, Hai Lin and Panos Antsaklis The support of the National
Science Foundation (Grant No. IIS-1724070, 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

This paper considers the problem of decentralized analysis and control
synthesis to verify and ensure properties like stability and dissipativity of
a large-scale networked system comprised of linear subsystems interconnected
in an arbitrary topology. In particular, we design systematic networked system
analysis and control synthesis processes that can be executed in a
decentralized manner at the subsystem level with minimal information sharing
among the subsystems. Compared to our most recent work on the same topic, we
consider a substantially more generalized problem setup in this paper and
develop decentralized processes to verify and ensure a broader range of
networked system properties. We show that for such decentralized processes:
(1) optimizing the used subsystem indexing scheme can substantially reduce the
required inter-subsystem information-sharing sessions, and (2) in some network
topologies, information sharing among only neighboring subsystems is
sufficient (hence, distributed!). Moreover, the proposed networked system
analysis and control synthesis processes are compositional/resilient to
subsystem removals, which enable them to conveniently and efficiently handle
situations where new subsystems are being added/removed to/from an existing
network. We also provide significant insights into our decentralized approach
so that it can be quickly adopted to verify and ensure properties beyond the
stability and dissipativity of networked systems. En route to developing such
decentralized techniques, we have also derived new centralized solutions for
dissipative observer and dynamic output feedback controller design problems.
Subsequently, we also specialize all the derived results for discrete-time
networked systems. We conclude this paper by providing several simulation
results demonstrating the proposed novel decentralized analysis and control
synthesis processes and dissipativity-based results.

##  I Introduction

Analysis and control synthesis of large-scale networked systems comprised of
dynamically coupled subsystems has gained a renewed attention due to various
emerging applications in infrastructure networks [1, 2]. A prime example of
this is found in vehicular networks where often a group of autonomous vehicles
that co-ordinate with each other to maintain a particular formation can lead
to saving energy as well as reducing congestion and improving safety in the
transportation infrastructure [3, 4]. Another example application of large-
scale networked systems is the power grid, which needs to be constantly
analyzed and controlled in a distributed manner while accounting for the
penetration of highly varying renewable energy sources and smart/unknown loads
[5, 6, 7].

Numerous distributed control solutions have already been proposed in the
literature concerning such large-scale networked dynamical systems to enforce
stability while also optimizing various performance objectives of interest [8,
9, 10, 11, 12]. These solutions synthesize local controllers (i.e., at the
subsystem level) that only require the state information of a subset of other
subsystems in the network to operate. However, many of such distributed
control solutions assume the existence of a central entity with the knowledge
of the entire networked system to execute the control synthesis process in a
centralized manner. Inherently, such a centralized setup has several
disadvantages: (1) feasibility, privacy and security concerns related to
collecting all subsystem information at a centralized entity, (2) having to
re-evaluate the entire analysis and/or control synthesis when new subsystems
are added (or removed) to/from the network, and (3) scalability concerns
arising due to the complexity of the problem that needs to be solved at the
centralized entity.

Several decentralized control synthesis approaches have also been proposed to
address such limitations in the literature. In such methods, controllers are
derived locally at subsystems without the explicit knowledge of the dynamics
of the other subsystems. As pointed out in [1], such decentralized control
synthesis approaches can be categorized into three groups: (i) Approaches that
induce and exploit weak coupling between subsystems [13, 14, 15]; (ii)
Hierarchical approaches that compute (in a centralized manner) and enforce
additional conditions on local control synthesis [16, 17]; (iii) Approaches
that decompose a centralized control synthesis process using numerical
techniques such as methods of multipliers and Sylvester’s criterion [18, 19,
20, 2, 1]. In many networked systems of interest, assuming or enforcing weak
coupling among subsystems is not practical [2]. Moreover, existing
hierarchical approaches are computationally intensive and still involve a
considerable centralized component. Taking these limitations into account, the
recent work in [2, 1] has developed a decentralized analysis and control
synthesis framework for networked dynamical systems inspired by Sylvester’s
criterion.

In particular, both [2, 1] assume each subsystem dynamics to be linear and
coupled with a subset of neighboring subsystems (determined by the network
topology) through their state values. In [2], the network topology is assumed
to be cascaded bi-directional, and distributed analysis and control synthesis
techniques have been developed to verify and ensure the passivity of the
networked system. This approach is then further extended in [1] considering
arbitrary bi-directional network topologies to verify and ensure a general
quadratic dissipativity property (known as $(Q,S,R)$-dissipativity [21]) over
the networked system in a decentralized manner.

Compared to [1], the recent work in [22, 23] assume each subsystem dynamics to
be non-linear and coupled with a set of neighboring subsystems through their
output (not the state) values. Then, dissipativity properties of the
subsystems are exploited to derive centralized stability and performance
verification techniques in a compositional manner. Therefore, unlike in [2,
1], the techniques proposed in [22, 23] are only applicable for centralized
analysis of networked systems.

On the other hand, compared to [1], a few application-specific (structure)
decentralized control synthesis approaches can also be found in the
literature. For example, [24, 25, 26, 27] and [28] respectively address the
decentralized control synthesis problems for vehicular platoons, switched
networked systems, distributed parameter systems, micro-grids and unmanned
aerial systems (see also the review in [29] and references therein). However,
these approaches either exploit special structural properties specific to the
considered problem setup or belong to one of the earlier mentioned three
groups.

It should also be noted that several works in the literature refer to
centrally executed distributed (or fully-local) control synthesis as
“decentralized control synthesis” (deviating from our nomenclature). For
instance, [30, 31] and [32, 33] respectively propose Youla parametrization and
orthogonal functions based approaches for centrally synthesize distributed
controllers - where as [1] propose Sylvester’s criterion based approach for
decentrally synthesize distributed controllers.

##### Contributions

Taking these concerns into account, in this paper, we further generalize the
decentralized analysis and control synthesis approach proposed in [1, 34].
Compared to [34], our main contributions can be outlined as follows:

  1. 1.

We provide a comprehensive collection of linear matrix inequality (LMI) based
solutions for several standard control problems associated with continuous-
time linear time-invariant systems (CT-LTI).

  2. 2.

To the best of the authors’ knowledge, this collection includes novel (only
known to date) LMI-based solutions for $(Q,S,R)$-dissipative observer
synthesis and $(Q,S,R)$-dissipative dynamic output feedback controller
synthesis problems for CT-LTI systems.

  3. 3.

We consider a fully-coupled continuous-time networked system (CTNS) model (in
terms of subsystem states, inputs, and disturbances) and only introduce
decoupling assumptions critical to addressing each interested CTNS control
problem in a decentralized manner.

  4. 4.

We have relaxed the bi-directionality assumption made regarding the network
topology.

  5. 5.

The distributed observer synthesis problem has been considered, and a
decentralized solution has been proposed.

  6. 6.

The distributed dynamic output feedback controller synthesis problem has been
considered, and a decentralized solution has been proposed.

  7. 7.

We propose decentralized analysis and control synthesis approaches to verify
and enforce: (1) exponential stability and (2) optimal
$\mathcal{H}_{2}/\mathcal{H}_{\infty}$ gains of a CTNS;

  8. 8.

We also study the effect of subsystem indexing (i.e., the order in which the
proposed decentralized processes are executed) on the required total
communications among the subsystems and derive a cost function that can be
optimized to avoid some costly inter-subsystem communications;

  9. 9.

We provide significant insights into our approach so that it may be quickly
adopted to address similar LMI-based control problems associated with CTNSs.

  10. 10.

All the derived results for CTNSs have also been specialized for discrete-time
networked systems (DTNSs).

##### Organization

This paper is organized as follows. In Section II, we summarize a
comprehensive collection of linear matrix inequality (LMI) conditions that
arise in the analysis and control synthesis of continuous-time linear time-
invariant (CT-LTI) systems. The details of the construction of the considered
continuous-time networked system (CTNS) and the problem formulation are
discussed in Section III. Next, in Section IV, we define a class of matrices
specific to a given network topology (called “network matrices”) and discuss
several related theoretical results along with a decentralized algorithm to
analyze the positive-definiteness of such network matrices. Subsequently, in
Section V, we provide the details of the proposed decentralized analysis and
local control synthesis processes for the CTNS. The discrete-time versions of
the concepts and results provided in Sections II, III and V are summarized in
subsequent Sections VI, VII and VIII, respectively. Finally, Section IX
discusses several simulation results before concluding the paper in Section X.

##### Notation

The sets of real and natural numbers are denoted by $\mathbb{R}$ and
$\mathbb{N}$, respectively. An $n$-dimensional real vector is denoted by
$\mathbb{R}^{n}$. 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
$A=[A_{ij}]_{i\in\mathbb{N}_{n},j\in\mathbb{N}_{m}}$ (or simply as $[A_{ij}]$)
where $A_{ij}$ is the $(i,j)$th block of $A$. Similarly,
$[A_{ij}]_{j\in\mathbb{N}_{m}}$ represents a block row matrix,
$\text{diag}(A_{ii}:i\in\mathbb{N}_{n})$ represents a block diagonal matrix
and unless defined otherwise,
$A_{i}\triangleq\\{A_{ii}\\}\cup\\{A_{ij},j\in\mathbb{N}_{i-1}\\}\cup\\{A_{ji}:j\in\mathbb{N}_{i}\\}$.
The transpose of a matrix $A$ is denoted by $A^{\top}$ and
$(A^{\top})^{-1}=A^{-\top}$, The zero matrix is denoted by $\mathbf{0}$ and
the standard identity matrix is denoted by $\mathbf{I}$ (dimensions will be
clear form 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, $A>0\iff A=A^{\top}>0$ (i.e., symmetry is
implied by the positive definiteness). The symbol $\star$ is used to represent
redundant conjugate block matrices (e.g., $\begin{bmatrix}A&B\\\
\star&C\end{bmatrix}=\begin{bmatrix}A&B\\\ B^{\top}&C\end{bmatrix}$ and
$\begin{bmatrix}A_{ij}&B_{ij}\\\
\star&C_{ij}\end{bmatrix}=\begin{bmatrix}A_{ij}&B_{ij}\\\
B_{ji}^{\top}&C_{ij}\end{bmatrix}$). The symmetric part of a matrix $A$ is
denoted by $\mathcal{H}_{s}(A)\triangleq A+A^{\top}$ and
$\mathcal{H}_{s}(A_{ij})\triangleq A_{ij}+A_{ji}^{\top}$. $\mathcal{L}_{2e}$
is the extended $\mathcal{L}_{2}$ space (i.e., the space of signals with
finite $\mathcal{L}_{2}$ norms). Given sets $A$ and $B$, $A\backslash B$
indicates the set subtraction operation that results in the set of elements in
$A$ that are not in $B$. The notation $\mathbf{1}_{\\{\cdot\\}}$ is used to
represent the indicator function and
$e_{ij}\triangleq\mathbf{I}\cdot\mathbf{1}_{\\{i=j\\}}$.

##  II Preliminaries: Continuous-Time Linear Time Invariant (CT-LTI) Systems

In this section, we present a comprehensive collection of linear matrix
inequality (LMI) conditions that arise when analyzing or synthesizing
controllers for continuous-time linear time-invariant (CT-LTI) systems. We
particularly focus on LMIs because: (1) we will subsequently propose a
systematic approach to decentralize such LMIs (in Sec. IV) and (2) LMIs can be
efficiently and conveniently solved using standard convex optimization
techniques [35]. We start with stating two well-known lemmas that will be
useful when deriving LMIs.

######  Lemma 1.

(Schur’s complement [36]) Let $W=\scriptsize\begin{bmatrix}\Theta&\Phi\\\
\Phi^{\top}&\Gamma\end{bmatrix}$ be a symmetric $2\times 2$ block matrix. Then
if:
(i) $\Theta$ is invertible, $W>0\iff\Theta>0,\
\Gamma-\Phi^{\top}\Theta^{-1}\Phi>0$,
(ii) $\Gamma$ is invertible, $W>0\iff\Gamma>0,\
\Theta-\Phi\Gamma^{-1}\Phi^{\top}>0$.

######  Lemma 2.

(Congruence principle [36]) A matrix $W>0$ if and only if $P^{\top}WP>0$ where
$P$ is a full-rank matrix.

Consider the CT-LTI system given by

$\displaystyle\dot{x}(t)=Ax(t)+Bu(t),$ (1) $\displaystyle y(t)=Cx(t)+Du(t),$

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

###  II-A Analysis of CT-LTI Systems

####  II-A1 Stability

A well-known necessary and sufficient LMI condition for the _stability_ of (1)
(under $u(t)=\mathbf{0}$) is given in the following proposition.

######  Proposition 1.

[37] The CT-LTI system (1) under $u(t)=\mathbf{0}$ is globally exponentially
stable iff $\exists P>0$ such that

$-A^{\top}P-PA>0.$ (2)

In the interest of brevity, we omit providing standard definitions of uniform
and exponential stability, which can be found in [37, 38]. In the remainder of
this paper, by ‘stability,’ we simply refer to the global exponential
stability.

####  II-A2 $(Q,S,R)$-Dissipativity

Similar to the stability, the _dissipativity_ property introduced in the
seminal paper [21] is an important property of dynamical systems. In this
paper, we particularly consider the quadratic dissipativity property called
$(Q,S,R)$-dissipativity [39] defined below.

######  Definition 1.

[39] The CT-LTI system (1) is $(Q,S,R)$-dissipative (from $u(t)$ to $y(t)$) if
there exists a positive definite function
$V(x):\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$ called the storage
function such that for all $t_{1}\geq t_{0}\geq 0,x(t_{0})\in\mathbb{R}^{n}$
and $u(t)\in\mathbb{R}^{p}$, 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, where $Q\in\mathbb{R}^{m\times m},S\in\mathbb{R}^{m\times p}$,
$R\in\mathbb{R}^{p\times p}$ are given.

Analyzing and enforcing this $(Q,S,R)$-dissipativity property on large-scale
networked systems is a central objective of this paper as through appropriate
choices of $Q,S$ and $R$ matrices, it can capture a wide range of dynamical
properties of interest as summarized in the following remark.

######  Remark 1.

[39] The dynamical system (1) satisfying Def.1:

  1. 1.

is _passive_ iff $Q=0,S=\frac{1}{2}\mathbf{I},R=0$;

  2. 2.

is _strictly input passive_ iff $Q=0,S=\frac{1}{2}\mathbf{I},R=-\nu\mathbf{I}$
where $\nu>0$ ($\nu$ is an input feedforward passivity index);

  3. 3.

is _strictly output passive_ iff
$Q=-\rho\mathbf{I},S=\frac{1}{2}\mathbf{I},R=0$ where $\rho>0$ ($\rho$ is an
output feedback passivity index);

  4. 4.

is _strictly passive_ iff
$Q=-\rho\mathbf{I},S=\frac{1}{2}\mathbf{I},R=-\nu\mathbf{I}$ where
$\rho,\nu>0$;

  5. 5.

is _$\mathcal{L}_{2}$ -stable_ iff
$Q=-\frac{1}{\gamma}\mathbf{I},S=0,R=-\gamma\mathbf{I}$ where $\gamma\geq 0$
($\gamma$ is an _$\mathcal{L}_{2}$ -gain_ of the system);

  6. 6.

_conic_ iff $Q=-\mathbf{I},S=c\mathbf{I},R=(r^{2}-c^{2})\mathbf{I}$ where
$c\in\mathbb{R}$ and $r>0$ ($c$ and $r$ are conic parameters); and

  7. 7.

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 LMI condition for the $(Q,S,R)$-dissipativity of
(1) is established in the following proposition.

######  Proposition 2.

The CT-LTI system (1) is $(Q,S,R)$-dissipative (with $-Q>0,R=R^{\top}$) from
$u(t)$ to $y(t)$ iff $\exists P>0$ such that

$\begin{bmatrix}-A^{\top}P-PA&-PB+C^{\top}S&C^{\top}\\\
\star&D^{\top}S+S^{\top}D+R&D^{\top}\\\ \star&\star&-Q^{-1}\end{bmatrix}\geq
0.$ (3)

###### Proof.

Using Lm. 1 in (3), we get:

$\begin{bmatrix}-A^{\top}P-PA+\hat{Q}&-PB+\hat{S}\\\
\star&\hat{R}\end{bmatrix}\geq 0,$ (4)

with $\hat{Q}=C^{\top}QC,\ \ \hat{S}=C^{\top}S+C^{\top}QD\mbox{ and }\ \
\hat{R}=D^{\top}QD+D^{\top}S+S^{\top}D+R$. Hence the proof is complete [39,
Lm. 2] (note also that (4) can be used instead of (3) if $Q\geq 0$). ∎

####  II-A3 $\mathcal{H}_{2}$-Norm

Let $\mathcal{G}:\mathcal{L}_{2e}\rightarrow\mathcal{L}_{2e}$ represent the
transfer matrix of the CT-LTI system (1) from $u(t)$ to $y(t)$. If $A$ in (1)
is Hurwitz, $\mathcal{G}(s)=C(s\mathbf{I}-A)^{-1}B+D$ and its
$\mathcal{H}_{2}$-norm is

$\|\mathcal{G}\|_{\mathcal{H}_{2}}^{2}\triangleq\sup_{h>0}\frac{1}{2\pi}\int_{-\infty}^{\infty}\mbox{tr}\left(\mathcal{G}^{\top}(j\omega+h)\mathcal{G}(j\omega+h)\right)d\omega.$
(5)

As shown in [40], under $D=\mathbf{0}$, the $\mathcal{H}_{2}$-norm of
$\mathcal{G}$ is

$\|\mathcal{G}\|_{\mathcal{H}_{2}}^{2}=\mbox{tr}\left(B^{\top}MB^{\top}\right)=\mbox{tr}\left(CNC^{\top}\right),$
(6)

where $M,N>0$ with $MA+A^{\top}M+C^{\top}C=0$ and $AN+NA^{\top}+BB^{\top}=0$ .
Consider the following proposition.

######  Proposition 3.

[40, pp. 58] The $\mathcal{H}_{2}$-Norm of the transfer matrix of the CT-LTI
system (1) (i.e., $\|\mathcal{G}\|_{\mathcal{H}_{2}}$) is
$\|\mathcal{G}\|_{\mathcal{H}_{2}}<\gamma<\infty$ iff $D=\mathbf{0}$ and
$\exists P,Q>0$ and $\gamma\in\mathbb{R}_{>0}$ such that

$\begin{bmatrix}-AP-PA^{\top}&-PC^{\top}\\\
\star&\gamma\mathbf{I}\end{bmatrix}>0,\ \begin{bmatrix}P&B\\\
\star&Q\end{bmatrix}>0,\ \mbox{tr}\left(Q\right)<\gamma,$ (7)

or

$\begin{bmatrix}-A^{\top}P-PA&-PB\\\ \star&\gamma\mathbf{I}\end{bmatrix}>0,\
\begin{bmatrix}P&C^{\top}\\\ \star&Q\end{bmatrix}>0,\
\mbox{tr}\left(Q\right)<\gamma.$ (8)

####  II-A4 $H_{\infty}$-Norm

The $\mathcal{H}_{\infty}$-norm of $\mathcal{G}$ is formally defined as

$\|\mathcal{G}\|_{\mathcal{H}_{\infty}}^{2}\triangleq\frac{1}{2\pi}\int_{-\infty}^{\infty}\mbox{tr}\left(G^{\top}(j\omega)G(j\omega)\right)d\omega$
(9)

($\|\mathcal{G}\|_{\mathcal{H}_{\infty}}=\|\mathcal{G}\|_{\mathcal{L}_{2}}$).
As shown in [38, pp. 210] and [40, pp.49],

$\|\mathcal{G}\|_{\mathcal{H}_{\infty}}^{2}=\sup_{u\in\mathcal{L}_{2},u\neq
0}\frac{\|y\|_{\mathcal{L}_{2}}^{2}}{\|u\|_{\mathcal{L}_{2}}^{2}}=\sup_{\omega\in\mathbb{R}}\|G(j\omega)\|_{2}^{2}.$
(10)

Now, consider the following proposition.

######  Proposition 4.

[40, pp.50] The $\mathcal{H}_{\infty}$-Norm of the transfer matrix of the CT-
LTI system (1) (i.e., $\|\mathcal{G}\|_{\mathcal{H}_{\infty}}$) is
$\|\mathcal{G}\|_{\mathcal{H}_{\infty}}<\gamma$ iff $\exists P>0$ and
$\gamma\in\mathbb{R}_{>0}$ such that

$\begin{bmatrix}-A^{\top}P-PA&-PB&-C^{\top}\\\
\star&\gamma\mathbf{I}&-D^{\top}\\\
\star&\star&\gamma\mathbf{I}\end{bmatrix}>0.$ (11)

####  II-A5 Controllability

Regarding the _controllability_ [37] of the CT-LTI system (1), consider the
following proposition.

######  Proposition 5.

[41, pp.76] [40, pp. 86] A CT-LTI system (1) is: (1) stable and controllable
iff $\exists P>0$ such that

$-AP-PA^{\top}-BB^{\top}=0,$ (12)

and (2) stabilizable iff $\exists P>$ such that

$-AP-PA^{\top}+BB^{\top}>0$ (13)

(here, under $K=-\frac{1}{2}B^{\top}P^{-1}$, $A+BK$ is Hurwitz).

####  II-A6 Observability

Regarding the _observability_ [37] of the CT-LTI system (1), consider the
following proposition.

######  Proposition 6.

[41, pp.76][40, pp. 87] A CT-LTI system (1) is: (1) stable and observable iff
$\exists P>0$ such that

$-A^{\top}P-PA-C^{\top}C=0,$ (14)

and (2) detectable iff $\exists P>$ such that

$-A^{\top}P-PA+C^{\top}C>0$ (15)

(here, under $L=\frac{1}{2}P^{-1}C^{\top}$, $A-LC$ is Hurwitz).

###  II-B Full-State Feedback (FSF) Controller Synthesis for CT-LTI Systems

Consider the CT-LTI system (1) with noise $w(t)\in\mathbb{R}^{q}$:

$\displaystyle\dot{x}(t)=Ax(t)+Bu(t)+Ew(t),$ (16) $\displaystyle
y(t)=Cx(t)+Du(t)+Fw(t).$

Under full-state feedback (FSF) control $u(t)=Kx(t)$, the closed-loop CT-LTI
system takes the form

$\displaystyle\dot{x}(t)=(A+BK)x(t)+Ew(t),$ (17) $\displaystyle
y(t)=(C+DK)x(t)+Fw(t).$

####  II-B1 Stabilization

The following proposition gives an LMI condition that leads to synthesizing a
FSF controller $K$ such that the closed-loop system (17) is stabilized.

######  Proposition 7.

Under $D=w(t)=\mathbf{0}$, the closed-loop CT-LTI system (17) is stable iff
$\exists M>0$ and $L$ such that

$-MA^{\top}-AM-L^{\top}B^{\top}-BL>0$ (18)

and $K=LM^{-1}$.

###### Proof.

The proof is complete by following the steps: (1) apply Prop. 1 for (17) to
get an LMI in $P$ and $K$, (2) transform the obtained LMI using Lm. 2 with
$P^{-1}$ (pre- and post-multiply by $P^{-1}$), and (3) change the LMI
variables using $M\triangleq P^{-1}$ and $L\triangleq KP^{-1}$. Details are
omitted for brevity. ∎

####  II-B2 $(Q,S,R)$-Dissipativation

The following proposition provides an LMI condition that leads to synthesize a
FSF controller $K$ such that the closed-loop system (17) is
$(Q,S,R)$-dissipative from $w(t)$ to $y(t)$.

######  Proposition 8.

Under $D=\mathbf{0}$, the closed-loop CT-LTI system (17) is
$(Q,S,R)$-dissipative (with $-Q>0$, $R=R^{\top}$) from $w(t)$ to $y(t)$ iff
$\exists M>0$ and $L$ such that

$\begin{bmatrix}-\mathcal{H}_{s}(AM+BL)&-E+MC^{\top}S&MC^{\top}\\\
\star&\mathcal{H}_{s}(F^{\top}S)+R&F^{\top}\\\
\star&\star&-Q^{-1}\end{bmatrix}>0$ (19)

and $K=LM^{-1}$.

###### Proof.

The proof is complete by following the steps: (1) apply Prop. 2 for (17) to
get an LMI in $P$ and $K$, (2) transform the LMI using Lm. 2 with
$\text{diag}(P^{-1},\mathbf{I},\mathbf{I})$, and (3) change the LMI variables
using $M\triangleq P^{-1}$ and $L\triangleq KP^{-1}$. ∎

####  II-B3 $\mathcal{H}_{2}$-Optimal Control

Under FSF control $u(t)=Kx(t)$, the goal of $\mathcal{H}_{2}$-optimal control
is to synthesize a controller $K$ that minimizes the $\mathcal{H}_{2}$-norm of
the closed-loop system (17) (from $w(t)$ to $y(t)$). For this purpose, the
following proposition provides an LMI based approach.

######  Proposition 9.

Under $F=\mathbf{0}$, the $\mathcal{H}_{2}$-optimal FSF controller $K$ for the
closed-loop system (17) is found by solving the LMI:

$\displaystyle\min_{M,Q,L,\gamma}$ $\displaystyle\ \gamma$ (20) sub. to:
$\displaystyle\ M>0,\ Q>0,\ \gamma>0$
$\displaystyle\begin{bmatrix}-\mathcal{H}_{s}(AM+BL)&-MC^{\top}-L^{\top}D^{\top}\\\
\star&\gamma\mathbf{I}\end{bmatrix}>0,$ $\displaystyle\begin{bmatrix}M&E\\\
\star&Q\end{bmatrix}>0,\ \mbox{tr}\left(Q\right)<\gamma,$

and $K=LM^{-1}$.

###### Proof.

The proof is complete by applying Prop. 3 to (17) and executing the change of
variables: $M=P$ and $L=KP$. ∎

####  II-B4 $\mathcal{H}_{\infty}$-Optimal Control

Similarly, the goal of $\mathcal{H}_{\infty}$-optimal control is to synthesize
a controller $K$ that minimizes the $\mathcal{H}_{\infty}$-norm of the closed-
loop system (17) (from $w(t)$ to $y(t)$).

######  Proposition 10.

The $\mathcal{H}_{\infty}$-optimal FSF controller $K$ for the closed-loop
system (17) is found by solving the LMI:

$\displaystyle\min_{M,L,\gamma}$ $\displaystyle\ \gamma$ (21) sub. to:
$\displaystyle\ M>0,\gamma>0,$
$\displaystyle\begin{bmatrix}-\mathcal{H}_{s}(AM+BL)&-E&-MC^{\top}-L^{\top}D^{\top}\\\
\star&\gamma\mathbf{I}&-F^{\top}\\\
\star&\star&\gamma\mathbf{I}\end{bmatrix}>0,$

and $K=LM^{-1}$.

###### Proof.

The proof is complete by following the steps: (1) apply Prop. 4 to (17), (2)
transform the main LMI using Lm. 2 with
$\text{diag}(P^{-1},\mathbf{I},\mathbf{I})$, and (3) change the LMI variables
using $M\triangleq P^{-1}$ and $L=KP^{-1}$. ∎

###  II-C Observer Design for CT-LTI Systems

For state feedback control $u(t)=Kx(t)$ (unlike output feedback control
$u(t)=Ky(t)$), the controller requires the state information $x(t)$ of the CT-
LTI system (16). However, typically, the state $x(t)$ (unlike the output
$y(t)$) is not available to the controller. Therefore, an _observer_ is
required to keep an _estimate_ of the state $x(t)$ as $\hat{x}(t)$.

####  II-C1 Luenberger Observer

For the CT-LTI system (16), consider a _Luenberger observer_ implemented at
the controller:

$\dot{\hat{x}}=\hat{A}\hat{x}(t)+\hat{B}u(t)+Ly(t),$ (22)

that has the estimation error ($e(t)\triangleq x(t)-\hat{x}(t)$) dynamics:

$\displaystyle\dot{e}(t)=$ $\displaystyle\hat{A}e(t)+(A-\hat{A}-LC)x(t)\ $
(23) $\displaystyle+(B-\hat{B}-LD)u(t)+(E-LF)w(t).$

The Luenberger observer parameters: $\hat{A},\hat{B}$ and $L$ can be selected
according to the following proposition.

######  Proposition 11.

Under $w(t)=0$ and Luenberger observer (22) parameters: $\hat{A}\triangleq
A-LC$ and $\hat{B}\triangleq B-LD$, the estimation error dynamics (23) are
stable iff $\exists P>0$ and $K$ such that

$-A^{\top}P-PA+C^{\top}K^{\top}+KC>0$ (24)

and $L=P^{-1}K$.

###### Proof.

Under $w(t)=0$ and the given observer parameter choices, the estimation error
dynamics (23) reduces to:

$\dot{e}(t)=(A-LC)e(t).$ (25)

The proof is complete by applying Prop. 1 to (25) and then executing a change
of variables using $K=PL$. ∎

The Luenberger observer design proposed above assumes the noise-less case of
(16) (i.e., (1)). This assumption is relaxed in the $(Q,S,R)$-dissipative and
$\mathcal{H}_{2}/\mathcal{H}_{\infty}$-optimal observer designs described in
subsequent subsections. However, before getting into those details, first,
consider the Luenberger observer (22) parameters:

$\hat{A}\triangleq A-LC\ \ \ \ \mbox{ and }\ \ \ \ \hat{B}\triangleq B-LD,$
(26)

under which the estimation error dynamics (23) take the form

$\displaystyle\dot{e}(t)=$ $\displaystyle\ (A-LC)e(t)+(E-LF)w(t),$ (27)
$\displaystyle z(t)=$ $\displaystyle\ Ge(t)+Jw(t).$

where $z(t)$ is a pre-defined performance metric.

####  II-C2 $(Q,S,R)$-Dissipative Observer

The $(Q,S,R)$-dissipative observer synthesizes the Luenberger observer gain
$L$ such that (27) is $(Q,S,R)$-dissipative from $w(t)$ to $z(t)$. For this
purpose, the following proposition can be used.

######  Proposition 12.

The estimation error dynamics (27) is $(Q,S,R)$-dissipative (with $-Q>0$,
$R=R^{\top}$) from $w(t)$ to $z(t)$ iff $\exists P>0$ and $K$ such that

$\begin{bmatrix}-\mathcal{H}_{s}(PA-KC)&-PE+KF+G^{\top}S&G^{\top}\\\
\star&\mathcal{H}_{s}(J^{\top}S)+R&J^{\top}\\\
\star&\star&-Q^{-1}\end{bmatrix}>0$ (28)

and $L=KP^{-1}$.

###### Proof.

The proof is complete by applying Prop. 2 to (27) and executing a change of
variables using $K=PL$. ∎

####  II-C3 $\mathcal{H}_{2}$-Optimal Observer

The goal of the $\mathcal{H}_{2}$-optimal observer is to synthesize the
Luenberger observer gain $L$ that minimizes the $\mathcal{H}_{2}$-norm of (27)
(from $e(t)$ to $z(t)$).

######  Proposition 13.

Under $J=0$, the $\mathcal{H}_{2}$-optimal observer gain $L$ for (27) is found
by solving the LMI:

$\displaystyle\min_{P,Q,K,\gamma}$ $\displaystyle\ \gamma$ (29) sub. to:
$\displaystyle\ P>0,\ Q>0,\ \gamma>0,$
$\displaystyle\begin{bmatrix}-\mathcal{H}_{s}(PA-KC)&-PE+KF\\\
\star&\gamma\mathbf{I}\end{bmatrix}>0,$
$\displaystyle\begin{bmatrix}P&G^{\top}\\\ \star&Q\end{bmatrix}>0,\
\mbox{tr}\left(Q\right)<\gamma,$

and $L=P^{-1}K$.

###### Proof.

The proof is complete by applying Prop. 3 (8) to (27) and executing a change
of variables using $K=PL$. ∎

####  II-C4 $\mathcal{H}_{\infty}$-Optimal Observer

Similarly, the goal of the $\mathcal{H}_{\infty}$-optimal observer is to
synthesize the Luenberger observer gain $L$ that minimizes the
$\mathcal{H}_{\infty}$-norm of (27) (from $e(t)$ to $z(t)$).

######  Proposition 14.

The $\mathcal{H}_{\infty}$-optimal observer gain $L$ for (27) is found by
solving the LMI:

$\displaystyle\min_{P,K,\gamma}$ $\displaystyle\ \gamma$ (30) sub. to:
$\displaystyle\ P>0,\ \gamma>0,$
$\displaystyle\begin{bmatrix}-\mathcal{H}_{s}(PA-KC)&-PE+KF&-G^{\top}\\\
\star&\gamma\mathbf{I}&-J^{\top}\\\
\star&\star&\gamma\mathbf{I}\end{bmatrix}>0,$

and $L=P^{-1}K$.

###### Proof.

The proof is complete by applying Prop. 4 to (27) and executing a change of
variables using $K=PL$. ∎

###  II-D Dynamic Output Feedback (DOF) Controller Synthesis for CT-LTI
Systems

Consider the CT-LTI system (1) with noise $w(t)\in\mathbb{R}^{q}$ and
performance $z(t)\in\mathbb{R}^{l}$:

$\displaystyle\dot{x}(t)=Ax(t)+Bu(t)+Ew(t),$ (31) $\displaystyle
y(t)=Cx(t)+Du(t)+Fw(t),$ $\displaystyle z(t)=Gx(t)+Hu(t)+Jw(t).$

Under $D=\mathbf{0}$ and dynamic output feedback (DOF) control (from $y(t)$ to
$u(t)$):

$\displaystyle\dot{\zeta}(t)=A_{c}\zeta(t)+B_{c}y(t),$ (32) $\displaystyle
u(t)=C_{c}\zeta(t)+D_{c}y(t),$

where $\zeta(t)\in\mathbb{R}^{r}$, the closed-loop CT-LTI system (31) takes
the form:

$\displaystyle\dot{\theta}(t)=\bar{A}\theta(t)+\bar{B}w(t),$ (33)
$\displaystyle z(t)=\bar{C}\theta(t)+\bar{D}w(t),$

with
$\theta(t)=\begin{bmatrix}x^{\top}(t)&\zeta^{\top}(t)\end{bmatrix}^{\top}$ and

$\displaystyle\bar{A}\triangleq\begin{bmatrix}A+BD_{c}C&BC_{c}\\\
B_{c}C&A_{c}\end{bmatrix},\ \bar{B}\triangleq\begin{bmatrix}E+BD_{c}F\\\
B_{c}F\end{bmatrix},$
$\displaystyle\bar{C}\triangleq\begin{bmatrix}G+HD_{c}C&HC_{c}\end{bmatrix},\
\bar{D}\triangleq\begin{bmatrix}J+HD_{c}F\end{bmatrix}.$

In parallel to Sec. II-B, in the subsequent subsections, we provide LMI
conditions for DOF controller synthesis (i.e., to design
$A_{c},B_{c},C_{c},D_{c}$ in (32)) so as to stabilize, $(Q,S,R)$-dissipativate
or optimize $\mathcal{H}_{2}$/$\mathcal{H}_{\infty}$-norm of the closed-loop
system (33). Before getting into those details, first, consider the unique
change of variables (CoVs) process given below.

####  II-D1 Change of Variables (CoVs)

Inspired by [42, 40], let

$\displaystyle P\triangleq\begin{bmatrix}X&M\\\ M^{\top}&\ast\end{bmatrix},\ $
$\displaystyle P^{-1}\triangleq\begin{bmatrix}Y&N\\\
N^{\top}&\ast\end{bmatrix},$ (34)
$\displaystyle\Pi_{x}\triangleq\begin{bmatrix}\mathbf{I}&X\\\
\mathbf{0}&M^{\top}\end{bmatrix},\ $
$\displaystyle\Pi_{y}\triangleq\begin{bmatrix}Y&\mathbf{I}\\\
N^{\top}&\mathbf{0}\end{bmatrix},$

where $X,Y,M,N$ are some matrices with appropriate dimensions ($\ast$
represents irrelevant matrices). Based on this definition, it is easy to
establish the following three properties:

  1. 1.

Matrices $X,Y,M,N$ satisfy:

$XY+MN^{\top}=\mathbf{I};$ (35)

  2. 2.

Matrices $P,P^{-1},\Pi_{x},\Pi_{y}$ satisfy:

$P\Pi_{y}=\Pi_{x},\ \ \Pi_{y}=P^{-1}\Pi_{x},\ \
\Pi_{y}^{\top}=\Pi_{x}^{\top}P^{-1};$ (36)

  3. 3.

If $M$ is full-rank,

$P>0\iff\exists X,Y>0\mbox{ such that }\begin{bmatrix}Y&\mathbf{I}\\\
\mathbf{I}&X\end{bmatrix}>0.$ (37)

We point out that the last property given above can be proven by subsequently
using Lm. 2 with $\Pi_{x}^{\top}P^{-1}$, (36) and (34) as:

$P>0\iff\Pi_{x}^{\top}P^{-1}\Pi_{x}=\Pi_{y}^{\top}\Pi_{x}=\begin{bmatrix}Y&\mathbf{I}\\\
\mathbf{I}&X\end{bmatrix}>0.$

Note also that, given any two matrices $X,Y$ such that $\mathbf{I}-XY$ is non-
singular, two unique full-rank matrices $M,N$ can always be found such that
(35) using LU-decomposition [36].

Further, under such given $X,Y,M,N$ matrices, a set of matrices
$\\{A_{n},B_{n},C_{n},D_{n}\\}$ can be uniquely transformed respectively to
and from another set of matrices $\\{A_{c},B_{c},C_{c},D_{c}\\}$ using

$\displaystyle D_{c}\triangleq$ $\displaystyle\ D_{n},$ (38) $\displaystyle
C_{c}\triangleq$ $\displaystyle\ (C_{n}-D_{n}CY)N^{-\top},$ $\displaystyle
B_{c}\triangleq$ $\displaystyle\ M^{-1}(B_{n}-XBD_{n}),$ $\displaystyle
A_{c}\triangleq$ $\displaystyle\ M^{-1}(A_{n}-B_{n}CY-
XBC_{n}-X(A-BD_{n}C)Y)N^{-\top}$

and

$\displaystyle D_{n}\triangleq$ $\displaystyle\ D_{c},$ (39) $\displaystyle
C_{n}\triangleq$ $\displaystyle\ D_{c}CY+C_{c}N^{\top},$ $\displaystyle
B_{n}\triangleq$ $\displaystyle\ XBD_{c}+MB_{c},$ $\displaystyle
A_{n}\triangleq$ $\displaystyle\
MA_{c}N^{\top}+MB_{c}CY+XBC_{c}N^{\top}+X(A+BD_{c}C)Y.$

As we will see in the sequel, we formulate each DOF controller synthesis
problem as an LMI problem in matrix parameters $X,Y,A_{n},B_{n},C_{n},D_{n}$
using: (1) intermediate variables in (34), (2) Lm. 2, and (3) relationships in
(33),(36), (37) and (39). More explicitly, the following relationships (that
can be proven by direct substitution) will be pivotal in this task:

$\displaystyle\Pi_{x}^{\top}\bar{A}\Pi_{y}=$ $\displaystyle\
\begin{bmatrix}AY+BC_{n}&A+BD_{n}C\\\ A_{n}&XA+B_{n}C\end{bmatrix},$ (40)
$\displaystyle\Pi_{x}^{\top}\bar{B}=$ $\displaystyle\
\begin{bmatrix}E+BD_{n}F\\\ XE+B_{n}F\end{bmatrix},$
$\displaystyle\bar{C}\Pi_{y}=$ $\displaystyle\
\begin{bmatrix}GY+HC_{n}&G+HD_{n}C\end{bmatrix}.$

Upon solving each such formulated LMI problem (in
$X,Y,A_{n},B_{n},C_{n},D_{n}$), the relationships in (35) and (38) can be used
to obtain the matrices $M,N,A_{c},B_{c},C_{c},D_{c}$ (i.e., the DOF controller
(32)).

####  II-D2 Stabilization

The following proposition gives an LMI condition that leads to synthesizing a
DOF controller (32) such that the closed-loop system (33) is stabilized.

######  Proposition 15.

Under $D=w(t)=\mathbf{0}$, the closed-loop CT-LTI system (33) is stable iff
$\exists X,Y>0$ and $A_{n},B_{n},C_{n},D_{n}$ such that

$\displaystyle\begin{bmatrix}Y&\mathbf{I}\\\ \mathbf{I}&X\end{bmatrix}>0,$
(41)
$\displaystyle\begin{bmatrix}-\mathcal{H}_{s}(AY+BC_{n})&-A-BD_{n}C-A_{n}^{\top}\\\
\star&-\mathcal{H}_{s}(XA+B_{n}C)\end{bmatrix}>0,$ (42)

and $A_{c},B_{c},C_{c},D_{c}$ (32) are found by CoVs (35) and (38).

###### Proof.

Applying Prop. 1 to (33) give the LMI conditions necessary and sufficient for
the stabilization of (33) as: $\exists P>0$ such that

$-\bar{A}^{\top}P-P\bar{A}>0.$ (43)

Using the CoVs in (34) and (37), the LMI $P>0$ can be transformed to (41).
Finally, (43) can be transformed to (42) by applying Lm. 2 with
$\Pi_{x}^{\top}P^{-1}$ and substituting from (36):

$\displaystyle-\bar{A}^{\top}P-P\bar{A}>0\iff-\Pi_{x}^{\top}P^{-1}\bar{A}^{\top}\Pi_{x}-\Pi_{x}^{\top}\bar{A}P^{-1}\Pi_{x}>0,$
$\displaystyle\iff-\Pi_{y}^{\top}\bar{A}^{\top}\Pi_{x}-\Pi_{x}^{\top}\bar{A}\Pi_{y}>0\iff\mbox{\eqref{Eq:Pr:StabilizationUnderDOF2}}.$

Note that the last step above results from (40). ∎

$\begin{bmatrix}-\mathcal{H}_{s}(AY+BC_{n})&-A-BD_{n}C-A_{n}^{\top}&-E-BD_{n}F+(YG^{\top}+C_{n}^{\top}H^{\top})S&YG^{\top}+C_{n}^{\top}H^{\top}\\\
\star&-\mathcal{H}_{s}(XA+B_{n}C)&-XE-
B_{n}F+(G^{\top}+C^{\top}D_{n}^{\top}H^{\top})S&G^{\top}+C^{\top}D_{n}^{\top}H^{\top}\\\
\star&\star&\mathcal{H}_{s}((J^{\top}+F^{\top}D_{n}^{\top}H^{\top})S)+R&J^{\top}+F^{\top}D_{n}^{\top}H^{\top}\\\
\star&\star&\star&-Q^{-1}\end{bmatrix}>0$ (44)

$\begin{bmatrix}-\mathcal{H}_{s}(AY+BC_{n})&-A-BD_{n}C-A_{n}^{\top}&-E-BD_{n}F\\\
\star&-\mathcal{H}_{s}(XA+B_{n}C)&-XE-B_{n}F\\\
\star&\star&\mathbf{I}\end{bmatrix}>0\ \ \mbox{ and }\ \
\begin{bmatrix}Y&\mathbf{I}&YG^{\top}+C_{n}^{\top}H^{\top}\\\
\star&X&G^{\top}+C^{\top}D_{n}^{\top}H^{\top}\\\ \star&\star&Q\end{bmatrix}>0$
(45)

$\begin{bmatrix}-\mathcal{H}_{s}(AY+BC_{n})&-A-BD_{n}C-A_{n}^{\top}&-E-BD_{n}F&-YG^{\top}-C_{n}^{\top}H^{\top}\\\
\star&-\mathcal{H}_{s}(XA+B_{n}C)&-XE-
B_{n}F&-G^{\top}-C^{\top}D_{n}^{\top}H^{\top}\\\
\star&\star&\gamma\mathbf{I}&-J^{\top}-F^{\top}D_{n}^{\top}H^{\top}\\\
\star&\star&\star&\gamma I\end{bmatrix}>0$ (46)

####  II-D3 $(Q,S,R)$-Dissipativation

The following proposition provides an LMI condition that leads to synthesize a
DOF controller (32) such that the closed-loop system (33) is
$(Q,S,R)$-dissipative from $w(t)$ to $z(t)$.

######  Proposition 16.

Under $D=\mathbf{0}$, the closed-loop CT-LTI system (33) is
$(Q,S,R)$-dissipative (with $-Q>0$, $R=R^{\top}$) from $w(t)$ to $z(t)$ iff
$\exists X,Y>0$ and $A_{n},B_{n},C_{n},D_{n}$ such that

$\begin{bmatrix}Y&\mathbf{I}\\\ \mathbf{I}&X\end{bmatrix}>0\mbox{ and
\eqref{Eq:Pr:DissipativationUsingDOF2}},$ (47)

and $A_{c},B_{c},C_{c},D_{c}$ (32) are found by CoVs (35) and (38).

###### Proof.

The proof starts with applying Prop. 2 to (33) to obtain the LMI conditions
necessary and sufficient for the $(Q,S,R)$-dissipativation of (33) as:
$\exists P>0$ such that

$\begin{bmatrix}-\bar{A}^{\top}P-P\bar{A}&-P\bar{B}+\bar{C}^{\top}S&\bar{C}^{\top}\\\
\star&\bar{D}^{\top}S+S^{\top}\bar{D}+R&\bar{D}^{\top}\\\
\star&\star&-Q^{-1}\end{bmatrix}>0.$ (48)

Similar to the proof of Prop. 15, using the CoVs in (34) and (37), the LMI
$P>0$ can be transformed to (47). To obtain (44) from (48), first, Lm. 2 is
applied with $\text{diag}(\Pi_{x}^{\top}P^{-1},\mathbf{I},\mathbf{I}\\}$ and
then the results are substituted using (36) to obtain:

$\begin{bmatrix}-\mathcal{H}_{s}(\Pi_{x}^{\top}\bar{A}\Pi_{y})&-\Pi_{x}^{\top}\bar{B}+\Pi_{y}^{\top}\bar{C}^{\top}S&\Pi_{y}^{\top}\bar{C}^{\top}\\\
\star&\mathcal{H}_{s}(\bar{D}^{\top}S)+R&\bar{D}^{\top}\\\
\star&\star&-Q^{-1}\end{bmatrix}>0.$ (49)

Finally, the above matrix inequality can be transformed to get the LMI in (44)
using (40). ∎

####  II-D4 $\mathcal{H}_{2}$-Optimal Control

The goal of $\mathcal{H}_{2}$-optimal control here is to synthesize a DOF
controller (32) that minimizes the $\mathcal{H}_{2}$-norm of the closed-loop
system (33) from $w(t)$ to $z(t)$.

######  Proposition 17.

Under $D=\mathbf{0}$, the $\mathcal{H}_{2}$-optimal DOF controller (32) for
the closed-loop system (33) is found by solving the LMI:

$\displaystyle\min_{\begin{subarray}{c}X,Y,Q,\gamma\\\
A_{n},B_{n},C_{n},D_{n}\end{subarray}}$ $\displaystyle\gamma$ (50) sub. to:
$\displaystyle X>0,\ Y>0,\ Q>0,\ \gamma>0,\
\mbox{\eqref{Eq:Pr:H2ControlUnderDOF2}},$ $\displaystyle J+HD_{n}F=0,\
\mbox{tr}\left(Q\right)<\gamma,$

and $A_{c},B_{c},C_{c},D_{c}$ (32) are found by CoVs (35) and (38).

###### Proof.

The proof follows similar steps as that of Prop. 16. ∎

####  II-D5 $\mathcal{H}_{\infty}$-Optimal Control

Similarly, the goal of $\mathcal{H}_{\infty}$-optimal control is to synthesize
a DOF controller (32) that minimizes the $\mathcal{H}_{\infty}$-norm of the
closed-loop system (33) (from $w(t)$ to $z(t)$).

######  Proposition 18.

Under $D=0$, the $\mathcal{H}_{\infty}$-optimal DOF controller (32) for the
closed-loop system (33) is found by solving the LMI:

$\displaystyle\min_{\begin{subarray}{c}X,Y,\gamma\\\
A_{n},B_{n},C_{n},D_{n}\end{subarray}}$ $\displaystyle\ \gamma$ (51) sub. to:
$\displaystyle\ X>0,Y>0,\gamma>0,\begin{bmatrix}Y&\mathbf{I}\\\
\mathbf{I}&X\end{bmatrix}>0,\eqref{Eq:Pr:HInfControlUnderDOF2}.$

and $A_{c},B_{c},C_{c},D_{c}$ (32) are found by CoVs (35) and (38).

###### Proof.

The proof follows similar steps as that of Prop. 16. ∎

We conclude this section by summarizing all the established theoretical
results in Tab. I.

TABLE I: Summary of Theoretical Results (CT-LTI Systems).

Proposition # (For CT-LTI Systems) | Concept
---|---
Stability |  |  $(Q,S,R)$-
---
Dissipativity
|  $\mathcal{H}_{2}$-
---
Norm
|  $\mathcal{H}_{\infty}$-
---
Norm
Task |  | CT-LTI System
---
Analysis
1 | 2 | 3 | 4
| FSF Controller
---
Synthesis
7,5 | 8 | 9 | 10
| Observer
---
Design
11,6 | 12 | 13 | 14
| DOF Controller
---
Synthesis
15 | 16 | 17 | 18

##  III The Continuous-Time Networked System (CTNS)

In this section, we provide the details of the considered continuous-time
networked system (CTNS) and outline the interested research problem.

###  III-A Subsystems of the CTNS

####  III-A1 Dynamics

We consider a CTNS $\mathcal{G}_{N}$ comprised of $N$ interconnected
subsystems $\\{\Sigma_{i}:i\in\mathbb{N}_{N}\\}$ (e.g., see Fig. 1). The
dynamics of the $i$th subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ are given by

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

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 represent the state, input, disturbance and output specific to
the subsystems $\Sigma_{i}$ at time $t\in\mathbb{R}_{\geq 0}$.

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

####  III-A2 Neighbors

In (52), we denote $\bar{\mathcal{E}}_{i}\triangleq\mathcal{E}_{i}\cup\\{i\\}$
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 subsystem $\Sigma_{i}$ (i.e., $j\in\mathcal{E}_{i}$) iff $j\neq
i$ and matrices $A_{ij},B_{ij},C_{ij},D_{ij},E_{ij},F_{ij}$ in (52) are not
all zero matrices.

On the other hand, we also define a set of “out-neighbors” for the subsystem
$\Sigma_{i}$ as
$\mathcal{F}_{i}\triangleq\\{j:j\in\mathbb{N}_{N},\mathcal{E}_{j}\ni i\\}$
with $\bar{\mathcal{F}}_{i}\triangleq\mathcal{F}_{i}\cup\\{i\\}$. Formally,
any subsystem $\Sigma_{j}$ is an “out-neighbor” of subsystem $\Sigma_{i}$
(i.e., $j\in\mathcal{F}_{i}$) iff $j\neq i$ and matrices
$A_{ji},B_{ji},C_{ji},D_{ji},E_{ji},F_{ji}$ (in (52) written for subsystem
$\Sigma_{j}$) are not all zero matrices. Finally, for notational convenience,
let us denote $\mathcal{C}_{i}\triangleq\mathcal{E}_{i}\cup\mathcal{F}_{i}$
with $\bar{\mathcal{C}}_{i}\triangleq\mathcal{C}\cup\\{i\\}$.

It is worth noting that in this paper (similar to our preliminary work [43])
we do not constrain ourselves to cascaded network topologies where
$\mathcal{E}_{i}\equiv\mathcal{F}_{i}=\\{i-1,i+1\\}\cap\mathbb{N}_{N},\forall
i\in\mathbb{N}_{N}$ as in [2] or to bi-directional network topologies where
$\mathcal{E}_{i}\equiv\mathcal{F}_{i},\forall i\in\mathbb{N}_{N}$ as in [1].
Moreover, as can be seen in (52), here we consider a general form of a
networked dynamical system as opposed to [2] or [1] where coupling between any
two subsystems $\Sigma_{i}$ and $\Sigma_{j}$ was possible only through the
state variables $x_{i},x_{j}$ (i.e., via matrices $A_{ij},A_{ji}$ in (52)).

####  III-A3 Local Controllers and Observers

One of the main objectives of this paper is to design distributed (i.e.,
local) controllers (e.g., FSF) and observers (e.g., Luenberger) at subsystems
of the considered CTNS in a decentralized manner.

In particular, at a subsystem $\Sigma_{i}$:

  1. 1.

a local FSF controller may take the form

$u_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}K_{ij}x_{j}(t),$ (53)

where $\\{K_{ij}:j\in\bar{\mathcal{E}}_{i}\\}$ is the set of local FSF
controller parameters that subsystem $\Sigma_{i}$ has to design;

  2. 2.

a local Luenberger observer may take the form

$\dot{\hat{x}}_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}\hat{A}_{ij}\hat{x}_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}\hat{B}_{ij}u_{j}+\sum_{j\in\bar{\mathcal{E}}_{i}}L_{ij}y_{j}(t),$
(54)

where $\\{(\hat{A}_{ij},\hat{B}_{ij},L_{ij}):j\in\bar{\mathcal{E}}_{i}\\}$ is
the set of local Luenberger observer parameters that subsystem $\Sigma_{i}$
has to design;

  3. 3.

a local DOF controller may take the form

$\displaystyle\dot{\zeta}_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}A_{c,ij}\zeta_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}B_{c,ij}y_{j}(t),$
(55) $\displaystyle
u_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}C_{c,ij}\zeta_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}D_{c,ij}y_{j}(t),$

where $\\{(A_{c,ij},B_{c,ij},C_{c,ij},D_{c,ij}):j\in\bar{\mathcal{E}}_{i}\\}$
is the set of local DOF controller parameters that subsystem $\Sigma_{i}$ has
to design.

####  III-A4 Local Performance Metrics

As we saw in Sec. II, a pre-defined performance metric (often denoted as
$z(t)$, e.g., see (27),(31)) is required when designing controllers or
observers in a $(Q,S,R)$-dissipative or
$\mathcal{H}_{2}/\mathcal{H}_{\infty}$-optimal sense. Therefore, a pre-defined
local performance metric is required at each subsystem $\Sigma_{i}$ when
designing local controllers and observers in a $(Q,S,R)$-dissipative or
$\mathcal{H}_{2}/\mathcal{H}_{\infty}$-optimal sense.

In particular, at a subsystem $\Sigma_{i}$, we use the following pre-defined
local performance metrics:

  1. 1.

For local FSF controller design:

$y_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}C_{ij}x_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}D_{ij}u_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}F_{ij}w_{j}(t);$
(56)

  2. 2.

For local Luenberger observer design:

$z_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}G_{ij}(x_{j}(t)-\hat{x}_{j}(t))+\sum_{j\in\bar{\mathcal{E}}_{i}}J_{ij}w_{j}(t);$
(57)

  3. 3.

For local DOF controller design:

$z_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}G_{ij}x_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}H_{ij}u_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}J_{ij}w_{j}(t).$
(58)

###  III-B The CTNS

####  III-B1 Dynamics

By writing (52) for all $i\in\mathbb{N}_{N}$, the dynamics of the networked
system $\mathcal{G}_{N}$ can be obtained as

$\displaystyle\dot{x}(t)=$ $\displaystyle\ Ax(t)+Bu(t)+Ew(t),$ (59)
$\displaystyle y(t)=$ $\displaystyle\ Cx(t)+Du(t)+Fw(t),$

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
$x(t)=[x_{i}^{\top}(t)]_{i\in\mathbb{N}_{N}}^{\top}\in\mathbb{R}^{n}$,
$u(t)=[u_{i}^{\top}(t)]_{i\in\mathbb{N}_{N}}^{\top}\in\mathbb{R}^{p}$,
$w(t)=[w_{i}^{\top}(t)]_{i\in\mathbb{N}_{N}}^{\top}\in\mathbb{R}^{q}$ and
$y(t)=[y_{i}^{\top}(t)]_{i\in\mathbb{N}_{N}}^{\top}\in\mathbb{R}^{m}$ (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 $t\in\mathbb{R}_{\geq
0}$.

####  III-B2 Network Topology

Note that the block structure of the matrices $A,B,C,D,F,F$ in (59) determines
the topology of the CTNS (i.e., how different subsystems are coupled) and vice
versa. In the following Sec. IV, we will revisit this topic and extensively
study the properties of such “network” matrices.

####  III-B3 Controllers and Observers

By composing each local controller/observer forms in (53),(54), (55) for all
$i\in\mathbb{N}_{N}$, we can respectively obtain the network level (i.e.,
global):

  1. 1.

FSF controller

$u(t)=Kx(t);$ (60)

  2. 2.

Luenberger observer

$\dot{\hat{x}}(t)=\hat{A}x(t)+\hat{B}u(t)+Ly(t);$ (61)

  3. 3.

DOF controller

$\displaystyle\dot{\zeta}(t)=A_{c}\zeta(t)+B_{c}y(y),$ (62) $\displaystyle
u(t)=C_{c}\zeta(t)+D_{c}y(y),$

where matrices $K,\hat{A},\hat{B},L,A_{c},B_{c},C_{c},D_{c}$ are all $N\times
N$ block matrices comprised of the corresponding local design parameters
(e.g., $K=[K_{ij}]_{i,j\in\mathbb{N}_{N}}$).

####  III-B4 Performance Metrics

Similarly, by composing each pre-defined local controller/observer performance
metric forms in (56),(57),(58) for all $i\in\mathbb{N}_{N}$, we can
respectively obtain the global performance metrics considered for:

  1. 1.

FSF controller design as:

$y(t)=Cx(t)+Dy(t)+Fw(t);$ (63)

  2. 2.

Luenberger observer design as:

$z(t)=G(x(t)-\hat{x}(t))+Jw(t);$ (64)

  3. 3.

DOF controller design as:

$z(t)=Gx(t)+Hu(t)+Jw(t).$ (65)

Here also matrices $C,D,F,G,H,J$ are all $N\times N$ block matrices comprised
of the corresponding pre-defined local performance metric parameters (e.g.,
$C=[C_{ij}]_{i,j\in\mathbb{N}_{N}}$).

###  III-C The Research Problem

Note that the forms of the CTNS (59), global controllers/observers (60)-(62)
and global performance metrics (63)-(65) are respectively identical to the
general CT-LTI system (e.g., (16)), controllers/observers (e.g., (22),(32))
and performance metrics (e.g., (31),(27)) considered in Sec. II. Therefore,
all the LMI-based control solutions (Prop. 1-18) discussed in Sec. II are
directly applicable for the CTNS (59).

For example, based on Prop. 1, the CTNS (59) is globally exponentially stable
if there exists a matrix $P=P^{\top}>0$ such that $-A^{\top}P-PA>0$, where now
$A=[A_{ij}]_{i,j\in\mathbb{N}_{N}}$ is a block matrix with a particular
structure determined by the network topology (e.g., $A_{ij}=\mathbf{0}$ if
$j\not\in\mathcal{E}_{i}$).

Intuitively, verifying/enforcing such LMI conditions requires the knowledge of
the entire networked system and thus calls for a centralized entity. Moreover,
the complete verification/enforcement process may have to be repeated whenever
new subsystems are introduced into the networked system. To address these
challenges, we make the objective of this paper to design a systematic
decentralized and compositional approach to verify/enforce different LMI
conditions of interest (corresponding to different properties of interest,
e.g., see Prop. 1-18) regarding the networked system.

Towards this goal, a critical feature that we will exploit is that LMI
conditions of interest now involve matrices of a particular structure
determined by the network topology (as also pointed out earlier). The next
section of the paper focuses on such network-related matrices (which we will
define as “network matrices”) and derives a decentralized and compositional
test criterion to evaluate their positive definiteness. As we will see in the
subsequent section (Sec. IV), such a test criterion can effortlessly be
adopted to verify/enforce LMI conditions for networked systems in a
decentralized and compositional manner.

##  IV Decentralized Analysis of Networked Systems

As mentioned above, this section is dedicated to establishing several
theoretical and algorithmic results regarding evaluating the positive
definiteness of a particular class of matrices.

###  IV-A Preliminary Concepts

####  IV-A1 Network Matrices

We start by defining a class of matrices we named “network matrices” [43],
that corresponds to a given networked system topology (e.g., $\mathcal{G}_{N}$
in Fig. 1).

######  Definition 2.

Given a networked dynamical system $\mathcal{G}_{n},n\in\mathbb{N}$, any
$n\times n$ block matrix $\Theta=[\Theta_{ij}]_{i,j\in\mathbb{N}_{n}}$ is a
_network matrix_ if: (1) any information specific to the subsystem $i$ is
embedded only in its $i$th block row or block column, and (2)
$j\not\in\mathcal{C}_{i}(\triangleq\mathcal{E}_{i}\cup\mathcal{F}_{i})\implies\Theta_{ij}=\Theta_{ji}=\mathbf{0}$
for all $i,j\in\mathbb{N}_{n}$.

Based on this definition, note that all the $N\times N$ block matrices: (1)
$A,B,C,D,E,F$ in (59), $K,\hat{A},\hat{B},L,A_{c},B_{c},C_{c},D_{c}$ in
(60)-(62) and (3) $C,D,F,G,H,J$ in (63)-(58) are network matrices of the
considered networked dynamical system $\mathcal{G}_{N}$. Note also that any
$n\times n$ block diagonal matrix
$\Theta=\text{diag}(\Theta_{ii}:i\in\mathbb{N}_{n})$ will be a network matrix
of any arbitrary network with $n\in\mathbb{N}$ subsystems if $\Theta_{ii}$ is
specific only to the subsystem $i$. The following lemma summarizes several
interesting and useful properties of such network matrices.

######  Lemma 3.

Given a networked dynamical system $\mathcal{G}_{n},n\in\mathbb{N}$, a few
corresponding block network matrices
$\Theta,\Phi,\\{\Psi^{kl}:k,l\in\mathbb{N}_{m}\\}$ and some arbitrary block
matrix 111Note that $\Psi$ is a block matrix of block network matrices.
$\Psi=[\Psi^{kl}]_{k,l\in\mathbb{N}_{m}}$ (with appropriate block structures):

  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.

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

  4. 4.

$P\Theta P^{\top}$ is a network matrix that corresponds to a re-indexed
version of the original networked system $\mathcal{G}_{n}$ whenever $P$ is a
block symmetric permutation matrix.

###### Proof.

According to Def. 2, any block matrix $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}$ is a
network matrix if $W_{ij}$ is specific only to the subsystems $i$ and $j$.
This view of Def. 2 can be used to prove the Cases 1-3 respectively as:

  1. 1.

$W=\Theta^{\top}\implies W_{ij}=\Theta_{ji}^{\top}$ and
$W=\alpha\Theta+\beta\Phi\implies W_{ij}=\alpha\Theta_{ij}+\beta\Phi_{ij}$;

  2. 2.

$W=\Phi\Theta\implies W_{ij}=\Phi_{ii}\Theta_{ij}$ and
$W=\Theta\Phi\implies W_{ij}=\Theta_{ij}\Phi_{jj}$;

  3. 3.

$W=[[\Psi^{kl}_{ij}]_{k,l\in\mathbb{N}_{m}}]_{i,j\in\mathbb{N}_{n}}\implies
W_{ij}=[\Psi^{kl}_{ij}]_{k,l\in\mathbb{N}_{m}}$;

(where in each case, $W_{ij}$ is specific only to the subsystems $i$ and $j$).
The proof of Case 4 is complete by noting that $P\Theta P$ executes identical
block row and block column operations on $\Theta$ and $P\Theta
P^{\top}=P\Theta P$. ∎

The above lemma allows us to analyze custom block matrices by claiming them to
be “network matrices” under some additional conditions. For example, if $A,P$
are block network matrices and $P$ is block diagonal, then: (1)
$W=-A^{\top}P-PA$ (see Prop. 1) is a network matrix, and (2) if
$\scriptsize\Psi=\begin{bmatrix}P&A^{\top}P\\\ PA&P\end{bmatrix}$ is some
block matrix (see Prop. 19), its _block element-wise_ (BEW) form

$W\triangleq\text{BEW}(\Psi)\triangleq\begin{bmatrix}\begin{bmatrix}P_{ii}e_{ij}&A_{ji}^{\top}P_{jj}\\\
P_{ii}A_{ij}&P_{ii}e_{ij}\end{bmatrix}\end{bmatrix}_{i,j\in\mathbb{N}_{N}}$

is a network matrix222Recall the notation:
$e_{ij}\triangleq\mathbf{I}\cdot\mathbf{1}_{\\{i=j\\}}$..

####  IV-A2 Positive Definiteness

We next provide several useful lemmas on the positive definiteness property of
matrices (in addition to Lm. 1 and Lm. 2).

######  Lemma 4.

(Sylvester’s criterion [37]) A symmetric matrix $W$ is positive definite if
and only if its determinants of the leading principal minors are positive.

######  Lemma 5.

(Cholesky decomposition [36]) A symmetric matrix $W$ is positive definite (or
positive semi-definite) if and only if there exists a lower-triangular matrix
$L$ with positive (or non-negative) diagonal entries such that $W=LL^{\top}$.

The following lemma is parallel to Lm. 3-Case 3.

######  Lemma 6.

Let $\Psi=[\Psi^{kl}]_{k,l\in\mathbb{N}_{m}}$ be an $m\times m$ block matrix
where each of the constituent matrices $\\{\Psi^{kl}:k,l\in\mathbb{N}_{m}\\}$
is also an $n\times n$ block matrix (with appropriate dimensions). Then,
$\Psi>0$ iff
$W=\text{BEW}(\Psi)\triangleq[[\Psi^{kl}_{ij}]_{k,l\in\mathbb{N}_{m}}]_{i,j\in\mathbb{N}_{n}}>0$.

###### Proof.

Notice that $W$ is the block element-wise form of $\Psi$
($W=\text{BEW}(\Psi)$). Therefore, $W$ can be constructed from $\Psi$ by
executing a series of simultaneous row swap and column swap operations on
$\Psi$. In other words, we can find a permutation matrix $P$ so that $W=P\Psi
P$. Note that this permutation matrix $P$ will be symmetric as all the
required row/column operations are simple independent row/column swap
operations. Therefore, $W=P\Psi P^{\top}$. Since all permutation matrices are
full-rank, we can apply Lm. 2 to arrive at result: $\Psi>0\iff W=P\Psi
P^{\top}>0$. ∎

###  IV-B The Main Theoretical Result

We are now ready to establish our main theoretical result as a lemma, which
will be exploited throughout the remainder of this paper. We also acknowledge
that different versions of this lemma have already appeared in [1, 34] and
[44], but without rigorous proofs. Therefore, here we provide a concise
version of it along with a complete proof.

######  Lemma 7.

A symmetric $N\times N$ block matrix $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ iff
$\tilde{W}_{ii}>0,\forall i\in\mathbb{N}_{N}$ where

$\tilde{W}_{ij}=W_{ij}-\sum_{k\in\mathbb{N}_{j-1}}\tilde{W}_{ik}\tilde{W}_{kk}^{-1}\tilde{W}_{jk}^{\top},\
\ \ \ \forall j\in\mathbb{N}_{i}.$ (66)

###### Proof.

To apply Lm. 5, we first determine a lower-triangular block matrix
$L=[L_{ij}]_{i,j\in\mathbb{N}_{N}}$ such that $W=LL^{\top}$. Note that

$W=LL^{\top}\iff
W_{ij}=\sum_{k\in\mathbb{N}_{N}}L_{ik}L_{jk}^{\top}=\sum_{k\in\mathbb{N}_{\min\\{i,j\\}}}L_{ik}L_{jk}^{\top},$
(67)

for any $i,j\in\mathbb{N}_{N}$ (the last step is due to
$L_{ik}=\mathbf{0},\forall k>i$ and $L_{jk}=\mathbf{0},\forall k>j$). In (67),
to make the term $L_{ij}$ the subject, we consider the case $i\geq j$:

$W_{ij}=\sum_{k\in\mathbb{N}_{j}}L_{ik}L_{jk}^{\top}=L_{ij}L_{jj}^{\top}+\sum_{k\in\mathbb{N}_{j-1}}L_{ik}L_{jk}^{\top},$
(68)

which gives (also using the fact that $L_{ij}=\mathbf{0}$, for $i<j$)

$L_{ij}=(W_{ij}-\sum_{k\in\mathbb{N}_{j-1}}L_{ik}L_{jk}^{\top})L_{jj}^{-\top}\mathbf{1}_{\\{i\geq
j\\}}\ \ \ \ \forall i,j\in\mathbb{N}_{N}.$ (69)

With $L=[L_{ij}]_{i,j\in\mathbb{N}_{N}}$ derived in (69) we get $W=LL^{\top}$.
Therefore, according to Lm. 5, $W>0$ if and only if the diagonal elements of
$L$ are positive, i.e., if and only if the diagonal elements of lower-
triangular matrices $\\{L_{ii}:i\in\mathbb{N}_{N}\\}$ are positive. Re-using
Lm. 5, it is easy to see that the latter will occur if and only if
$L_{ii}L_{ii}^{\top}>0,\forall i\in\mathbb{N}_{N}$. In all,

$W>0\iff L_{ii}L_{ii}^{\top}>0,\ \ \ \ \forall i\in\mathbb{N}_{N}.$ (70)

Now, for the case $j\leq i$ (i.e., $j\in\mathbb{N}_{i}$), we can state (69) as

$L_{ij}=\tilde{W}_{ij}L_{jj}^{-\top},$ (71)

where we define $\tilde{W}_{ij}$ as

$\tilde{W}_{ij}\triangleq
W_{ij}-\sum_{k\in\mathbb{N}_{j-1}}L_{ik}L_{jk}^{\top},\ \ \ \
j\in\mathbb{N}_{i}.$ (72)

From (71) (or (69)), when $j=i$, we get the relationship

$L_{ii}L_{ii}^{\top}=\tilde{W}_{ii}.$ (73)

From (73) and (70), it is clear that $W>0\iff\tilde{W}_{ii}>0,\forall
i\in\mathbb{N}_{N}$. Therefore, we now only need to prove that (72) $\iff$
(66). For this purpose, we first simplify the $L_{ik}L_{jk}^{\top}$ term in
(72) using (71) as

$\displaystyle L_{ik}L_{jk}^{\top}=$ $\displaystyle\
\tilde{W}_{ik}L_{kk}^{-\top}(\tilde{W}_{jk}L_{kk}^{-\top})^{\top}=\tilde{W}_{ik}L_{kk}^{-\top}L_{kk}^{-1}\tilde{W}_{jk}^{\top}$
$\displaystyle=$ $\displaystyle\
\tilde{W}_{ik}(L_{kk}L_{kk}^{\top})^{-1}\tilde{W}_{jk}^{\top}=\tilde{W}_{ik}\tilde{W}_{kk}^{-1}\tilde{W}_{jk}^{\top}.$
(74)

Finally, applying (74) in (72), we can obtain (66). ∎

According to the above lemma, 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 tests (iterations). In particular, at the $i$th iteration, we
now only need to test whether $\tilde{W}_{ii}>0$. Since
$\tilde{W}_{ii}=W_{ii}-\sum_{k\in\mathbb{N}_{i-1}}\tilde{W}_{ik}\tilde{W}_{kk}^{-1}\tilde{W}_{ik}^{\top}$
(66), computing $\tilde{W}_{ii}$ only requires the following sets of matrices:

  1. 1.

$\\{W_{ij}:j\in\mathbb{N}_{i}\\}$ (extracted from $W$ in iteration $i$);

  2. 2.

$\\{\tilde{W}_{ij}:j\in\mathbb{N}_{i-1}\\}$ (computed using (66) in iteration
$i$);

  3. 3.

$\\{\\{\tilde{W}_{jk}:k\in\mathbb{N}_{j}\\}:j\in\mathbb{N}_{i-1}\\}$ (computed
in previous $(i-1)$ iterations).

The following corollary of Lm. 7 provide more insights on how the information
computed in previous $(i-1)$ iterations are used when testing
$\tilde{W}_{ii}>0$ at the $i$th iteration.

######  Corollary 1.

A symmetric $N\times N$ block matrix $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ iff
$\tilde{W}_{ii}>0,\forall i\in\mathbb{N}_{N}$ where

$\displaystyle\tilde{W}_{ii}\triangleq$ $\displaystyle\
W_{ii}-\tilde{W}_{i}\mathcal{D}_{i}\tilde{W}_{i}^{\top},$ (75)
$\displaystyle\tilde{W}_{i}\triangleq$ $\displaystyle\
[\tilde{W}_{i1},\tilde{W}_{i2},\ldots,\tilde{W}_{i,i-1}]\triangleq
W_{i}(\mathcal{D}_{i}\mathcal{A}_{i}^{\top})^{-1},$ $\displaystyle
W_{i}\triangleq$ $\displaystyle\ [W_{i1},W_{i2},\ldots,W_{i,i-1}],$
$\displaystyle\mathcal{D}_{i}\triangleq$ $\displaystyle\
\text{diag}([\tilde{W}_{11}^{-1},\tilde{W}_{22}^{-1},\ldots,\tilde{W}_{i-1,i-1}^{-1}]),$
$\displaystyle\mathcal{A}_{i}\triangleq$ $\displaystyle\
\begin{bmatrix}\tilde{W}_{11}&\mathbf{0}&\cdots&\mathbf{0}\\\
\tilde{W}_{21}&\tilde{W}_{22}&\cdots&\mathbf{0}\\\
\vdots&\vdots&\vdots&\vdots\\\
\tilde{W}_{i-1,1}&\tilde{W}_{i-1,2}&\cdots&\tilde{W}_{i-1,i-1}\end{bmatrix}.$

###### Proof.

By simplifying (66) for each $j\in\mathbb{N}_{i-1}$ and re-arranging its terms
to make the term $W_{ij}$ the subject, we can obtain a system of $(i-1)$
equations that can be jointly represented by
$W_{i}=\tilde{W}_{i}\mathcal{D}_{i}\mathcal{A}_{i}^{\top}$ (using the block
matrices defined in (75)). Therefore,
$\tilde{W}_{i}=W_{i}(\mathcal{D}_{i}\mathcal{A}_{i}^{\top})^{-1}$. Finally, by
writing (66) for $j=i$, we get
$\tilde{W}_{ii}=W_{ii}-\sum_{k\in\mathbb{N}_{i-1}}\tilde{W}_{ik}\tilde{W}_{kk}^{-1}\tilde{W}_{ik}^{\top}=W_{ii}-\tilde{W}_{i}\mathcal{D}_{i}\tilde{W}_{i}^{\top},$
and hence the proof is complete via Lm. 7. ∎

###  IV-C Application to Networked Systems Analysis

Let the $N\times N$ block matrix $W$ considered in Co. 1 be a network matrix
(see Def. 2) corresponding to some networked system $\mathcal{G}_{N}$ (59).
The following remarks now can be made regarding using Co. 1 in a such network
setting.

####  IV-C1 Decentralized and Compositional Nature

The nature of Co. 1 implies that testing/enforcing $W>0$ can be achieved in a
decentralized manner over $\mathcal{G}_{N}$ by sequentially testing/enforcing
$\tilde{W}_{ii}>0$ at each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$.

Moreover, during a such process, at a subsystem $\Sigma_{i}$, it only requires
to execute some local computations using some information obtained from the
subsystems that came before it (i.e., from
$\\{\Sigma_{j}:j\in\mathbb{N}_{i-1}\\}$). Therefore, testing/enforcing $W>0$
can be achieved in a compositional manner.

In other words, adding a new subsystem to $\mathcal{G}_{N}$ while ensuring the
positive definiteness of some overall network matrix (now corresponding to
$\mathcal{G}_{N+1}$) can be efficiently and conveniently achieved without
having to re-evaluate the local tests/enforcements at the existing subsystems
in $\mathcal{G}_{N}$.

####  IV-C2 Resilience to Subsystem Removals

The aforementioned compositionality property implies that the proposed
decentralized approach to test/enforce the positive definiteness of a network
matrix (via. Co. 1) is resilient to _subsystem additions_. It turns out that
the proposed approach is also resilient to _subsystem removals_. To understand
this, first, note that a removal of a subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$ will change the network matrix $W$ into a
smaller $(N-1)\times(N-1)$ block network matrix $\bar{W}$. Algebraically,
$\bar{W}$ can be obtained from $W$ by removing its $i$th block row and block
column. The following Lm. 8 proves that $W>0\implies\bar{W}$ for any
$i\in\mathbb{N}_{N}$, i.e., if the network matrix $W$ is positive definite,
any possible residual network matrix $\bar{W}$ will also be positive definite.

######  Lemma 8.

Given a symmetric $N\times N$ block matrix $W>0$, any $(N-1)\times(N-1)$ block
matrix $\bar{W}$ obtained from $W$ by removing its $i$th block row and block
column, will retain the positive definiteness, i.e., $\bar{W}>0$, for any
$i\in\mathbb{N}_{N}$.

###### Proof.

Note that, for any $i\in\mathbb{N}_{N}$, there exists a symmetric permutation
matrix $P$ such that

$PWP^{\top}=\begin{bmatrix}\bar{W}&W_{i}^{\top}\\\ W_{i}&W_{ii}\end{bmatrix},$

where $[W_{i},W_{ii}]$ contains all the blocks of the $i$th block row of $W$.
According to the Sylvester’s criterion (see Lm. 4),
$PWP^{\top}>0\implies\bar{W}>0$. Moreover, as $P$ is a full-rank matrix (bu
definition), from Lm. 2, $W>0\iff PWP^{\top}>0$. Therefore, by combining these
two results, we get $W>0\implies\bar{W}>0$. ∎

This result implies that the proposed approach to enforce/test the positive
definiteness of a network matrix (via. Co. 1) is resilient to _subsystem
removals_. Repeated use of this result implies that if a network matrix of
some network is positive definite, even if several subsystems were removed
from that network, the network matrix of the residual network will remain
positive definite. Therefore, this eliminates the need to re-evaluate the
local tests upon such subsystem removals from a network.

####  IV-C3 The Algorithm

Note that the $j$th block row of the matrix $\mathcal{A}_{i}$ in (75) can be
obtained from the information seen at the subsystem
$\Sigma_{j},j\in\mathbb{N}_{i-1}$ (when $\tilde{W}_{jj}>0$ was tested). In
essence, this matrix $\mathcal{A}_{i}$ can be seen as a compilation of
messages received at the subsystem $\Sigma_{i}$ from previous/existing $(i-1)$
subsystems in the network. On the other hand, the matrix $\mathcal{D}_{i}$ in
(75) is fully-determined by this message matrix $\mathcal{A}_{i}$. Note also
that some components of the matrix $W_{i}$ in (75) may still be unknown to the
subsystem $i$ if the network is asymmetric. However, such unknown components
can be obtained by requesting them from previous/existing $(i-1)$ subsystems
in the network. Hence additional communications may be required to create the
matrix $W_{i}$. Finally, note that $W_{ii}$ is known at (in fact, is intrinsic
to) the subsystem $i$. Therefore, it is now clear how $\tilde{W}_{ii}$ (of
which the positive definiteness needs to be tested) can be obtained using
(75). The proposed overall decentralized and compositional approach to
test/enforce the positive-definiteness of a network matrix $W$ (based on Co.
1) is summarized in the following Alg. 1

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: (75).

12: Compute: $\tilde{W}_{i}\triangleq
W_{i}(\mathcal{D}_{i}\mathcal{A}_{i}^{\top})^{-1}$ $\triangleright$ From (75).

13: Compute: $\tilde{W}_{ii}\triangleq
W_{ii}-\tilde{W}_{i}\mathcal{D}_{i}\tilde{W}_{i}^{\top}$ $\triangleright$ From
(75).

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

###  IV-D Inter-Subsystem Communications

####  IV-D1 Redundant Communications

Even though Alg. 1 is decentralized and compositional, it is, in general, not
distributed. This limitation is evident from the fact that a subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$ having to communicate with all the subsystems
that came before it (i.e., with $\\{\Sigma_{j}:j\in\mathbb{N}_{i-1}\\}$) so as
to construct the matrix $\mathcal{A}_{i}$ (75) when executing Alg. 1.

Nevertheless, we prove the following corollary of Lm. 7 to show that some
communication sessions between subsystems are redundant and thus can be
avoided - depending on the network topology and the used subsystem indexing
scheme.

######  Corollary 2.

A symmetric $N\times N$ block network matrix
$W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ iff $\tilde{W}_{ii}>0,\forall
i\in\mathbb{N}_{N}$ where

$\tilde{W}_{ij}=W_{ij}-\sum_{k\in\mathbb{N}_{j-1}\backslash\mathbb{N}_{L_{ij}-1}}\tilde{W}_{ik}\tilde{W}_{kk}^{-1}\tilde{W}_{jk}^{\top},\
\ \forall j\in\mathbb{N}_{i},$ (76)

with
$\mathcal{L}_{ij}\triangleq\max\\{\min\\{\bar{\mathcal{C}}_{i}\\},\min\\{\bar{\mathcal{C}}_{j}\\}\\}$.

###### Proof.

According to (75), $\mathcal{D}_{i}\mathcal{A}_{i}$ is upper-triangular.
Therefore, $(\mathcal{D}_{i}\mathcal{A}_{i})^{-1}$ is also upper-triangular.
Consequently, due to the relationship
$\tilde{W}_{i}=W_{i}(\mathcal{D}_{i}\mathcal{A}_{i})^{-1}$ (75), whenever the
first $n\in\mathbb{N}_{i-1}$ blocks of $W_{i}$ (75) are zero blocks, the first
$n\in\mathbb{N}_{i-1}$ blocks of $\tilde{W}_{i}$ (75) are also zero blocks.

Since $W$ is a network matrix, $W_{ij}=\mathbf{0}$ for all
$i,j\in\mathbb{N}_{N}$ such that $j\not\in\mathcal{C}_{i}$ (recall that
$\mathcal{C}_{i}\triangleq\mathcal{E}_{i}\cup\mathcal{F}_{i}$, see Def. 2).
Therefore, the first $\min\\{\bar{\mathcal{C}}_{i}\\}-1$ blocks of $W_{i}$ are
zero blocks. Consequently, the first $\min\\{\bar{\mathcal{C}}_{i}\\}-1$
blocks of $\tilde{W}_{i}$ are also zero blocks. Simply, for any
$i\in\mathbb{N}_{N}$, $\tilde{W}_{ij}=\mathbf{0},\forall
j<\min\\{\bar{\mathcal{C}}_{i}\\}$. Similarly, for any $i\in\mathbb{N}_{N}$,
$\tilde{W}_{ik}=\mathbf{0},\forall k<\min\\{\bar{\mathcal{C}}_{i}\\}$ and for
any $j\in\mathbb{N}_{N}$, $\tilde{W}_{jk}=\mathbf{0},\forall
k<\min\\{\bar{\mathcal{C}}_{j}\\}$.

Therefore,
$\tilde{W}_{ik}\tilde{W}_{kk}^{-1}\tilde{W}_{jk}^{\top}=\mathbf{0},\forall
k<\mathcal{L}_{ij}$ where
$\mathcal{L}_{ij}\triangleq\max\\{\min\\{\bar{\mathcal{C}}_{i}\\},\min\\{\bar{\mathcal{C}}_{j}\\}\\}$.
By applying this result in (66), we can directly obtain (76). ∎

Compared to (66), in (76), there may be less number of terms in the summation
as $\mathcal{L}_{ij}\geq 1$. This implies a possible reduction in the
communications required at the subsystem $\Sigma_{i}$ compared to what is
required in (66). For example, in a bi-directional network, if a subsystem
$\Sigma_{j},j\in\mathbb{N}_{i-1}$ is such that
$\min\\{\bar{\mathcal{C}}_{j}\\}>j-1$ (i.e., $\Sigma_{j}$ is the least indexed
subsystem among its neighbors, which also implies $\mathcal{L}_{ij}>j-1$),
then the subsystem $\Sigma_{i}$ only need to get $\tilde{W}_{jj}$ value from
subsystem $\Sigma_{j}$ as $\tilde{W}_{jk}=\mathbf{0},\forall
k\in\mathbb{N}_{j-1}$ under $\mathcal{L}_{ij}>j-1$.

At this point, it should be clear that even though the proposed positive
definiteness testing/enforcing criterion (i.e., Alg. 1) is not distributed in
general, depending on the network topology and the used subsystem indexing
scheme, some communication sessions can be avoided. We next propose a
communication cost function to find an optimum indexing scheme for a given
network topology.

####  IV-D2 Communication Cost

According Co. 2, to analyze/enforce $\tilde{W}_{ii}>0$ at a subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$, it requires each previous subsystem
$\Sigma_{j},j\in\mathbb{N}_{i-1}$ to send the information
$\\{\tilde{W}_{jk}:k\in\mathbb{N}_{j}\\}$ and out of these block matrices,
$\mathcal{L}_{ij}-1$ (at most $j-1$) number of block matrices will be
$\mathbf{0}$, making them redundant. Let us assume the communication cost
associated with a such block matrix $\tilde{W}_{jk}$ as $\gamma_{ijk}$. On the
other hand, the subsystem $\Sigma_{i}$ might also require additional
information from each subsystem $\Sigma_{j},j\in\mathbb{N}_{i-1}$ just to
compute $\\{W_{ij}:j\in\mathbb{N}_{i-1}\\}$ \- if there are unknown components
in these block matrices. Note that this requirement arises only if
$j\in\mathcal{F}_{i}$. Let us assume the communication cost associated with
the block matrix $W_{ij}$ as $\beta_{ij}$. Taking these costs into account, we
can formulate a communication cost function as:

$J(X)=\sum_{i\in\mathbb{N}_{N}}\sum_{j\in\mathbb{N}_{i-1}}\alpha_{ij}\big{(}\beta_{ij}\mathbf{1}_{\\{j\in\mathcal{F}_{i}\\}}+\gamma_{ijj}+\sum_{k=\mathcal{L}_{ij}}^{j-1}\gamma_{ijk}\big{)},$
(77)

associated with executing Alg. 1 over the considered network topology under
the used subsystem indexing scheme $X$. In (77), $\alpha_{ij}\in\mathbb{R}$
represents the unit cost of communication from subsystem $j$ to $i$. Simply,
we can set $\alpha_{ij}=\mathbf{1}_{\\{j\not\in\mathcal{E}_{i}\\}}$ to
penalize the communications that happen over subsystems that are not
neighbors. Here, the indexing scheme $X$ can be any permutation of
$\mathbb{N}_{N}$ and it determines the neighbor sets (e.g.,
$\mathcal{E}_{i},\mathcal{F}_{i},\mathcal{C}_{i}$) and $\mathcal{L}_{ij}$
values for all $i,j\in\mathbb{N}_{N}$. Therefore, the objective function
$J(X)$ formulated above (77) can be used to determine an optimal subsystem
indexing scheme that minimizes costly inter-subsystem communications.

The following corollary shows that, in some networks, under some subsystem
indexing schemes, we can altogether avoid communications between subsystems
that are not neighbors, i.e., we can execute the proposed decentralized
process (Alg. 1) in a distributed manner. For example, Figure 2 shows such a
network configuration.

######  Corollary 3.

Under $\alpha_{ij}\triangleq\mathbf{1}_{\\{j\not\in\mathcal{E}_{i}\\}}$ in
(77), if there exists a subsystem indexing scheme $X$ that makes
$\mathcal{E}_{i}\supseteq\mathbb{N}_{i-1}$ at each subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$, then, $J(X)=0$, i.e., no communications are
needed between non-neighbors.

###### Proof.

Note that $\mathcal{E}_{i}\supseteq\mathbb{N}_{i-1}\iff
j\in\mathcal{E}_{i},\forall j\in\mathbb{N}_{i-1}$. Moreover, since
$\alpha_{ij}\triangleq\mathbf{1}_{\\{j\not\in\mathcal{E}_{i}\\}}$,
$j\in\mathcal{E}_{i}\iff\alpha_{ij}=0$. Combining these two relationships, we
obtain that

$\mathcal{E}_{i}\supseteq\mathbb{N}_{i-1}\iff\alpha_{ij}=0,\forall
j\in\mathbb{N}_{i-1}.$

Therefore, if there exists a subsystem indexing scheme $X$ that makes
$\mathcal{E}_{i}\supseteq\mathbb{N}_{i-1},\forall i\in\mathbb{N}_{N}$, it
implies that $\alpha_{ij}=0,\forall j\in\mathbb{N}_{i-1},\forall
i\in\mathbb{N}_{N}$. Applying this in (77) we get the communication cost
$J(X)$ as $J(X)=0$. ∎

Figure 2: A network configuration where the proposed decentralized process in
Alg. 1 can be executed without communications between subsystems that are not
neighbors, i.e., in a distributed manner.

##### Communication Cost Optimization

Now, let us re-state the communication cost function $J(X)$ (77) as

$J(X)=\sum_{i\in\mathbb{N}_{N}}\sum_{j\in\mathbb{N}_{i-1}}\bar{\alpha}_{ij},$
(78)

where we define

$\bar{\alpha}_{ij}\triangleq\alpha_{ij}(\beta_{ij}\mathbf{1}_{\\{i\in\mathcal{E}_{j}\\}}+\gamma_{ijj}+\sum_{k=\mathcal{L}_{ij}}^{j-1}\gamma_{ijk}),$
(79)

and $G\triangleq[\bar{\alpha}_{ij}]_{i,j\in\mathbb{N}_{N}}$, and make the
assumption given below.

######  Assumption 1.

Each $\bar{\alpha}_{ij}$ value (79) required in (78) is only dependent on the
nature of the interconnection between subsystems $i$ and $j$, but not on
specific $i$ and $j$ values.

For example, As. 1 holds if
$\bar{\alpha}_{ij}=\mathbf{1}_{\\{j\not\in\mathcal{E}_{i}\\}}$ but not if
$\bar{\alpha}_{ij}=j\mathbf{1}_{\\{j\in\mathcal{E}_{i}\\}}$. Pertaining to the
definition given in (79), note that As. 1 holds if $\gamma_{ijk}=0,\forall k$
but not if $\gamma_{ijk}=\gamma_{ijj},\forall k$.

Under As. 1, it is easy to see that the problem of finding the optimal
subsystem indexing scheme $X$ that minimizes $J(X)$ (78) is identical to the
problem of finding the optimal permutation matrix $P\in\mathbb{R}^{N\times N}$
that minimizes the sum of all upper triangular elements of $PGP^{\top}$
(recall that $G=[\bar{\alpha}_{ij}]_{i,j\in\mathbb{N}_{N}}$ is a known
matrix). Conveniently, the latter problem belongs to a well-known class of
combinatorial optimization problems called “linear ordering problems” [45, 46,
47, 48, 49].

While linear ordering problems are NP-hard [46], when $N$ is small, exact
solutions can be obtained using brute-force or branch-and-bound algorithms
[48]. Even if $N$ is large, local optimal solutions can be obtained using
efficient and systematic heuristic algorithms [45, 46] that start with a
greedy solution and execute local optimizations until a convergence is
achieved. In general, linear ordering problems can be formulated as standard
linear integer programs [47], and therefore, can also be solved using
commercial solvers.

Finally, to make a remark on situations where As. 1 does not hold, consider
the simple case where $\gamma_{ijk}=\gamma_{ijj},\forall k$ with
$\mathcal{L}_{ij}=1$ in (79). This transforms (79) into the form:

$\bar{\alpha}_{ij}=\underbrace{\bar{\alpha}_{ij}\beta_{ij}\mathbf{1}_{\\{i\in\mathcal{E}_{j}\\}}}_{\triangleq\,\bar{\alpha}_{ij}^{(1)}}+j\underbrace{\bar{\alpha}_{ij}\gamma_{ijj}}_{\triangleq\,\bar{\alpha}_{ij}^{(2)}}=\bar{\alpha}_{ij}^{(1)}+j\bar{\alpha}_{ij}^{(2)}.$
(80)

Consequently, the communication cost function $J(X)$ in (78) is non-linear,
and thus, the corresponding “ordering problem” is also non-linear. To the best
of the authors’ knowledge, no existing literature directly studies such non-
linear ordering problems. However, even for such non-linear ordering problems,
we still can use a brute force or a heuristic algorithm (like before) that
starts with a greedy solution and executes local optimizations to obtain a
locally optimal solution.

Figure 3 shows two example network configurations with the overall inter-
subsystem communication cost function $J(X)$ (77) defined as:
$\alpha_{ij}=\mathbf{1}_{j\not\in\mathcal{E}_{i}}$, $\beta_{ij}=2n_{j}^{2}$,
$\gamma_{ijj}=n_{j}^{2}$, $\gamma_{ijk}=n_{j}n_{k}$ and $n_{i}=2,\forall
i\in\mathbb{N}_{N}$. In there, apart from a random nominal subsystem indexing
scheme, both the worst and the best possible subsystem indexing schemes
obtained by optimizing $J(X)$ (via a brute force algorithm) are indicated
respectively using black, red and blue colored texts. Note that, in these two
examples, optimizing the subsystem indexing scheme has respectively lead to
$75.0\%$ and $59.0\%$ savings in the communication cost function $J(X)$ (77).

(a) Ex. 1: $J(X)={\color[rgb]{1,0,0}220},192,{\color[rgb]{0,0,1}48}$.

(b) Ex. 2: $J(X)={\color[rgb]{1,0,0}220},156,{\color[rgb]{0,0,1}64}$.

Figure 3: Two example network configurations and their communication cost
values under different subsystem indexing schemes: (1) worst (red), (2)
nominal (black), (3) best (blue).

###  IV-E Enforcing Matrix Equatlities

In this section, so far, we have focused on establishing a decentralized
approach for testing/enforcing matrix inequalities of the form $W>0$, i.e.,
Alg. 1. In the same spirit, we conclude this section by providing a simple
decentralized approach for enforcing matrix equalities of the form $V=0$,
i.e., Alg. 2.

Algorithm 2 Enforcing $V=0$ in a Network Setting.

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

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

3: if $i=1$ then

4: Enforce: $V_{11}=0$

5: else

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

7: Receive: Required info. to compute $V_{ij},V_{ji}$

8: End receiving

9: Enforce: $V_{ij}=0,V_{ji}=0,\forall j\in\mathbb{N}_{i-1},V_{ii}=0$

10: end if

11:End execution

Consider the case where $V=0$ is a linear matrix equality (LME) in terms of a
single controllable matrix variable $X$:

$V\equiv AXB+C=0,$ (81)

with matrices $A,B,C$ and $X$ being network matrices corresponding to some
network $\mathcal{G}_{N}$. For this particular case, the following lemma
provides conditions that ensure the applicability of Alg. 2 to enforce the LME
(81).

######  Lemma 9.

A unique solution for the linear matrix equality $V=0$ (81) can be found via
Alg. 2 iff $A,B$ are block diagonal with non-singular blocks and: (1) $X$ is a
strictly non-block diagonal, or (2) $C,X$ are both block diagonal.

###### Proof.

The proof is complete by considering the $(i,j)$th element of both sides in
(81), $i,j\in\mathbb{N}_{N}$, which respectively gives unique solutions for
$X$ as: (1)

$V_{ij}\equiv A_{ii}X_{ij}B_{jj}+C_{ij}=0\iff
X_{ij}=-A_{ii}^{-1}C_{ij}B_{jj}^{-1},$

or (2) (recall $e_{ij}\triangleq\mathbf{I}\cdot\mathbf{1}_{\\{i=j\\}}$)

$V_{ij}\equiv A_{ii}X_{ij}B_{jj}+C_{ii}e_{ij}=0\iff
X_{ij}=-A_{ii}^{-1}C_{ii}e_{ij}B_{jj}^{-1}.$

∎

We conclude this section by pointing out that enforcing LME conditions of the
form (81) via Alg. 2 is not only decentralized and compositional but also
distributed.

##  V Distributed Analysis and Control Synthesis of CTNS

In this section, we design decentralized, compositional and possibly
distributed techniques for different analysis and control synthesis tasks
discussed in Sec. II over CTNSs introduced in Sec. III using the algorithms
proposed in Sec. IV.

TABLE II: Summary of Results for Decentralized Analysis and Control Synthesis
Techniques for Continuous-Time Networked Systems (CTNS)

Task | Concept |  Global LMI Prop. # | Preliminary Conditions | Assumptions on Global CTNS Parameters and Global LMI Variables | Local LMI variables, Local LMIs and Local LMEs (at subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$)
---|---|---|---|---|---
Network Matrices |  Scaler LMI Var./Obj. | LMI Variables |  Scaler LMI Var./Obj. |  LMIs: BEW Forms of (via Alg. 1): |  LMEs: BEW Form of (via Alg.2):
CTNS Parameters | LMI Variables
General | Diagonal | General | Diagonal
LTI System Analysis | Stability | 1 | $u=0,w=0$ | $A$ |  |  | $P$ |  | $P_{ii}$ |  | (2) |
Dissipativity | 2 | $Q<0,R=R^{\top},w=0,u\rightarrow y$ | $A,B,S,R$ | $C,D,Q$ |  | $P$ |  | $P_{ii}$ |  | (3) |
$\mathcal{H}_{2}$-Norm | 3 | ${\color[rgb]{1,0,0}D=0},w=0,u\rightarrow y$ | $A,B,C$ |  | $Q$ | $P$ | $\gamma$ | $Q_{i},P_{ii}$ | $\gamma_{i}$ | (7) or (8) |
$H_{\infty}$-Norm | 4 | $w=0,u\rightarrow y$ | $A,B,C,D$ |  |  | $P$ | $\gamma$ | $P_{ii}$ | $\gamma_{i}$ | (11) |
Stabilizability | 5 |  | $A$ | $B$ |  | $P$ |  | $P_{ii}$ |  | (13) |
Detectability | 6 |  | $A$ | $C$ |  | $P$ |  | $P_{ii}$ |  | (15) |
FSF Controller Synthesis | Stability | 7 | $D=0,w=0$ | $A$ | $B$ | $L$ | $M$ |  | $L_{i},M_{ii}$ |  | (18) | $K=LM^{-1}$
Dissipativity | 8 | $D=0,Q<0,R=R^{\top},w\rightarrow y$ | $A,E,S,R$ | $B,C,F,Q$ | $L$ | $M$ |  | $L_{i},M_{ii}$ |  | (19) | $K=LM^{-1}$
$\mathcal{H}_{2}$-Norm | 9 | ${\color[rgb]{1,0,0}F=0},w\rightarrow y$ | $A,C,E$ | $B,D$ | $L,Q$ | $M$ | $\gamma$ | $L_{i},Q_{i},M_{ii}$ | $\gamma_{i}$ | (20) | $K=LM^{-1}$
$H_{\infty}$-Norm | 10 | $w\rightarrow y$ | $A,C,E,F$ | $B,D$ | $L$ | $M$ | $\gamma$ | $L_{i},M_{ii}$ | $\gamma_{i}$ | (21) | $K=LM^{-1}$
Observer Design | Stability | 11 | $w=0$ | $A$ | $C,D$ | $K$ | $P$ |  | $K_{i},P_{ii}$ |  | (24) |  $L=P^{-1}K$, (26)
Dissipativity | 12 | $Q<0,R=R^{\top},w\rightarrow z$ | $A,E,S,R$ | $C,D,F,G,J,Q$ | $K$ | $P$ |  | $K_{i},P_{ii}$ |  | (28) |  $L=P^{-1}K$, (26)
$\mathcal{H}_{2}$-Norm | 13 | ${\color[rgb]{1,0,0}J=0},w\rightarrow z$ | $A,E,G$ | $C,D,F$ | $K,Q$ | $P$ | $\gamma$ | $K_{i},Q_{i},P_{ii}$ | $\gamma_{i}$ | (29) |  $L=P^{-1}K$, (26)
$H_{\infty}$-Norm | 14 | $w\rightarrow z$ | $A,E,G,J$ | $C,D,F$ | $K$ | $P$ | $\gamma$ | $K_{i},P_{ii}$ | $\gamma_{i}$ | (30) |  $L=P^{-1}K$, (26)
DOF Controller Synthesis | Stability | 15 | $D=0,w=0$ | $A$ | $B,C$ | $A_{n},B_{n},C_{n},D_{n}$ | $X,Y$ |  | $A_{n,i},B_{n,i},C_{n,i},D_{n,i},X_{ii},Y_{ii}$ |  | (41),(42) | (38)
Dissipativity | 16 | $D=0,Q<0,R=R^{\top},w\rightarrow z$ | $A,E,G,J,R$ | $B,C,F,H,S,Q$ | $A_{n},B_{n},C_{n},D_{n}$ | $X,Y$ |  | $A_{n,i},B_{n,i},C_{n,i},D_{n,i},X_{ii},Y_{ii}$ |  | (47),(44) | (38)
$\mathcal{H}_{2}$-Norm | 17 | $D=0,w\rightarrow z$ | $A,E,G,J$ | $B,C,F,H$ | $A_{n},B_{n},C_{n},D_{n},Q$ | $X,Y$ | $\gamma$ | $A_{n,i},B_{n,i},C_{n,i},D_{n,i},Q_{i}X_{ii},Y_{ii}$ | $\gamma_{i}$ | (50), (45) |  $J+HD_{n}F=0$, (38)
$H_{\infty}$-Norm | 18 | $D=0,w\rightarrow z$ | $A,E,G,J$ | $B,C,F,H$ | $A_{n},B_{n},C_{n},D_{n}$ | $X,Y$ | $\gamma$ | $A_{n,i},B_{n,i},C_{n,i},D_{n,i},X_{ii},Y_{ii}$ | $\gamma_{i}$ | (51), (46) | (38)

TABLE III: Summary of Results for Decentralized Analysis and Control Synthesis
Techniques for Discrete-Time Networked Systems (DTNS)

Task | Concept |  Global LMI Prop. # | Preliminary Conditions | Assumptions on Global DTNS Parameters and Global LMI Variables | Local LMI variables, Local LMIs and Local LMEs (at subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$)
---|---|---|---|---|---
Network Matrices |  Scaler LMI Var./Obj. | LMI Variables |  Scaler LMI Var./Obj. |  LMIs: BEW Forms of (via Alg. 1): |  LMEs: BEW Form of (via Alg.2):
DTNS Parameters | LMI Variables
General | Diagonal | General | Diagonal
LTI System Analysis | Stability | 19 | $u=0,w=0$ | $A$ |  |  | $P$ |  | $P_{ii}$ |  | (104) |
Dissipativity | 20 | $Q<0,R=R^{\top},w=0,u\rightarrow y$ | $A,B,S,R$ | $C,D,Q$ |  | $P$ |  | $P_{ii}$ |  | (106) |
$\mathcal{H}_{2}$-Norm | 21 | $w=0,u\rightarrow y$ | $A,B,C,{\color[rgb]{1,0,0}D}$ |  | $Q$ | $P$ | $\gamma$ | $Q_{i},P_{ii}$ | $\gamma_{i}$ | (108) or (109) |
$H_{\infty}$-Norm | 22 | $w=0,u\rightarrow y$ | $A,B,C,D$ |  |  | $P$ | $\gamma$ | $P_{ii}$ | $\gamma_{i}$ | (111) or (111) |
Stabilizability | 23 |  | $A$ | $B$ |  | $P$ |  | $P_{ii}$ |  | (114) |
Detectability | 24 |  | $A$ | $C$ |  | $P$ |  | $P_{ii}$ |  | (116) |
FSF Controller Synthesis | Stability | 25 | $D=0,w=0$ | $A$ | $B$ | $L$ | $M$ |  | $L_{i},M_{ii}$ |  | (119) | $K=LM^{-1}$
Dissipativity | 26 | $D=0,Q<0,R=R^{\top},w\rightarrow y$ | $A,E,S,R$ | $B,C,F,Q$ | $L$ | $M$ |  | $L_{i},M_{ii}$ |  | (120) | $K=LM^{-1}$
$\mathcal{H}_{2}$-Norm | 27 | $w\rightarrow y$ | $A,C,E,{\color[rgb]{1,0,0}F}$ | $B,D$ | $L,Q$ | $M$ | $\gamma$ | $L_{i},Q_{i},M_{ii}$ | $\gamma_{i}$ | (121) | $K=LM^{-1}$
$H_{\infty}$-Norm | 28 | $w\rightarrow y$ | $A,C,E,F$ | $B,D$ | $L$ | $M$ | $\gamma$ | $L_{i},M_{ii}$ | $\gamma_{i}$ | (122) | $K=LM^{-1}$
Observer Design | Stability | 29 | $w=0$ | $A$ | $C,D$ | $K$ | $P$ |  | $K_{i},P_{ii}$ |  | (125) |  $L=P^{-1}K$, (26)
Dissipativity | 30 | $Q<0,R=R^{\top},w\rightarrow z$ | $A,E,S,R$ | $C,D,F,G,J,Q$ | $K$ | $P$ |  | $K_{i},P_{ii}$ |  | (128) |  $L=P^{-1}K$, (26)
$\mathcal{H}_{2}$-Norm | 31 | $w\rightarrow z$ | $A,E,G,{\color[rgb]{1,0,0}J}$ | $C,D,F$ | $K,Q$ | $P$ | $\gamma$ | $K_{i},Q_{i},P_{ii}$ | $\gamma_{i}$ | (129) |  $L=P^{-1}K$, (26)
$H_{\infty}$-Norm | 32 | $w\rightarrow z$ | $A,E,G,J$ | $C,D,F$ | $K$ | $P$ | $\gamma$ | $K_{i},P_{ii}$ | $\gamma_{i}$ | (130) |  $L=P^{-1}K$, (26)
DOF Controller Synthesis | Stability | 33 | $D=0,w=0$ | $A$ | $B,C$ | $A_{n},B_{n},C_{n},D_{n}$ | $X,Y$ |  | $A_{n,i},B_{n,i},C_{n,i},D_{n,i},X_{ii},Y_{ii}$ |  | (134),(135) | (38)
Dissipativity | 34 | $D=0,Q<0,R=R^{\top},w\rightarrow z$ | $A,E,G,J,R$ | $B,C,F,H,S,Q$ | $A_{n},B_{n},C_{n},D_{n}$ | $X,Y$ |  | $A_{n,i},B_{n,i},C_{n,i},D_{n,i},X_{ii},Y_{ii}$ |  | (137),(142) | (38)
$\mathcal{H}_{2}$-Norm | 35 | $D=0,w\rightarrow z$ | $A,E,G,J$ | $B,C,F,H$ | $A_{n},B_{n},C_{n},D_{n},Q$ | $X,Y$ | $\gamma$ | $A_{n,i},B_{n,i},C_{n,i},D_{n,i},Q_{i}X_{ii},Y_{ii}$ | $\gamma_{i}$ | (140), (143) | (38)
$H_{\infty}$-Norm | 36 | $D=0,w\rightarrow z$ | $A,E,G,J$ | $B,C,F,H$ | $A_{n},B_{n},C_{n},D_{n}$ | $X,Y$ | $\gamma$ | $A_{n,i},B_{n,i},C_{n,i},D_{n,i},X_{ii},Y_{ii}$ | $\gamma_{i}$ | (141), (144) | (38)

##### Notation

Double and single subscripts used in local LMI variables (Column 10) denote
variable matrices as follows: (1) Double subscripts, e.g., $P_{ii}$ is the
$i$th diagonal block of $P\triangleq\text{diag}(P_{ii}:i\in\mathbb{N}_{N})$;
and (2) Single subscript, e.g., $Q_{i}$ is the collection of blocks
$L_{i}\triangleq\\{L_{ii}\\}\cup\\{L_{ij}:j\in\mathbb{N}_{i-1}\\}\cup\\{L_{ji}:j\in\mathbb{N}_{i-1}\\}$
taken from $L\triangleq[L_{ij}]_{i,j\in\mathbb{N}_{N}}$. The acronym BEW
(Column 12-13) stands for “Block Element-Wise” (see Lms. 3 and 6).

###  V-A Summary of Results (Table II)

Recall that, in Sec. II, we formulated different analysis and control
synthesis tasks of interest for CT-LTI systems (1) as LMI problems in Props.
1-18. As pointed out in Sec. III, each of these LMI problems are directly
applicable to globally analyze and synthesize controllers for CTNSs (59).
Moreover, we showed in Section IV that, under certain conditions, such global
LMI (and LME) conditions that arise in network settings can be enforced in a
decentralized, compositional and possibly distributed manner using Alg. 1 (and
2) based on the established Lm. 7 (and 9). For example, to decentrally
analyze/enforce the global LMI condition $W>0$ via Alg. 1, $W$ requires to be
a network matrix corresponding to the considered CTNS (see Def. 2 and Lm. 3).
In all, under certain conditions, different analysis and control synthesis
tasks of interest for CTNSs (59) can be executed in a decentralized manner via
solving specifically formulated local versions of the LMI problems in Props.
1-18 using Algs. 1 and 2.

In the interest of brevity, rather than explicitly stating all such local
(decentralized) versions of the LMI problems in Props. 1-18, we have
summarized their key details in Tab. II. In particular, Tab. II-Col. 4
provides the preliminary conditions required of the CTNS to apply the
interested proposition (given in Col. 2) while Cols. 5-9 provide the
assumptions required regarding various CTNS parameters and LMI variables to
decentralize the interested proposition. Note that, while all the CTNS
parameters and LMI variables are, by definition, network matrices (except for
the scalar LMI variables given in Tab. II-Col. 9), some of them may need to be
block diagonal network matrices in order to decentralize the LMI problem in
the interested proposition.

For example, consider the stability analysis (i.e., Prop. 1 and Tab. II-Row 1)
which depends on finding a matrix $P>0$ such that $W\equiv-A^{\top}P-PA>0$. In
this case, to decentralize the enforcement of the LMI $W>0$, $W$ needs to be a
network matrix. As $A$, by definition, is a general (non-block diagonal)
network matrix, based on Lm. 3, for $W$ to be a network matrix, $P$ needs to
be a block diagonal network matrix.

Table II Cols. 10-13 summarize the details of the decentralized LMIs that
needs to be solved locally. In particular, the variables involved in the local
LMI problem solved at the subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ are given
in Tab. II Cols. 10-11. According to the used notation (see below Tabs. II-
III), note that, such a local LMI problem may involve finding: (1) the $i$-th
diagonal block of a block diagonal network matrix (denoted using double
subscripts, e.g., $P_{ii}$), (2) a certain collection of blocks (matrices) of
a general network matrix (denoted using single subscripts, e.g.,
$L_{i}\triangleq\\{L_{ii}\\}\cup\\{L_{ij}:j\in\mathbb{N}_{i-1}\\}\cup\\{L_{ji}:j\in\mathbb{N}_{i-1}\\}$).
On the other hand, the local LMIs and LMEs that need to be enforced at
subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ are given in Tab. II Cols. 12-13.
Note that, such local LMIs and LMEs are basically the BEW forms of global LMIs
and LMEs (given in Props. 1-18) enforced through decentralized Algs. 1 and 2.

For example, consider the FSF stabilization (i.e., Prop. 7 and Tab. II-Row 7)
which depends on finding a matrix $M>0$ and $L$ such that $W\equiv-
MA^{\top}-AM-L^{\top}B^{\top}-BL>0$ (where FSF controller gain $K=LM^{-1}$).
According to Lm. 3, when $A,L$ are considered as general network matrices, we
need to constrain $M,B$ to be block diagonal network matrices so as to ensure
$W$ is a network matrix (to enable decentralization). Upon decentralization,
the local LMI variables at subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ are
$M_{ii}$ and
$L_{i}\triangleq\\{L_{ii}\\}\cup\\{L_{ij}:j\in\mathbb{N}_{i-1}\\}\cup\\{L_{ji}:j\in\mathbb{N}_{i-1}\\}$,
which respectively corresponds to the global LMI variables $M$ and $L$. The
local LMI that needs to be enforced is found when enforcing

$W\equiv[-M_{ii}A_{ji}^{\top}-A_{ij}M_{jj}-L_{ji}^{\top}B_{jj}^{\top}-B_{ii}L_{ij}]_{i,j\in\mathbb{N}_{N}}>0$

via Alg. 1 (i.e., its Step 14: $\tilde{W}_{ii}>0$). Similarly, the local LME
that needs to be enforced is found when enforcing

$V\equiv K-LM^{-1}=[K_{ij}-L_{ij}M_{jj}^{-1}]_{i,j\in\mathbb{N}_{N}}=0$

via Alg. 2 (i.e., its Step 9: $K_{ij}=L_{ij}M_{jj}^{-1}$,
$K_{ji}=L_{ji}M_{ii}^{-1},\forall j\in\mathbb{N}_{i-1}$,
$K_{ii}=L_{ii}M_{ii}^{-1}$). Finally, $K_{ij},K_{ji}$ and $K_{ii}$ matrices
determined above (collectively denoted as $K_{i}$) give the necessary local
FSF controller gains (53).

###  V-B Stability Related Decentralized Results

Formally, the aforementioned decentralized stability analysis and FSF
stabilization techniques can be summarized respectively as in the following
two theorems (correspond to Props. 1 and 7, and Tab. II-Rows 1 and 7).

######  Theorem 1.

(Stability analysis) The CTNS (59) under $u(t)=\mathbf{0}$ and
$w(t)=\mathbf{0}$ is stable if at each subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\mathbb{P}_{1}:\ \ \text{Find}\ \ P_{ii}\ \ \text{such that}\ \ P_{ii}>0,\ \
\tilde{W}_{ii}>0,$ (82)

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

$W_{ij}=-A_{ji}^{\top}P_{jj}-P_{ii}A_{ij}.$ (83)

###### Proof.

Let us define $P\triangleq\text{diag}(P_{ii}:i\in\mathbb{N}_{N})$ and
$W\triangleq-A^{\top}P-PA$. According to Prop. 1, we need to find $P>0$ such
that $W>0$ to establish the global stability. Under the above definitions, it
easy to see that $W_{ij}=-A_{ji}^{\top}P_{jj}-P_{ii}A_{ij}$, i.e., (83). Note
also that $W$ is a symmetric network matrix (see Def. 2). Therefore, $W>0$ can
be tested in a decentralized manner by applying Alg. 1 and testing
$\tilde{W}_{ii}>0$ at each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$. Note
also that $P_{ii}>0,\forall i\in\mathbb{N}_{N}\implies P>0$. Therefore, the
existence of a matrix $P>0$ such that $W>0$ can be evaluated in a
decentralized manner by solving $\mathbb{P}_{1}$ (82) at each subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$. ∎

######  Theorem 2.

(FSF Stabilization) The CTNS (59) (where $B$ is block diagonal) under
$D=\mathbf{0}$, $w(t)=\mathbf{0}$ and local FSF control (53) is stable if at
each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\mathbb{P}_{2}:\ \ \text{Find}\ \ M_{ii},\ \ L_{i}\ \ \text{such that}\ \
M_{ii}>0,\ \ \tilde{W}_{ii}>0,$ (84)

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

$W_{ij}=-M_{ii}A_{ji}^{\top}-A_{ij}M_{jj}-L_{ji}^{\top}B_{jj}^{\top}-B_{ii}L_{ij}.$
(85)

The local FSF controller gains $K_{i}$ are computed using $M_{ii}$ and
$L_{i}$, from Alg. 2 (Steps 3-10) when enforcing
$V=[V_{ij}]_{i,j\in\mathbb{N}_{N}}=0$ with

$V_{ij}=K_{ij}-L_{ij}M_{jj}^{-1}.$ (86)

###### 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 $V\triangleq K-LM^{-1}$.
According to Prop. 7, we need to find $M>0$ and $L$ such that $W>0$ to ensure
the closed loop stability, and then, the controller gains can be found using
$K=LM^{-1}$. Under the above definitions, it easy to see that
$W_{ij}=-M_{ii}A_{ji}^{\top}-A_{ij}M_{jj}-L_{ji}^{\top}B_{jj}^{\top}-B_{ii}L_{ij}$,
i.e., (85) and $V_{ij}=K_{ij}-L_{ij}M_{jj}^{-1}$, i.e., (86). Note also that
$W$ is a symmetric network matrix (see Def. 2) while $V$ is of the form (81)
and satisfies the conditions stated in Lm. 9. Consequently, $W>0$ can be
enforced in a decentralized manner by applying Alg. 1 and enforcing
$\tilde{W}_{ii}>0$ at each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$. Since
$M_{ii}>0,\forall i\in\mathbb{N}_{N}\implies M>0$, the existence of a matrix
$M>0$ and $L$ such that $W>0$ can be evaluated in a decentralized manner by
solving $\mathbb{P}_{2}$ (84). Subsequently, $V=0$ can be enforced in a
decentralized manner by applying Alg. 1 to determine the local FSF controller
gains $K_{i}$ at each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$. ∎

Note that, the problems $\mathbb{P}_{1}$ (82) and $\mathbb{P}_{2}$ (84) are
LMI problems due to the applicability of Lm. 1 to simplify the matrix
inequality $\tilde{W}_{ii}>0$ in Alg. 1 (Step 14). Therefore, such problems
can be solved conveniently and efficiently using readily available LMI
software toolboxes [35].

To conclude the discussion on stability-based decentralized results, in what
follows, we provide two more theorems, respectively, regarding decentralized
stable observer design and DOF stabilization techniques (correspond to Props.
11 and 15, and Tab. II-Rows 11 and 15).

######  Theorem 3.

(Stable observer design) For the CTNS (59) (where $C,D$ are block diagonal)
under $w(t)=\mathbf{0}$, the local Luenberger observers (54) render the state
estimation error dynamics (22) stable if at each subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\mathbb{P}_{3}:\ \ \text{Find}\ \ P_{ii},\ \ K_{i}\ \ \text{such that}\ \
P_{ii}>0,\ \ \tilde{W}_{ii}>0,$ (87)

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

$W_{ij}=-A_{ji}^{\top}P_{jj}-P_{ii}A_{ij}+C_{ii}^{\top}K_{ji}^{\top}+K_{ij}C_{jj}.$
(88)

The local Luenberger observer parameters $L_{i},\hat{A}_{i}$ and $\hat{B}_{i}$
(54) are computed using $P_{ii}$ and $K_{i}$, from Alg. 2 (Steps 3-10) when
enforcing $V^{(k)}=[V_{ij}^{(k)}]_{i,j\in\mathbb{N}_{N}}=0$, for $k=1,2,3,$
where

$\displaystyle V_{ij}^{(1)}=$ $\displaystyle\ L_{ij}-P_{ii}^{-1}K_{ij},$ (89)
$\displaystyle V_{ij}^{(2)}=$ $\displaystyle\
\hat{A}_{ij}-A_{ij}+L_{ij}C_{jj},$ $\displaystyle V_{ij}^{(3)}=$
$\displaystyle\ \hat{B}_{ij}-B_{ij}+L_{ij}D_{jj}.$

###### Proof.

The proof starts by defining
$P\triangleq\text{diag}(P_{ii}:i\in\mathbb{N}_{N})$,
$W\triangleq-A^{\top}P-PA+C^{\top}K+KC$, $V^{(1)}\triangleq L-P^{-1}K$,
$V^{(2)}\triangleq\hat{A}-A+LC$, $V^{(3)}\triangleq\hat{B}-B+LD$ and proceeds
using Prop. 11 in a similar manner to the proof of Th. 2. Therefore, here we
omit providing explicit details. ∎

######  Theorem 4.

(DOF Stabilization) The CTNS (59) (where $B,C$ are block diagonal) under
$D=\mathbf{0}$, $w(t)=\mathbf{0}$ and local DOF control (55) is stable if at
each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\displaystyle\mathbb{P}_{4}:\ \ \ \ \ \ \text{Find}$ $\displaystyle\ \
X_{ii},\ Y_{ii},\ A_{n,i},\ B_{n,i},\ C_{n,i},\ D_{n,i}$ (90) such that
$\displaystyle\ \ X_{ii}>0,\ Y_{ii}>0,\ \tilde{W}_{ii}^{(1)}>0,\
\tilde{W}_{ii}^{(2)}>0,$

is feasible, where $\tilde{W}_{ii}^{(k)}$ is computed from Alg. 1 (Steps:
3-16) when enforcing $W^{(k)}=[W_{ij}^{(k)}]_{i,j\in\mathbb{N}_{N}}>0$, for
$k=1,2,$ with

$\displaystyle W_{ij}^{(1)}=\begin{bmatrix}Y_{ii}e_{ij}&e_{ij}\\\
e_{ij}&X_{ij}e_{ij}\end{bmatrix},\ \ \ \ W_{ij}^{(2)}=$ (91)
$\displaystyle\begin{bmatrix}-\mathcal{H}_{s}(A_{ij}Y_{jj}+B_{ii}C_{n,ij})&-A_{ij}-B_{ii}D_{n,ij}C_{jj}-A_{n,ji}^{\top}\\\
\star&-\mathcal{H}_{s}(X_{ii}A_{ij}+B_{n,ij}C_{jj})\end{bmatrix}.$

The local DOF controller parameters $A_{c,i},B_{c,i},C_{c,i}$ and $D_{c,i}$
are computed using $X_{ii},Y_{ii},A_{n,i},B_{n,i},C_{n,i}$ and $D_{n,i}$, from
Alg. 2 (Steps 3-10) when enforcing
$V^{(k)}=[V_{ij}^{(k)}]_{i,j\in\mathbb{N}_{N}}=0$, for $k=1,2,3,4$ with

$\displaystyle V_{ij}^{(1)}=$ $\displaystyle\ D_{c,ij}-D_{n,ij},$ (92)
$\displaystyle V_{ij}^{(2)}=$ $\displaystyle\
C_{c,ij}-(C_{n,ij}-D_{n,ij}C_{jj}Y_{jj})N^{-\top}_{jj},$ $\displaystyle
V_{ij}^{(3)}=$ $\displaystyle\
B_{c,ij}-M^{-1}_{ii}(B_{n,ij}-X_{ii}B_{ii}D_{n,ij}),$ $\displaystyle
V_{ij}^{(4)}=$ $\displaystyle\
A_{c,ij}-M^{-1}_{ii}(A_{n,ij}-B_{n,ij}C_{jj}Y_{jj}-X_{ii}B_{ii}C_{n,ij}$
$\displaystyle\ -X_{ii}(A_{ij}-B_{ii}D_{n,ij}C_{jj})Y_{jj})N^{-\top}_{jj},$

where $M_{ii}$ and $N_{ii},i\in\mathbb{N}_{N}$ are any two matrices that
satisfy

$X_{ii}Y_{ii}+M_{ii}N_{ii}^{\top}=\mathbf{I}.$ (93)

###### Proof.

The proof starts by defining (inspired by (41), (42), (38) and (35)):

$\displaystyle X\triangleq\text{diag}(X_{ii}:i\in\mathbb{N}_{N}),\ \
Y\triangleq\text{diag}(Y_{ii}:i\in\mathbb{N}_{N}),$ $\displaystyle
W^{(1)}\triangleq\text{BEW}\left(\begin{bmatrix}Y&\mathbf{I}\\\
\mathbf{I}&X\end{bmatrix}\right),\ \ W^{(2)}\triangleq$
$\displaystyle\text{BEW}\left(\begin{bmatrix}-\mathcal{H}_{s}(AY+BC_{n})&-A-BD_{n}C-A_{n}^{\top}\\\
\star&-\mathcal{H}_{s}(XA+B_{n}C)\end{bmatrix}\right),$ $\displaystyle
V^{(1)}\triangleq D_{c}-D_{n},\ \ V^{(2)}\triangleq
C_{c}-(C_{n}-D_{n}CY)N^{-\top},$ $\displaystyle V^{(3)}\triangleq
B_{c}-M^{-1}(B_{n}-XBD_{n}),\ \ V^{(4)}\triangleq$ $\displaystyle
A_{c}-M^{-1}(A_{n}-B_{n}CY-XBC_{n}-X(A-BD_{n}C)Y)N^{-\top},$ $\displaystyle
M\triangleq\text{diag}(M_{ii}:i\in\mathbb{N}_{N}),\ \
N\triangleq\text{diag}(Y_{ii}:i\in\mathbb{N}_{N}).$

and proceeds using Prop. 15 in a similar manner to the proof of Th. 2 (except
for the use of Lm. 6: $W=\text{BEW}(\Psi)>0\iff\Psi>0$). Hence, here we omit
providing explicit details. ∎

$W_{ij}=\begin{bmatrix}-\mathcal{H}_{s}(A_{ij}M_{jj}+B_{ii}L_{ij})&-E_{ij}+M_{ii}C_{ii}^{\top}S_{ij}&M_{ii}C^{\top}_{ii}e_{ij}\\\
\star&\mathcal{H}_{e}(F_{ii}^{\top}S_{ij})+R_{ij}&F_{ii}^{\top}e_{ij}\\\
\star&\star&-Q_{ii}^{-1}e_{ij}\end{bmatrix}$ (94)

$W_{ij}=\begin{bmatrix}-\mathcal{H}_{s}(P_{ii}A_{ij}-K_{ij}C_{jj})&-P_{ii}E_{ij}+K_{ij}F_{jj}+G_{ii}^{\top}S_{ij}&G_{ii}^{\top}e_{ij}\\\
\star&\mathcal{H}_{s}(J_{ii}^{\top}S_{ij})+R_{ij}&J_{ii}^{\top}e_{ij}\\\
\star&\star&-Q_{ii}^{-1}e_{ij}\end{bmatrix}$ (95)

$\scriptsize
W_{ij}^{(2)}=\begin{bmatrix}-\mathcal{H}_{s}(A_{ij}Y_{jj}+B_{ii}C_{n,ij})&-A_{ij}-B_{ii}D_{n,ij}C_{jj}-A_{n,ji}^{\top}&-E_{ij}-B_{ii}D_{n,ij}F_{jj}+(Y_{ii}G_{ji}^{\top}+C_{n,ji}^{\top}H_{jj}^{\top})S_{jj}&Y_{ii}G_{ji}^{\top}+C_{n,ji}^{\top}H_{jj}^{\top}\\\
\star&-\mathcal{H}_{s}(X_{ii}A_{ij}+B_{n,ij}C_{jj})&-X_{ii}E_{ij}-B_{n,ij}F_{jj}+(G_{ji}^{\top}+C_{ii}^{\top}D_{n,ji}^{\top}H_{jj}^{\top})S_{jj}&G_{ji}^{\top}+C_{ii}^{\top}D_{n,ji}^{\top}H_{jj}^{\top}\\\
\star&\star&\mathcal{H}_{s}((J_{ji}^{\top}+F_{ii}^{\top}D_{n,ji}^{\top}H_{jj}^{\top})S_{jj})+R_{ij}&J_{ji}^{\top}+F_{ii}^{\top}D_{n,ji}^{\top}H_{jj}^{\top}\\\
\star&\star&\star&-Q_{ii}^{-1}e_{ij}\end{bmatrix}$ (96)

$\scriptsize
W^{(2)}\triangleq\text{BEW}\Big{(}\begin{bmatrix}-\mathcal{H}_{s}(AY+BC_{n})&-A-BD_{n}C-A_{n}^{\top}&-E-BD_{n}F+(YG^{\top}+C_{n}^{\top}H^{\top})S&YG^{\top}+C_{n}^{\top}H^{\top}\\\
\star&-\mathcal{H}_{s}(XA+B_{n}C)&-XE-
B_{n}F+(G^{\top}+C^{\top}D_{n}^{\top}H^{\top})S&G^{\top}+C^{\top}D_{n}^{\top}H^{\top}\\\
\star&\star&\mathcal{H}_{s}((J^{\top}+F^{\top}D_{n}^{\top}H^{\top})S)+R&J^{\top}+F^{\top}D_{n}^{\top}H^{\top}\\\
\star&\star&\star&-Q^{-1}\end{bmatrix}\Big{)}$ (97)

###  V-C Dissipativity Related Decentralized Results

In this subsection, analogous to Theorems 1-4, we provide
$(Q,S,R)$-dissipativity based results on:

  1. 1.

decentralized $(Q,S,R)$-dissipativity analysis,

  2. 2.

decentralized FSF $(Q,S,R)$-dissipativation,

  3. 3.

decentralized $(Q,S,R)$-dissipative observer design and

  4. 4.

decentralized DOF $(Q,S,R)$-dissipativation,

that respectively correspond to Props. 2, 8, 12 and 16 (Tab. II-Rows 2, 8, 12
and 16). In what follows, regarding the given specification matrices $Q,S,R$,
it is assumed that: (1) $Q$ is a block diagonal network matrix, (2) $-Q>0$,
and (3) $R=R^{\top}$.

######  Theorem 5.

(Dissipativity analysis) The CTNS (59) (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 problem

$\mathbb{P}_{5}:\ \ \text{Find}\ \ P_{ii}\ \ \text{such that}\ \ P_{ii}>0,\ \
\tilde{W}_{ii}>0,$ (98)

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

$W_{ij}=\begin{bmatrix}-\mathcal{H}_{e}(P_{ii}A_{ij})&-P_{ii}B_{ij}+C_{ii}^{\top}S_{ij}&C_{ii}^{\top}e_{ij}\\\
\star&\mathcal{H}_{e}(D_{ii}^{\top}S_{ij})+R_{ij}&D_{ii}^{\top}e_{ij}\\\
\star&\star&-Q_{ii}^{-1}e_{ij}\end{bmatrix}.$ (99)

###### Proof.

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

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

(inspired by (3)), and proceeds using Prop. 2 in a similar manner to the proof
of Th. 2 (except for the use of Lm. 6: $W=\text{BEW}(\Psi)>0\iff\Psi>0$).
Therefore, here we omit providing explicit details. ∎

######  Theorem 6.

(FSF Dissipativation) The CTNS (59) (where $B,C,F$ are block diagonal) under
$D=\mathbf{0}$ and local FSF control (53) is $(Q,S,R)$-dissipative from $w(t)$
to $y(t)$ if at each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\mathbb{P}_{6}:\ \ \text{Find}\ \ M_{ii},\ \ L_{i}\ \ \text{such that}\ \
M_{ii}>0,\ \ \tilde{W}_{ii}>0,$ (100)

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 (94).
The local FSF controller gains $K_{i}$ are computed using $M_{ii}$ and $L_{i}$
identically to Th. 2 via (86).

###### Proof.

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

$W\triangleq\text{BEW}\big{(}\begin{bmatrix}-\mathcal{H}_{s}(AM+BL)&-E+MC^{\top}S&MC^{\top}\\\
\star&\mathcal{H}_{s}(F^{\top}S)+R&F^{\top}\\\
\star&\star&-Q^{-1}\end{bmatrix}\big{)}$

(inspired by (19)), and proceeds using Prop. 8 and Lm. 6 in a similar manner
to the proof of Th. 5. Therefore, here we omit providing explicit details. ∎

######  Theorem 7.

(Dissipative observer design) For the CTNS (59) (where $C,D,F$ are block
diagonal), the local Luenberger observers (54) with the local performance
metrics (57) (such that $G,J$ in (64) are block diagonal) render the global
state estimation error dynamics (27) $(Q,S,R)$-dissipative from $w(t)$ to
$z(t)$ if at each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\mathbb{P}_{7}:\ \ \text{Find}\ \ P_{ii},\ \ K_{i}\ \ \text{such that}\ \
P_{ii}>0,\ \ \tilde{W}_{ii}>0,$ (101)

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 (95).
The local Luenberger observer parameters $L_{i},\hat{A}_{i}$ and $\hat{B}_{i}$
(54) are computed using $P_{ii}$ and $K_{i}$ identically to Th. 3 via (89).

###### Proof.

The proof starts by defining
$P\triangleq\text{diag}(P_{ii}:i\in\mathbb{N}_{N})$,
$W\triangleq\text{BEW}(\Psi)$, where (inspired by (28)):

$\Psi\triangleq\begin{bmatrix}-\mathcal{H}_{s}(PA-
KC)&-PE+KF+G^{\top}S&G^{\top}\\\
\star&\mathcal{H}_{s}(J^{\top}S)+R&J^{\top}\\\
\star&\star&-Q^{-1}\end{bmatrix},$

and proceeds using Prop. 12 and Lm. 6 in a similar manner to the proof of Th.
5. Therefore, here we omit providing explicit details. ∎

######  Theorem 8.

(DOF Dissipativation) The CTNS (59) (where $B,C,F$ are block diagonal) under
$D=\mathbf{0}$, local DOF control (55) and local performance metrics (58)
(such that $H$ in (65) is block diagonal), i.e., (33), is
$(Q,S,R)$-dissipative (where $S$ is block diagonal) from $w(t)$ to $z(t)$ if
at each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\displaystyle\mathbb{P}_{8}:\ \ \ \ \ \ \text{Find}$ $\displaystyle\ \
X_{ii},\ Y_{ii},\ A_{n,i},\ B_{n,i},\ C_{n,i},\ D_{n,i}$ (102) such that
$\displaystyle\ \ X_{ii}>0,\ Y_{ii}>0,\ \tilde{W}_{ii}^{(1)}>0,\
\tilde{W}_{ii}^{(2)}>0,$

is feasible, where $\tilde{W}_{ii}^{(k)}$ is computed from Alg. 1 (Steps:
3-16) when enforcing $W^{(k)}=[W_{ij}^{(k)}]_{i,j\in\mathbb{N}_{N}}>0$ for
$k=1,2,$ with $\scriptsize W_{ij}^{(1)}=\begin{bmatrix}Y_{ii}e_{ij}&e_{ij}\\\
e_{ij}&X_{ij}e_{ij}\end{bmatrix}$ and $W_{ij}^{(2)}$ given in (96). The local
DOF controller parameters $A_{c,i},B_{c,i},C_{c,i}$ and $D_{c,i}$ are computed
using $X_{ii},Y_{ii},A_{n,i},B_{n,i},C_{n,i}$ and $D_{n,i}$, identically to
Th. 4 via (92) and (93).

###### Proof.

The proof starts by defining (inspired by (47) and (44)):
$X\triangleq\text{diag}(X_{ii}:i\in\mathbb{N}_{N})$,
$Y\triangleq\text{diag}(Y_{ii}:i\in\mathbb{N}_{N})$, $\scriptsize
W^{(1)}\triangleq\text{BEW}\left(\begin{bmatrix}Y&\mathbf{I}\\\
\mathbf{I}&X\end{bmatrix}\right)$, and $W^{(2)}$ as in (97), and proceeds
using Prop. 16 and Lm. 6 in a similar manner to the proof of Th. 4. Hence,
here we omit providing explicit details. ∎

To conclude this section, we re-emphasizing that local problems
$\mathbb{P}_{1}$-$\mathbb{P}_{8}$ stated respectively in the theorems 1-8 are
LMI problems due to the applicability of Lm. 1 to simplify the matrix
inequality $\tilde{W}_{ii}>0$ in Alg. 1 (Step 14). Consequently, such problems
can be solved conveniently and efficiently using readily available LMI
software toolboxes [35]. Moreover, based on the remaining propositions given
in Sec. II, a similar set of theorems can be proposed to provide respective
decentralized techniques (as summarized in the respective remaining rows in
Tab. II).

##  VI Preliminaries: Discrete-Time Linear Time Invariant (DT-LTI) Systems

In parallel to Sec. II, in this section, we provide a comprehensive collection
of linear matrix inequality (LMI) conditions that arise when analyzing or
synthesizing controllers for discrete-time linear time-invariant (DT-LTI)
systems.

Consider the DT-LTI system given by

$\displaystyle x(t+1)=Ax(t)+Bu(t),$ (103) $\displaystyle y(t)=Cx(t)+Du(t),$

where $x(t)\in\mathbb{R}^{n},u(t)\in\mathbb{R}^{p},y(t)\in\mathbb{R}^{m}$ and
$t\in\mathbb{N}$.

###  VI-A Analysis of DT-LTI Systems

####  VI-A1 Stability

A necessary and sufficient LMI condition for the _stability_ of (103) is given
in the following proposition.

######  Proposition 19.

The DT-LTI system (103) under $u(t)=\mathbf{0}$ is stable iff $\exists P>0$
such that

$\begin{bmatrix}P&A^{\top}P\\\ PA&P\end{bmatrix}>0.$ (104)

###### Proof.

It is well-known (e.g., see [37]) that (103) under $u(t)=\mathbf{0}$ is stable
iff $\exists P>0$ such that

$P-A^{\top}PA>0.$ (105)

Therefore, we only need to prove the equivalence of (104) and (105). This is
achieved by applying Lm. 1 and using Lm. 2 with $\text{diag}(I,P)$ on (105)
as:

$\displaystyle P-A^{\top}PA>0\iff\begin{bmatrix}P&A^{\top}\\\
A&P^{-1}\end{bmatrix}>0\iff\begin{bmatrix}P&A^{\top}P\\\ PA&P\end{bmatrix}.$

∎

######  Remark 2.

When $A$ is a network matrix (and $P$ is a block diagonal network matrix), the
matrix term $A^{\top}PA$ in (105) will not be a network matrix (as opposed to
matrix terms $A^{\top}P$ and $PA$ in (104)). Therefore, (105) cannot be
implemented in a decentralized manner for analysis and control synthesis tasks
via Alg. 1. This is the motivation behind establishing a slightly different
stability condition in (104) (as opposed to (105)).

####  VI-A2 $(Q,S,R)$-dissipativity

For DT-LTI systems (103), the _$(Q,S,R)$ -dissipativity_ [39] property is
defined as follows.

######  Definition 3.

[39] The DT-LTI system (103) is $(Q,S,R)$-dissipative (from $u(t)$ to $y(t)$),
if there exists a positive definite function
$V(x):\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$ called the storage
function such that for all $t_{1},t_{0}\in\mathbb{N},t_{1}\geq t_{0}\geq
0,x(t_{0})\in\mathbb{R}^{n}$ and $u(t)\in\mathbb{R}^{p}$, the inequality

$V(x(t_{1}))-V(x(t_{0}))\leq\sum_{t=t_{0}}^{t_{1}-1}\begin{bmatrix}y(t)\\\
u(t)\end{bmatrix}^{\top}\begin{bmatrix}Q&S\\\
S^{\top}&R\end{bmatrix}\begin{bmatrix}y(t)\\\ u(t)\end{bmatrix}$

holds, where $Q\in\mathbb{R}^{m\times m},S\in\mathbb{R}^{m\times
p},R\in\mathbb{R}^{p\times p}$ are given.

At this point, it is worth noting that Remark 1 is equally applicable for DT-
LTI systems (103). In parallel to Prop. 19, a necessary and sufficient
condition for the $(Q,S,R)$-dissipativity of (103) is given in the following
proposition.

######  Proposition 20.

The DT-LTI system (103) is $(Q,S,R)$-dissipative (with $-Q>0$, $R=R^{\top}$)
from $u(t)$ to $y(t)$ iff $\exists P>0$ such that

$\begin{bmatrix}P&C^{\top}S&A^{\top}P&C^{\top}\\\
\star&D^{\top}S+S^{\top}D+R&B^{\top}P&D^{\top}\\\ \star&\star&P&\mathbf{0}\\\
\star&\star&\star&-Q^{-1}\end{bmatrix}>0.$ (106)

###### Proof.

It is well-known (e.g., see [39]) that (103) is $(Q,S,R)$-dissipative from
$u(t)$ to $y(t)$ iff $\exists P>0$ such that

$\begin{bmatrix}P-A^{\top}PA+\hat{Q}&-A^{\top}PB+\hat{S}\\\
\star&-B^{\top}PB+\hat{R}\end{bmatrix}\geq 0,$ (107)

where $\hat{Q}=C^{\top}QC,\ \ \hat{S}=C^{\top}S+C^{\top}QD,\ \
\hat{R}=D^{\top}QD+(D^{\top}S+S^{\top}D)+R$. Therefore, we only need to prove
the equivalence of (106) and (107) under $-Q>0$ and $R=R^{\top}$.

This is achieved by following the steps: (1) applying Lm. 1 to isolate $P$,
(2) using Lm. 2 with $\text{diag}(I,I,P)$ and (3) applying Lm. 1 to isolate
$Q$, on (107), to respectively obtain:

$\displaystyle\begin{bmatrix}P+C^{\top}QC&C^{\top}S+C^{\top}QD&A^{\top}\\\
\star&D^{\top}QD+\mathcal{H}_{s}(D^{\top}S)+R&B^{\top}\\\
\star&\star&P^{-1}\end{bmatrix}>0$
$\displaystyle\iff\begin{bmatrix}P+C^{\top}QC&C^{\top}S+C^{\top}QD&A^{\top}P\\\
\star&D^{\top}QD+\mathcal{H}_{s}(D^{\top}S)+R&B^{\top}P\\\
\star&\star&P\end{bmatrix}>0$
$\displaystyle\iff\begin{bmatrix}P&C^{\top}S&A^{\top}P&C^{\top}\\\
\star&\mathcal{H}_{s}(D^{\top}S)+R&B^{\top}P&D^{\top}\\\
\star&\star&P&\mathbf{0}\\\
\star&\star&\star&-Q^{-1}\end{bmatrix}>0\iff\mbox{\eqref{Eq:Pr:DTLTIQSRDissipativity}}.$

∎

As mentioned in Rm. 2, we remind that the motivation behind establishing a
slightly different $(Q,S,R)$-dissipativity condition in (106) (as opposed to
(107)) is to ensure the decentralized analysis and enforcement of (106) via
Alg. 1.

####  VI-A3 $\mathcal{H}_{2}$-Norm

Let $\mathcal{G}:\mathcal{L}_{2e}\rightarrow\mathcal{L}_{2e}$ be the transfer
matrix of the DT-LTI system (103) from $u(t)$ to $y(t)$. If $A$ in (103) is
Schur, $\mathcal{G}(z)=C(z\mathbf{I}-A)^{-1}B+D$. As shown in [40], the
$\mathcal{H}_{2}$-norm of $\mathcal{G}$ is

$\|\mathcal{G}\|_{\mathcal{H}_{2}}^{2}=\mbox{tr}\left(B^{\top}MB^{\top}+D^{\top}D\right)=\mbox{tr}\left(CNC^{\top}+DD^{\top}\right),$

where $M,N>0$ with $A^{\top}MA-M+C^{\top}C=0$ and $ANA^{\top}-N+BB^{\top}=0$.
Consider the following proposition.

######  Proposition 21.

[40, pp. 64] The $\mathcal{H}_{2}$-Norm of the transfer matrix of the DT-LTI
system (103) (i.e., $\|\mathcal{G}\|_{\mathcal{H}_{2}}$) is
$\|\mathcal{G}\|_{\mathcal{H}_{2}}<\gamma$ iff $\exists P,Q>0$ and
$\gamma\in\mathbb{R}_{>0}$ such that

$\begin{bmatrix}P&AP&B\\\ \star&P&\mathbf{0}\\\
\star&\star&\mathbf{I}\end{bmatrix}>0,\ \begin{bmatrix}Q&CP&D\\\
\star&P&\mathbf{0}\\\ \star&\star&\mathbf{I}\end{bmatrix}>0,\
\mbox{tr}\left(Q\right)<\gamma^{2},$ (108)

or

$\begin{bmatrix}P&AP&C^{\top}\\\ \star&P&\mathbf{0}\\\
\star&\star&\mathbf{I}\end{bmatrix}>0,\ \begin{bmatrix}Q&B^{\top}P&D^{\top}\\\
\star&P&\mathbf{0}\\\ \star&\star&\mathbf{I}\end{bmatrix}>0,\
\mbox{tr}\left(Q\right)<\gamma^{2}.$ (109)

####  VI-A4 $\mathcal{H}_{\infty}$-Norm

The $\mathcal{H}_{\infty}$-norm of $\mathcal{G}$ is [40, pp.49]:

$\|\mathcal{G}\|_{\mathcal{H}_{\infty}}^{2}=\sup_{u\in\mathcal{L}_{2},u\neq
0}\frac{\|y\|_{\mathcal{L}_{2}}^{2}}{\|u\|_{\mathcal{L}_{2}}^{2}}.$ (110)

Now, consider the following proposition.

######  Proposition 22.

[40, pp.50] The $\mathcal{H}_{\infty}$-Norm of the transfer matrix of the DT-
LTI system (103) (i.e., $\|\mathcal{G}\|_{\mathcal{H}_{\infty}}$) is
$\|\mathcal{G}\|_{\mathcal{H}_{\infty}}<\gamma$ iff $\exists P>0$ and
$\gamma\in\mathbb{R}_{>0}$ such that

$\begin{bmatrix}P&AP&B&\mathbf{0}\\\ \star&P&\mathbf{0}&PC^{\top}\\\
\star&\star&\gamma\mathbf{I}&D^{\top}\\\
\star&\star&\star&\gamma\mathbf{I}\end{bmatrix}>0,$ (111)

or

$\begin{bmatrix}P&PA&PB&\mathbf{0}\\\ \star&P&\mathbf{0}&C^{\top}\\\
\star&\star&\gamma\mathbf{I}&D^{\top}\\\
\star&\star&\star&\gamma\mathbf{I}\end{bmatrix}>0.$ (112)

####  VI-A5 Controllability

Regarding the _controllability_ [37] of the DT-LTI system (103), consider the
following proposition.

######  Proposition 23.

[40, pp. 86] A DT-LTI system (103) is: (1) stable and controllable iff
$\exists P>0$ such that

$P-APA^{\top}-BB^{\top}=0,$ (113)

and (2) stabilizable iff $\exists P>$ such that

$\begin{bmatrix}P&PA^{\top}\\\ \star&P+BB^{\top}\end{bmatrix}>0$ (114)

(with $K=-(2\mathbf{I}+B^{\top}P^{-1}B)^{-1}B^{\top}P^{-1}A$, $A+BK$ is
Schur).

####  VI-A6 Observability

Regarding the _observability_ [37] of the DT-LTI system (103), consider the
following proposition.

######  Proposition 24.

[40, pp. 87] A DT-LTI system (103) is: (1) stable and observable iff $\exists
P>0$ such that

$P-APA^{\top}-C^{\top}C=0,$ (115)

and (2) detectable iff $\exists P>$ such that

$\begin{bmatrix}P&PA\\\ \star&P+C^{\top}C\end{bmatrix}>0$ (116)

(under $L=-AP^{-1}C^{\top}(2\mathbf{I}+CP^{-1}C^{\top})^{-1}$, $A+LC$ is
Schur).

###  VI-B Full-State Feedback (FSF) Controller Synthesis for DT-LTI Systems

Consider the DT-LTI system (103) with noise $w(t)\in\mathbb{R}^{q}$:

$\displaystyle x(t+1)=Ax(t)+Bu(t)+Ew(t),$ (117) $\displaystyle
y(t)=Cx(t)+Du(t)+Fw(t).$

Under full-state feedback (FSF) control $u(t)=Kx(t)$, the closed-loop DT-LTI
system takes the form

$\displaystyle x(t+1)=(A+BK)x(t)+Ew(t),$ (118) $\displaystyle
y(t)=(C+DK)x(t)+Fw(t).$

In parallel to Sec. II-B, in the subsequent subsections, we provide LMI
conditions for FSF controller synthesis so as to stabilize,
$(Q,S,R)$-dissipativate or optimize
$\mathcal{H}_{2}$/$\mathcal{H}_{\infty}$-norm of the closed-loop system (118).

####  VI-B1 Stabilization

The following proposition gives an LMI condition that leads to synthesizing a
FSF controller $K$ such that the closed-loop system (118) is stabilized.

######  Proposition 25.

Under $D=w(t)=\mathbf{0}$, the closed-loop DT-LTI system (118) is stable iff
$\exists M>0$ and $L$ such that

$\begin{bmatrix}M&MA^{\top}+L^{\top}B^{\top}\\\ \star&M\end{bmatrix}>0$ (119)

and $K=LM^{-1}$.

###### Proof.

The proof is complete by following the steps: (1) apply Prop. 19 for (118) to
get an LMI in $P$ and $K$, (2) transform the obtained LMI using Lm. 2 with
$\text{diag}(P^{-1},P^{-1})$, and (3) change the LMI variables via
$M\triangleq P^{-1}$ and $L\triangleq KP^{-1}$. ∎

####  VI-B2 $(Q,S,R)$-Dissipativation

The following proposition provides an LMI condition that leads to synthesize a
FSF controller $K$ such that the closed-loop system (118) is
$(Q,S,R)$-dissipative from $w(t)$ to $y(t)$.

######  Proposition 26.

Under $D=\mathbf{0}$, the closed-loop DT-LTI system (118) is
$(Q,S,R)$-dissipative (with $-Q>0$, $R=R^{\top}$) from $w(t)$ to $y(t)$ iff
$\exists M>0$ and $L$ such that

$\begin{bmatrix}M&MC^{\top}S&MA^{\top}+L^{\top}B^{\top}&MC^{\top}\\\
\star&F^{\top}S+S^{\top}F+R&E^{\top}&F^{\top}\\\ \star&\star&M&\mathbf{0}\\\
\star&\star&\star&-Q^{-1}\end{bmatrix}>0$ (120)

and $K=LM^{-1}$.

###### Proof.

The proof is complete by following the steps: (1) apply Prop. 20 for (118) to
get an LMI in $P$ and $K$, (2) transform the LMI using Lm. 2 with
$\text{diag}(P^{-1},\mathbf{I},P^{-1},\mathbf{I})$, and (3) change the LMI
variables using $M\triangleq P^{-1}$ and $L\triangleq KP^{-1}$. ∎

####  VI-B3 $\mathcal{H}_{2}$-Optimal Control

Under FSF control $u(t)=Kx(t)$, the goal of $\mathcal{H}_{2}$-optimal control
is to synthesize a controller $K$ that minimizes the $\mathcal{H}_{2}$-norm of
the closed-loop system (118) (from $w(t)$ to $y(t)$). For this purpose, the
following proposition provides an LMI based approach.

######  Proposition 27.

The $\mathcal{H}_{2}$-optimal FSF controller $K$ for the closed-loop system
(118) is found by solving the LMI:

$\displaystyle\min_{M,Q,L,\gamma}$ $\displaystyle\ \gamma$ (121) sub. to:
$\displaystyle\ M>0,\ Q>0,\ \gamma>0,$
$\displaystyle\begin{bmatrix}M&AM+BL&E\\\ \star&M&\mathbf{0}\\\
\star&\star&\mathbf{I}\end{bmatrix}>0,$
$\displaystyle\begin{bmatrix}Q&CM+DL&F\\\ \star&M&\mathbf{0}\\\
\star&\star&\mathbf{I}\end{bmatrix}>0,\ \mbox{tr}\left(Q\right)<\gamma^{2},$

and $K=LM^{-1}$.

###### Proof.

The proof is complete by applying Prop. 21 to (118) and executing the change
of variables: $M=P$ and $L=KP$. ∎

####  VI-B4 $\mathcal{H}_{\infty}$-Optimal Control

Similarly, the goal of $\mathcal{H}_{\infty}$-optimal control is to synthesize
a controller $K$ that minimizes the $\mathcal{H}_{\infty}$-norm of the closed-
loop system (17) (from $w(t)$ to $y(t)$).

######  Proposition 28.

The $\mathcal{H}_{\infty}$-optimal FSF controller $K$ for the closed-loop
system (118) is found by solving the LMI:

$\displaystyle\min_{M,L,\gamma}$ $\displaystyle\ \gamma$ (122) sub. to:
$\displaystyle\ M>0,\gamma>0,$
$\displaystyle\begin{bmatrix}M&AM+BL&E&\mathbf{0}\\\
\star&M&\mathbf{0}&MC^{\top}+L^{\top}D^{\top}\\\
\star&\star&\gamma\mathbf{I}&F^{\top}\\\
\star&\star&\star&\gamma\mathbf{I}\end{bmatrix}>0,$

and $K=LM^{-1}$.

###### Proof.

The proof is complete by following the steps: (1) apply Prop. 22 to (118), (2)
transform the main LMI using Lm. 2 with
$\text{diag}(P^{-1},P^{-1},\mathbf{I},\mathbf{I})$, and (3) change the LMI
variables using $M\triangleq P^{-1}$ and $L=KP^{-1}$. ∎

###  VI-C Observer Design for DT-LTI Systems

For full-state feedback control $u(t)=Kx(t)$ the controller requires the state
information $x(t)$ of the DT-LTI system (117). Typically, $x(t)$ is not
available at the controller, and thus, an _observer_ is required to keep an
_estimate_ of $x(t)$ as $\hat{x}(t)$.

####  VI-C1 Luenberger Observer

For the DT-LTI system (117), consider a _Luenberger observer_ implemented at
the controller:

$\hat{x}(t+1)=\hat{A}\hat{x}(t)+\hat{B}u(t)+Ly(t),$ (123)

that has the estimation error ($e(t)\triangleq x(t)-\hat{x}(t)$) dynamics:

$\displaystyle e(t+1)=$ $\displaystyle\hat{A}e(t)+(A-\hat{A}-LC)x(t)\ $ (124)
$\displaystyle+(B-\hat{B}-LD)u(t)+(E-LF)w(t).$

The Luenberger observer parameters: $\hat{A},\hat{B}$ and $L$ can be selected
according to the following proposition.

######  Proposition 29.

Under $w(t)=0$ and Luenberger observer (123) parameters: $\hat{A}\triangleq
A-LC$ and $\hat{B}\triangleq B-LD$, the estimation error dynamics (124) are
stable iff $\exists P>0$ and $K$ such that

$\begin{bmatrix}P&A^{\top}P-C^{\top}K^{\top}\\\ \star&P\end{bmatrix}>0$ (125)

and $L=P^{-1}K$.

###### Proof.

Under $w(t)=0$ and the given observer parameter choices, the estimation error
dynamics (124) reduces to:

$e(t+1)=(A-LC)e(t).$ (126)

The proof is complete by applying Prop. 19 to (126) and then executing a
change of variables using $K=PL$. ∎

The Luenberger observer design proposed above assumes the noise-less case of
(117) (i.e., (103)). This assumption is relaxed in the $(Q,S,R)$-dissipative
and $\mathcal{H}_{2}/\mathcal{H}_{\infty}$-optimal observer designs described
in subsequent subsections. However, before getting into those details, first,
consider the Luenberger observer (123) parameters: $\hat{A}\triangleq A-LC$
and $\hat{B}\triangleq B-LD$ under which the estimation error dynamics (124)
take the form

$\displaystyle e(t+1)=$ $\displaystyle\ (A-LC)e(t)+(E-LF)w(t),$ (127)
$\displaystyle z(t)=$ $\displaystyle\ Ge(t)+Jw(t).$

where $z(t)$ is a pre-defined performance metric.

####  VI-C2 $(Q,S,R)$-Dissipative Observer

The $(Q,S,R)$-dissipative observer synthesizes the Luenberger observer gain
$L$ such that (127) is $(Q,S,R)$-dissipative from $w(t)$ to $z(t)$. For this
purpose, the following proposition can be used.

######  Proposition 30.

The estimation error dynamics (127) is $(Q,S,R)$-dissipative (with $-Q>0$,
$R=R^{\top}$) from $w(t)$ to $z(t)$ iff $\exists P>0$ and $K$ such that

$\begin{bmatrix}P&G^{\top}S&A^{\top}P-C^{\top}K^{\top}&G^{\top}\\\
\star&J^{\top}S+S^{\top}J+R&E^{\top}P-F^{\top}K^{\top}&J^{\top}\\\
\star&\star&P&\mathbf{0}\\\ \star&\star&\star&-Q^{-1}\end{bmatrix}>0$ (128)

and $L=KP^{-1}$.

###### Proof.

The proof is complete by applying Prop. 20 to (127) and executing a change of
variables using $K=PL$. ∎

####  VI-C3 $\mathcal{H}_{2}$-Optimal Observer

The goal of $\mathcal{H}_{2}$-optimal observer is to synthesize the Luenberger
observer gain $L$ that minimizes the $\mathcal{H}_{2}$-norm of (27) (from
$e(t)$ to $z(t)$).

######  Proposition 31.

The $\mathcal{H}_{2}$-optimal observer gain $L$ for (127) is found by solving
the LMI:

$\displaystyle\min_{P,Q,K,\gamma}$ $\displaystyle\ \gamma$ (129) sub. to:
$\displaystyle\ P>0,\ Q>0,\ \gamma>0,$ $\displaystyle\begin{bmatrix}P&PA-
KC&PE-KF\\\ \star&P&\mathbf{0}\\\ \star&\star&\mathbf{I}\end{bmatrix}>0,$
$\displaystyle\begin{bmatrix}Q&PG&J\\\ \star&P&\mathbf{0}\\\
\star&\star&\mathbf{I}\end{bmatrix}>0,\ \mbox{tr}\left(Q\right)<\gamma^{2},$

and $L=P^{-1}K$.

###### Proof.

The proof is complete by applying Prop. 21 to (127) and executing a change of
variables using $K=PL$. ∎

####  VI-C4 $\mathcal{H}_{\infty}$-Optimal Observer

The goal of $\mathcal{H}_{\infty}$-optimal observer is to synthesize the
Luenberger observer gain $L$ that minimizes the $\mathcal{H}_{\infty}$-norm of
(127) (from $e(t)$ to $z(t)$).

######  Proposition 32.

The $\mathcal{H}_{\infty}$-optimal observer gain $L$ for (127) is found by
solving the LMI:

$\displaystyle\min_{P,K,\gamma}$ $\displaystyle\ \gamma$ (130) sub. to:
$\displaystyle\ P>0,\ \gamma>0,$ $\displaystyle\begin{bmatrix}P&PA-KC&PE-
KF&\mathbf{0}\\\ \star&P&\mathbf{0}&G^{\top}\\\
\star&\star&\gamma\mathbf{I}&J^{\top}\\\
\star&\star&\star&\gamma\mathbf{I}\end{bmatrix}>0,$

and $L=P^{-1}K$.

###### Proof.

The proof is complete by applying Prop. 22(112) to (127) and executing a
change of variables using $K=PL$. ∎

###  VI-D Dynamic Output Feedback (DOF) Control Synthesis for DT-LTI Systems

Consider the DT-LTI system (103) with noise $w(t)\in\mathbb{R}^{q}$ and
performance $z(t)\in\mathbb{R}^{l}$:

$\displaystyle x(t+1)=Ax(t)+Bu(t)+Ew(t),$ (131) $\displaystyle
y(t)=Cx(t)+Du(t)+Fw(t),$ $\displaystyle z(t)=Gx(t)+Hu(t)+Jw(t).$

Under $D=\mathbf{0}$ and dynamic output feedback (DOF) control (from $y(t)$ to
$u(t)$):

$\displaystyle\zeta(t+1)=A_{c}\zeta(t)+B_{c}y(t),$ (132) $\displaystyle
u(t)=C_{c}\zeta(t)+D_{c}y(t),$

where $\zeta(t)\in\mathbb{R}^{r}$, the closed-loop DT-LTI system (131) takes
the form:

$\displaystyle\theta(t+1)=\bar{A}\theta(t)+\bar{B}w(t),$ (133) $\displaystyle
z(t)=\bar{C}\theta(t)+\bar{D}w(t),$

with
$\theta(t)=\begin{bmatrix}x^{\top}(t)&\zeta^{\top}(t)\end{bmatrix}^{\top}$ and

$\displaystyle\bar{A}\triangleq\begin{bmatrix}A+BD_{c}C&BC_{c}\\\
B_{c}C&A_{c}\end{bmatrix},\ \bar{B}\triangleq\begin{bmatrix}E+BD_{c}F\\\
B_{c}F\end{bmatrix},$
$\displaystyle\bar{C}\triangleq\begin{bmatrix}G+HD_{c}C&HC_{c}\end{bmatrix},\
\bar{D}\triangleq\begin{bmatrix}J+HD_{c}F\end{bmatrix}.$

In parallel to Sec. II-D, in the subsequent subsections, we provide LMI
conditions for DOF controller synthesis (i.e., to design
$A_{c},B_{c},C_{c},D_{c}$ in (132)) so as to stabilize,
$(Q,S,R)$-dissipativate or optimize
$\mathcal{H}_{2}$/$\mathcal{H}_{\infty}$-norm of the closed-loop system (133).
For this purpose, we can use the same change of variables process introduced
in section II-D1 due to the equivalence of coefficient matrices in (133) and
(33).

####  VI-D1 Stabilization

The following proposition gives an LMI condition that leads to synthesize a
DOF controller (132) such that the closed-loop system (133) is stabilized.

######  Proposition 33.

Under $D=w(t)=\mathbf{0}$, the closed-loop DT-LTI system (133) is stable iff
$\exists X,Y>0$ and $A_{n},B_{n},C_{n},D_{n}$ such that