# Distributed $H_{\infty}$ Edge Weight Synthesis for Cooperative Systems

Baris Taner1 and Kamesh Subbarao2 *Support Number1Baris Taner is with
Mechanical and Aerospace Engineering Department, University of Texas
Arlington, 76019, Arlington, Texas, USA<EMAIL_ADDRESS>Kamesh
Subbarao is with Mechanical and Aerospace Engineering Department, University
of Texas Arlington, 76019, Arlington, Texas, USA<EMAIL_ADDRESS>
###### Abstract

This paper studies distributed edge weight synthesis of a cooperative system
for a fixed topology to improve $H_{\infty}$ performance, considering that
disturbances are injected at interconnection channels. This problem is cast
into a linear matrix inequality problem by replacing original cooperative
system with an equivalent ideal cooperative system. Derivations of the method
relies on dissipative system framework. Proposed method provides an upper
bound for the induced $\mathcal{L}_{2}$ norm of the original lumped
cooperative system while reducing the computation time. A comparison for
computation time illustrates the advantage of the proposed method against the
lumped counterpart.

##  I INTRODUCTION

Synthesis and control of cooperative systems (CSs) are active areas of
research due to the attractive applications such as formation flight [1],
satellite clustering [2] and sensor fusion [3]. Some of these applications are
clustering a few agents and form a small CS, while the others make large-scale
CS, which might be geographically separated and life-critical [4].
Conventional way to approach these applications relies on lumped methods,
where CS is modelled as a single system. However, this approach impose a heavy
computational load in synthesis process even for relatively small CSs, which
motivates the distributed methodologies.

There is a vast body of work in the area of distributed control and decision
making for CSs. Some of the recent literature from different frameworks can be
given as follows. In [5], an optimal control framework is used to solve multi-
objective optimization problem for DC micro-grid using local-neighbour
information. In another work, distributed impedance control is synthesized for
event-triggered cooperative manipulation under disturbances using Lyapunov
stability theory [6]. Finally, a distributed robust controller synthesis
methodology from a dissipation perspective is utilized in [7] to consider
uncertainties with CS. The common basis for these methods is that, CS is
modelled in distributed fashion.

In literature, synthesis of edge weights and topology of a CS is cast as a
semi-definite programming (SDP). Following works can be given as examples,
which are minimizing effective resistance of CSs using optimal allocation of
the edge weights [8] and obtaining better dynamic properties for CSs by
meeting upper and lower bound constraints on maximum and Fiedler eigenvalues
[9]. Lumped problems for large scale CS can also be effectively solved using
sparsity promoting methods. In a recent work, topology design of an undirected
network is posed as an optimal control problem and solved using proximal
gradient and Newton methods [10]. Similar examples can be given from
distributed optimization of network systems, where cost function is separable
into smaller convex cost functions [11, 12]. All these works share a common
basis, which is the lumped modelling and distributed optimization of CS.

Motivation.To the best of authors knowledge, edge weight and topology
synthesis of CSs are only posed using lumped CSs with distributed optimization
methods in the literature. Yet this problem can be modelled in a distributed
fashion before referring to distributed optimization methods. By this way,
potentially faster converging algorithms will be available. Method originally
defined in [13] and recently implemented for controller synthesis in [7],
imposes a constraint on the interconnections by assuming the interconnections
are ideal. This assumption results in a symmetric adjacency matrix, which is a
limitation on the synthesis. This method is utilized in this paper for
distributed edge weight synthesis, where performance is measured in terms of
induced $\mathcal{L}_{2}$ norm of the system. Without the relaxations, this
problem is NP-hard [14]. Edge synthesis in this work is done on a CS, whose
individual agents are single input single output (SISO) stable systems. It
should be noted that topology of the CS is predefined and not synthesized.

Contributions. This work resolves limitations on adjacency matrices and
complexity due to NP-hard definition of the problem by redefining the CS. This
is done by embedding adjacency matrix into a synthetic system definition and
making ideal interconnections between original agents of the CS and the
synthetic one. This results in a unique communication topology, which allows
distributed CS modelling for any adjacency matrix. In addition, when
underlying structure is exploited, this method eliminates the conditions
leading to NP-hard problem by promoting sparsity in defining interconnection
constraints. Key contributions of this paper are as follows:

  * •

Adopts distributed CS modelling into an edge weight synthesis problem by
resolving its limitations on adjacency matrix.

  * •

Resolves complexity due to NP-hardness by introducing an equivalent sparse and
ideal CS definition to replace original CS. By this way, problem becomes
linear matrix inequality (LMI) optimization problem, which can be solved using
SDP solvers.

Paper is organized such that section II, expresses underlying Graph structure,
lumped CS modelling, $H_{\infty}$ nominal performance for lumped CS model,
distributed CS modelling and required definitions and lemmas to solve LMIs.
Following that, problem formulation is given in section III. Main result is
shared in section IV. This is followed by section V, where numerical
verifications of the method is given for four agent case in both lumped and
distributed fashions for comparison. In addition to that, distributed method
proposed in this paper is tested upto 10 vehicle case. Finally, conclusions
are provided in section VI.

##  II PRELIMINARIES

###  II-A Notation

Let $\mathbb{R}$, $\mathbb{N}$, $\mathbb{S}^{p}$ and $\mathbb{S}_{s}^{p}$
denotes real numbers, natural numbers, symmetric and skew symmetric matrices
of size $p$, respectively. $M_{ij}$ represents element in the $i^{th}$ row
$j^{th}$ column of matrix ${\bm{M}}$. Matrix inequality conditions are defined
with $<$ and $>$, which stands for $\leq$ and $\geq$, respectively. A matrix
${\bm{M}}:=\text{diag}(\cdot)$ is a block matrix, where diagonal entries are
the arguments of diag. A matrix ${\bm{M}}:=\text{col}(\cdot)$ is a block
matrix, where vertical entries are the arguments of col. Similarly,
${\bm{M}}:=\text{row}(\cdot)$ is a block matrix, where vertical entries are
the arguments of col. Finally, $\mathcal{F}_{L}$ and $\mathcal{F}_{U}$ stands
for lower and upper linear fractional transformation (LFT).

###  II-B Underlying Graph Structure

The interaction/communication among agents in the cooperative systems are
described by Graph $\mathcal{G}=(\mathcal{N},\mathcal{E})$, which consists of
node set $\mathcal{N}$ and edge set $\mathcal{E}$ [15]. Edge set
$\mathcal{E}\subset\mathcal{N}\times\mathcal{N}$ is given between nodes
$i\in\mathcal{N}$ and $j\in\mathcal{N}$ such that $(j,i)\in\mathcal{E}$
denotes node $i$ receives information from $j$. Adjacency matrix
${\bm{\Upsilon}}=[\upsilon_{ij}]\in\mathbb{R}^{N\times N}$ of $\mathcal{G}$ is
composed of weighting scalars ${\upsilon}$, where ${\upsilon}$ quantifies the
strength of the connection from node $j$ to node $i$. $N$ is the number agents
in the cooperative system (CS). Formally, ${{\Upsilon}}_{ij}$ is described as
in the following equation.

${\bm{\Upsilon}}_{ij}=\begin{cases}{\upsilon}>0,&j\neq i,\leavevmode\nobreak\
\leavevmode\nobreak\ (j,i)\in\mathcal{E}\\\
{\upsilon}=0,&otherwise\end{cases}$ (1)

###  II-C Construction of the Lumped CS

Construction of CS is described starting from agents, which are denoted as
${}^{i}{\bm{G}}$ for $i=\left[1,\cdots,N\right]$. The state space definition
of ${}^{i}{\bm{G}}$ is given as in (2), where
${}^{i}{\bm{x}}\in\mathbb{R}^{n_{xi}}$,
${}^{i}{\bm{w}}_{1}\in\mathbb{R}^{n_{w1i}}$,
${}^{i}{\bm{w}}\in\mathbb{R}^{n_{wi}}$,
${}^{i}{\bm{z}}_{1}\in\mathbb{R}^{n_{z1i}}$ and
${}^{i}{\bm{z}}\in\mathbb{R}^{n_{zi}}$. Agents share output information over
the input - output ports denoted as ${}^{i}{\bm{w}}\leavevmode\nobreak\
-\leavevmode\nobreak\ {}^{i}{\bm{z}}$ such that
${}^{i}{\bm{w}}={\bm{\Upsilon}}{}^{i}{\bm{z}}$. Signals over this port is
denoted as spatial signals. Channel through ports
${}^{i}{\bm{w}}_{1}\leavevmode\nobreak\ -\leavevmode\nobreak\
{}^{i}{\bm{z}}_{1}$ is the performance channel.

$\left[\begin{array}[]{c}{}^{i}\dot{\bm{x}}\\\ \hline\cr{}^{i}{\bm{y}}_{1}\\\
{}^{i}{\bm{y}}\end{array}\right]\leavevmode\nobreak\ =\leavevmode\nobreak\
\left[\begin{array}[]{c|cc}{}^{i}{\bm{A}}&{}^{i}{\bm{B}}_{1}&{}^{i}{\bm{B}}\\\
\hline\cr{}^{i}{\bm{C}}_{1}&{}^{i}{\bm{D}}_{1}&{}^{i}{\bm{E}}_{1}\\\
{}^{i}{\bm{C}}&{}^{i}{\bm{F}}_{1}&{\bm{0}}\end{array}\right]\left[\begin{array}[]{c}{}^{i}{\bm{x}}\\\
\hline\cr{}^{i}{\bm{w}}_{1}\\\ {}^{i}{\bm{w}}\end{array}\right]$ (2)

A system ${\bm{S}}$ is created that represents a group of agents, which is
simply described as ${\bm{S}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\text{diag}\\{^{i}{\bm{G}}\\}_{i=1}^{N}$ and given in (3), where
${\bm{x}}\in\mathbb{R}^{n_{x}}$, ${\bm{w}}_{1}\in\mathbb{R}^{n_{w1}}$,
${\bm{w}}\in\mathbb{R}^{n_{w}}$, ${\bm{z}}_{1}\in\mathbb{R}^{n_{z1}}$ and
${\bm{z}}\in\mathbb{R}^{n_{z}}$. Here $n_{x}=N\cdot n_{xi}$ and rest of the
signals are expanded likewise.

$\left[\begin{array}[]{c}\dot{\bm{x}}\\\ \hline\cr{\bm{z}}_{1}\\\
{\bm{z}}\end{array}\right]\leavevmode\nobreak\
=\left[\begin{array}[]{c|cc}{\bm{A}}&{\bm{B}}_{1}&{\bm{B}}\\\
\hline\cr{\bm{C}}_{1}&{\bm{D}}_{1}&{\bm{E}}_{1}\\\
{\bm{C}}&{\bm{F}}_{1}&{\bm{0}}\end{array}\right]\left[\begin{array}[]{c}{\bm{x}}\\\
\hline\cr{\bm{w}}_{1}\\\ {\bm{w}}\end{array}\right]$ (3)

A cooperative system, ${\bm{H}}$, can be constructed using ${\bm{\Upsilon}}$
to connect vehicles to each other as represented by lower LFT such that
${\bm{H}}=\mathcal{F}_{L}({\bm{S}},{\bm{\Upsilon}})$. The state space
representation of ${\bm{H}}$ is provided in (4).

$\displaystyle\left[\begin{array}[]{c|c}{\bm{\mathcal{A}}}&{\bm{\mathcal{B}}}_{1}\\\
\hline\cr{\bm{\mathcal{C}}}_{1}&{\bm{\mathcal{D}}}_{1}\end{array}\right]=\left[\begin{array}[]{c|c}{\bm{A}}+{\bm{B}}{\bm{\Upsilon}}{\bm{C}}&{\bm{B}}_{1}+{\bm{B}}{\bm{\Upsilon}}{\bm{F}}_{1}\\\
\hline\cr{\bm{C}}_{1}+{\bm{E}}_{1}{\bm{\Upsilon}}{\bm{C}}&{\bm{D}}_{1}+{\bm{E}}_{1}{\bm{\Upsilon}}{\bm{F}}_{1}\end{array}\right]$
(4)

###  II-D Feasibility Condition for $H_{\infty}$ Performance

Assuming system given in (4) for a given ${\bm{\Upsilon}}$ is well-posed and
asymptotically stable, then induced $\mathcal{L}_{2}$ norm of the system is
given by the scalar $\gamma$ and $\|{\bm{H}}\|_{2}<\gamma$ for all inputs
${}^{i}{\bm{w}}_{1}\in\ell_{2}^{n_{w1i}}$ [7]. This equivalently means that
matrix inequalities (MIs) in (5) holds true. Pre- and post-multiplying the
second inequality in (5) with
$\left[\begin{array}[]{cc}{\bm{x}}^{T}&{\bm{w}}_{1}^{T}\end{array}\right]$ and
$\left[\begin{array}[]{cc}{\bm{x}}^{T}&{\bm{w}}_{1}^{T}\end{array}\right]^{T}$,
yields $\gamma^{2}\|{\bm{w}}_{1}\|_{2}^{2}-\|{\bm{z}}_{1}\|_{2}^{2}<{\bm{0}}$.

$\displaystyle\mathcal{X}$ $\displaystyle\leavevmode\nobreak\
>\leavevmode\nobreak\ {\bm{0}}$ (5)
$\displaystyle\left[\begin{array}[]{c}{\bm{\mathcal{X}}}\\\
{\bm{0}}\end{array}\right]\left[\begin{array}[]{cc}{\bm{\mathcal{A}}}&{\bm{\mathcal{B}}}_{1}\end{array}\right]+\left[\begin{array}[]{c}{\bm{\mathcal{A}}}^{T}\\\
{\bm{\mathcal{B}}}_{1}^{T}\end{array}\right]\left[\begin{array}[]{cc}{\bm{\mathcal{X}}}&{\bm{0}}\end{array}\right]$
$\displaystyle\leavevmode\nobreak\ +\leavevmode\nobreak\
\left[\begin{array}[]{c}{\bm{\mathcal{C}}}_{1}^{T}\\\
{\bm{\mathcal{D}}}_{1}^{T}\end{array}\right]\left[\begin{array}[]{cc}{\bm{\mathcal{C}}}_{1}&{\bm{\mathcal{D}}}_{1}\end{array}\right]+\left[\begin{array}[]{cc}{\bm{0}}&{\bm{0}}\\\
{\bm{0}}&-\gamma^{2}{\bm{I}}\end{array}\right]$
$\displaystyle\leavevmode\nobreak\ <\leavevmode\nobreak\ {\bm{0}}$

Second inequality given in (5) depends non-linearly on matrix variables
${\bm{\mathcal{X}}}$ and ${\bm{\Upsilon}}$. This non-linear matrix inequality
falls into the class of bilinear matrix inequalities (BMIs). Linearization and
solution to this BMI can be obtained by multiple methods, where one of them is
non-linear transformation as in [16] and another one is given as in [17].
Drawback of these methods are computational burden to calculate feasible
solutions to ${\bm{\mathcal{X}}}$ and ${\bm{\Upsilon}}$ as the number of
agents and number of states of the agents grow. Number of unknowns to be
solved for this set of MIs is given by $(N\cdot n_{xi}(1+N\cdot
n_{xi})/2)+n_{\alpha}$, where $n_{\alpha}\in\mathbb{N}$ denotes the number of
independent variables used to define adjacency matrix to be synthesized. As
clearly seen, state size of each agent and number of agents in the system
quadratically increases the number of unknowns to be solved.

###  II-E Construction of Distributed CS for Arbitrary Interconnections

CS given in (4) can also be described in a distributed fashion and this is
realized by imposing constraints on the spatial signal channel of the agents.
Unlike in lumped CS, constraints cannot reveal themselves in the state space
representation, however, they are introduced as supply functions to the
storage function of the CS. This is done by introducing these constraints in
the MIs and this is summarized from the literature in the following text.

Distributed CS model relies on an interconnection constraint on spatial signal
ports of each agent. This constraint is characterized by a supply function
given as in (6). The relevance of the supply functions to synthesize edge
weight becomes clearer with the following definitions.

######  Definition II.1 (Dissipativity [18])

${\bm{W}}$ being input space, ${\bm{Z}}$ being output space and ${\bm{X}}$
being state space, let ${P}\leavevmode\nobreak\ :\leavevmode\nobreak\
{\bm{W}}\times{\bm{Z}}\rightarrow\mathbb{R}$ be a supply function. A system
${\bm{G}}$ with supply function ${P}$ is dissipative if there exists a non-
negative storage function $V\leavevmode\nobreak\ :\leavevmode\nobreak\
{\bm{X}}\rightarrow\mathbb{R}$ for admissible trajectories of $w$, $z$ and
$x$, such that

$V(x(t_{0}))+\int_{t_{0}}^{t_{1}}{P}(w(t),z(t))dt\geq
V(x(t_{1})),\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ t_{0}\leq t_{1}$ (6)

Considering the $H_{\infty}$ performance, a quadratic supply function is known
to be given by bounded real lemma [19, 7] that is given as in (7) for the
system ${}^{i}{\bm{G}}$.

$^{i}{\Psi}\leavevmode\nobreak\ =\leavevmode\nobreak\
\left[\begin{array}[]{c}{}^{i}{\bm{z}}_{1}\\\
\hline\cr{}^{i}{\bm{w}}_{1}\end{array}\right]^{T}\left[\begin{array}[]{c|c}-\frac{1}{\gamma^{2}}{\bm{I}}&{\bm{0}}\\\
\hline\cr{\bm{0}}&{\bm{I}}\end{array}\right]\left[\begin{array}[]{c}{}^{i}{\bm{z}}_{1}\\\
\hline\cr{}^{i}{\bm{w}}_{1}\end{array}\right]$ (7)

A general quadratic supply function can be given as in (8) to impose
interconnections for a cooperative system by mapping spatial signals to agents
as represented in [13, 7]. Spatial signals are mapped within CS by
${\bm{\Upsilon}}$, and in (8), ${\bm{\Upsilon}}$ is embedded in the
correlation matrix, ${}^{i}{\bm{Z}}\in\mathbb{R}^{n_{zi}+n_{wi}\times
n_{zi}+n_{wi}}$, which is partitioned as in (8). This definition will be
clarified as the paper progresses. For a CS, mapping of spatial signals is
illustrated in Figure 1.

Figure 1: Interconnection of agents.

$\displaystyle{}^{i}{\Phi}$ $\displaystyle\leavevmode\nobreak\
=\leavevmode\nobreak\ \left[\begin{array}[]{c}{}^{i}{\bm{z}}\\\
\hline\cr{}^{i}{\bm{w}}\end{array}\right]^{T}\left[\begin{array}[]{c|c}{}^{i}{\bm{Z}}_{11}&{}^{i}{\bm{Z}}_{12}\\\
\hline\cr{}^{i}{\bm{Z}}_{12}^{T}&{}^{i}{\bm{Z}}_{22}\end{array}\right]\left[\begin{array}[]{c}{}^{i}{\bm{z}}\\\
\hline\cr{}^{i}{\bm{w}}\end{array}\right]$ (8) $\displaystyle{}^{i}{\bm{Z}}$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\left[\begin{array}[]{c|c}{}^{i}{\bm{Z}}_{11}&{}^{i}{\bm{Z}}_{12}\\\
\hline\cr{}^{i}{\bm{Z}}_{21}&{}^{i}{\bm{Z}}_{22}\end{array}\right]$

Considering the supply functions ${}^{i}{\Psi}$ and ${}^{i}{\Phi}$, a general
supply function can be written as ${}^{i}{P}\leavevmode\nobreak\
=\leavevmode\nobreak\ {}^{i}{\Psi}+{}^{i}{\Phi}$ and $H_{\infty}$ performance
of the distributed CS can be checked using following MIs for
$i=\left[1,\cdots,N\right]$.

$\displaystyle{}^{i}{\bm{{X}}}$ $\displaystyle\leavevmode\nobreak\
>\leavevmode\nobreak\ {\bm{0}}$ (9)
$\displaystyle\left[\begin{array}[]{ccc}{}^{i}{\bm{{X}}}\leavevmode\nobreak\
{}^{i}{\bm{{A}}}+{}^{i}{\bm{{A}}}^{T}\leavevmode\nobreak\
{}^{i}{\bm{{X}}}&{}^{i}{\bm{{X}}}\leavevmode\nobreak\
{}^{i}{\bm{{B}}}_{1}&{}^{i}{\bm{{X}}}\leavevmode\nobreak\ {}^{i}{\bm{{B}}}\\\
{}^{i}{\bm{{B}}}_{1}^{T}\leavevmode\nobreak\
{}^{i}{\bm{{X}}}&-{\bm{I}}&{\bm{0}}\\\
{}^{i}{\bm{{B}}}^{T}\leavevmode\nobreak\
{}^{i}{\bm{{X}}}&{\bm{0}}&{\bm{0}}\end{array}\right]$
$\displaystyle\leavevmode\nobreak\ +\leavevmode\nobreak\ $
$\displaystyle\frac{1}{\gamma^{2}}\left[\begin{array}[]{c}{}^{i}{\bm{{C}}}_{1}^{T}\\\
{}^{i}{\bm{{D}}}_{1}^{T}\\\
{}^{i}{\bm{{E}}}_{1}^{T}\end{array}\right]^{T}\left[\begin{array}[]{ccc}{}^{i}{\bm{{C}}}_{1}&{}^{i}{\bm{{D}}}_{1}&{}^{i}{\bm{{E}}}_{1}\end{array}\right]$
$\displaystyle\leavevmode\nobreak\ +\leavevmode\nobreak\ $
$\displaystyle\left[\begin{array}[]{cc}{}^{i}{\bm{{C}}}^{T}&{\bm{0}}\\\
{}^{i}{\bm{{F}}}_{1}^{T}&{\bm{0}}\\\
{\bm{0}}&{\bm{I}}\end{array}\right]^{T}{}^{i}{\bm{Z}}\left[\begin{array}[]{ccc}{}^{i}{\bm{{C}}}&{}^{i}{\bm{{F}}}_{1}&{\bm{0}}\\\
{\bm{0}}&{\bm{0}}&{\bm{I}}\end{array}\right]$
$\displaystyle\leavevmode\nobreak\ <\leavevmode\nobreak\ {\bm{0}}$

##  III PROBLEM FORMULATION

This section provides the framework to synthesize the edge weights denoted as
${\upsilon}$ in a distributed fashion using dissipative system framework.

Input and output sizes of each agent are modified to simplify the algebraic
manipulations by introducing zeros to these vectors. New system is called as
${}^{i}\bar{\bm{G}}\in\mathbb{R}^{N\cdot(n_{z1i}+n_{zi})\times
N\cdot(n_{w1i}+n_{wi})}$. This extension is done as given in (10), where
$\star$ stands for ${\bm{w}}_{1}$, ${\bm{w}}$, ${\bm{z}}_{1}$ and ${\bm{z}}$.
Simply, if $1^{st}$ agent is being modified then
${}^{i}\bar{\bm{w}}_{1}^{T}\leavevmode\nobreak\ =\leavevmode\nobreak\
\left[{\bm{w}}_{1}^{T},\leavevmode\nobreak\
{\bm{0}}_{n_{w1i},1}^{T},\leavevmode\nobreak\ \ldots,\leavevmode\nobreak\
{\bm{0}}_{n_{w1i},1}^{T}\right]^{T}$. Same formulation applies to all vectors.
System matrices of ${}^{i}\bar{\bm{G}}$ is also extended to comply with the
input-output sizes except for ${}^{i}\bar{\bm{A}}$, which is
${}^{i}\bar{\bm{A}}\leavevmode\nobreak\ =\leavevmode\nobreak\ ^{i}{\bm{A}}$.
The extended system matrices are denoted as
${}^{i}\bar{\bm{B}}_{1}\in\mathbb{R}^{n_{xi}\times N\cdot n_{w1i}}$,
${}^{i}\bar{\bm{B}}\in\mathbb{R}^{n_{xi}\times N\cdot n_{wi}}$,
${}^{i}\bar{\bm{C}}_{1}\in\mathbb{R}^{N\cdot n_{z1i}\times n_{xi}}$,
${}^{i}\bar{\bm{C}}\in\mathbb{R}^{N\cdot n_{zi}\times n_{xi}}$,
${}^{i}\bar{\bm{D}}_{1}\in\mathbb{R}^{N\cdot n_{z1i}\times N\cdot n_{w1i}}$,
${}^{i}\bar{\bm{E}}_{1}\in\mathbb{R}^{N\cdot n_{z1i}\times N\cdot n_{wi}}$ and
${}^{i}\bar{\bm{F}}_{1}\in\mathbb{R}^{N\cdot n_{zi}\times N\cdot n_{w1i}}$.
Extension of these matrices are left to the reader for brevity of the paper.

$\displaystyle{}^{i}\bar{\bm{\star}}\leavevmode\nobreak\ =\leavevmode\nobreak$
$\displaystyle col\\{{\bm{\star}}_{j}\\}_{j=1}^{N},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ i=\left[1,\cdots,N\right]$ (10)
$\displaystyle{\bm{\star}}_{j}\leavevmode\nobreak\ =\leavevmode\nobreak$
$\displaystyle\begin{cases}{\bm{0}},&j\neq i\\\ {\bm{\star}},&j=i\end{cases}$

For the extended system, dissipativity is verified using supply function
${}^{i}{P}$ yet with a modification in ${}^{i}{\phi}$. A matrix is defined as
${}^{i}\tilde{\bm{Z}}\in\mathbb{R}^{(n_{zi}+n_{wi})\cdot
N\times(n_{zi}+n_{wi})\cdot N}$, which shares the same partition with
${}^{i}{\bm{Z}}$ as given in (8). ${}^{i}\tilde{\bm{Z}}$ reveals the effect of
edge weights as given in (11). The logic behind ${}^{i}{\tilde{\bm{Z}}}$ is
that, ${}^{i}{\bm{z}}$ is scaled by scalar ${\upsilon}_{ij}$ before
transferred to agent $j$ for all the agents, which is illustrated in Figure 1.
If this mapping is followed, ${}^{i}{\tilde{\bm{Z}}}$ constructs an
interconnection constraint on inputs and outputs of extended system
${}^{i}{\bar{\bm{G}}}$. Using the definition given in (11) MIs given in (12)
can be constructed.

$\begin{aligned} {}^{i}\tilde{\bm{Z}}\leavevmode\nobreak\
=\leavevmode\nobreak\ \left[\begin{array}[]{cc}{\bm{\Upsilon}}&{\bm{0}}\\\
{\bm{0}}&{\bm{I}}\end{array}\right]^{T}&\left[\begin{array}[]{cc}{}^{i}\bar{\bm{Z}}_{11}&{}^{i}\bar{\bm{Z}}_{12}\\\
{}^{i}\bar{\bm{Z}}_{12}^{T}&{}^{i}\bar{\bm{Z}}_{22}\end{array}\right]\left[\begin{array}[]{cc}{\bm{\Upsilon}}&{\bm{0}}\\\
{\bm{0}}&{\bm{I}}\end{array}\right]\end{aligned}$

(11)

$\displaystyle{}^{i}{\bm{{X}}}$ $\displaystyle\leavevmode\nobreak\
>\leavevmode\nobreak\ {\bm{0}}$ (12)
$\displaystyle\left[\begin{array}[]{ccc}{}^{i}{\bm{{X}}}\leavevmode\nobreak\
{}^{i}\bar{\bm{{A}}}+{}^{i}\bar{\bm{{A}}}^{T}\leavevmode\nobreak\
{}^{i}{\bm{{X}}}&{}^{i}{\bm{{X}}}\leavevmode\nobreak\
{}^{i}\bar{\bm{{B}}}_{1}&{}^{i}{\bm{{X}}}\leavevmode\nobreak\
{}^{i}\bar{\bm{{B}}}\\\ {}^{i}\bar{\bm{{B}}}_{1}^{T}\leavevmode\nobreak\
{}^{i}{\bm{{X}}}&-{\bm{I}}&{\bm{0}}\\\
{}^{i}\bar{\bm{{B}}}^{T}\leavevmode\nobreak\
{}^{i}{\bm{{X}}}&{\bm{0}}&{\bm{0}}\end{array}\right]\leavevmode\nobreak\
+\leavevmode\nobreak$
$\displaystyle\frac{1}{{\gamma}^{2}}\left[\begin{array}[]{c}{}^{i}\bar{\bm{{C}}}_{1}^{T}\\\
{}^{i}\bar{\bm{{D}}}_{1}^{T}\\\
{}^{i}\bar{\bm{{E}}}_{1}^{T}\end{array}\right]\left[\begin{array}[]{ccc}{}^{i}\bar{\bm{{C}}}_{1}&{}^{i}\bar{\bm{{D}}}_{1}&{}^{i}\bar{\bm{{E}}}_{1}\end{array}\right]\leavevmode\nobreak\
+\leavevmode\nobreak$
$\displaystyle\left[\begin{array}[]{cc}{}^{i}\bar{\bm{{C}}}^{T}&{\bm{0}}\\\
{}^{i}\bar{\bm{{F}}}_{1}^{T}&{\bm{0}}\\\
{\bm{0}}&{\bm{I}}\end{array}\right]{}^{i}\tilde{\bm{Z}}\left[\begin{array}[]{ccc}{}^{i}\bar{\bm{{C}}}&{}^{i}\bar{\bm{{F}}}_{1}&{\bm{0}}\\\
{\bm{0}}&{\bm{0}}&{\bm{I}}\end{array}\right]$
$\displaystyle\leavevmode\nobreak\ <\leavevmode\nobreak\ {\bm{0}}$

It is clear that MIs given in (9), is not suitable for adjacency weight
synthesis as components of ${\bm{\Upsilon}}$ does not appear explicitly. On
the other hand, synthesis of ${\bm{\Upsilon}}$ using the MIs given in (12)
requires us to solve a non-linear matrix inequality. In Section IV, this
problem will be given a solution.

##  IV MAIN RESULT

System given in Figure 1, is interpreted as $N$ agents sharing their
information with a media that sinks the spatial signals and then sends them
back again to the agents with some scaling. This media is specified with edge
weights ${\upsilon}_{ij}$ and ${\upsilon}_{ji}$ as a group within the dashed
red lines in Figure 2. Inputs and outputs of this group are ${}^{i}{z}$ and
${}^{i}{w}$, respectively.

Figure 2: A new interpretation of the communication.

This interpretation of the communication media allows us to assign a system
representation to the communication itself, which is denoted in this paper as
${}^{c}{\bm{G}}\in\mathbb{R}^{n_{z}\times n_{w}}$. Representation of
communication as a dynamic system is a concept used in dynamic Graph theory
[20], however, this paper assumes that the edge weights are constants, for
this reason ${}^{c}{\bm{G}}$ is given by a static feed-forward matrix as
provided in (13).

$\displaystyle{}^{c}{\bm{G}}:=\left[\begin{array}[]{c|c}{\bm{0}}&{\bm{0}}\\\
\hline\cr{\bm{0}}&\Upsilon\end{array}\right]$ (13)

CS then takes a new form, which has ideal interconnections and these
interconnections are between agents ${}^{i}{\bm{G}}$ and ${}^{c}{\bm{G}}$
only. The new interpretation of the CS is graphically given in Figure 3, where
dashed lines show the interconnections that are being made between agents upto
$N$.

Figure 3: Conversion of the original CS (left) to the modified CS (right).

Interconnections given in Figure 3 is defined as in (14), where
${}^{c}{\bm{w}}_{i}\in\mathbb{R}^{n_{zi}}$ and
${}^{c}{\bm{z}}_{i}\in\mathbb{R}^{n_{wi}}$ represents input to
${}^{c}{\bm{G}}$ from agent $i$ and output to agent $i$ from ${}^{c}{\bm{G}}$.

$\left[\begin{array}[]{c}{}^{i}{\bm{z}}\\\
{}^{i}{\bm{w}}\end{array}\right]\leavevmode\nobreak\ =\leavevmode\nobreak\
\left[\begin{array}[]{c}{}^{c}{\bm{w}}_{i}\\\
{}^{c}{\bm{z}}_{i}\end{array}\right]\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall\leavevmode\nobreak\
i=\left[1,\cdots,N\right],\leavevmode\nobreak\ \forall\leavevmode\nobreak\
t\geq 0$ (14)

######  Definition IV.1 (Neutrality of interconnections [13, 7])

Interconnections are considered to be neutral for neighbouring dissipative
agents with respect to supply functions as defined in the form of (8), if (14)
is satisfied along with the condition in (15), where ${}_{i}^{c}{\Phi}$ is the
supply function of agent ${}^{c}{\bm{G}}$ and ${}_{c}^{i}{\Phi}$ is the supply
function of agent ${}^{i}{\bm{G}}$ for $i=\left[1,\cdots,N\right]$.

$\displaystyle{}_{c}^{i}{\Phi}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}_{i}^{c}{\Phi}\leavevmode\nobreak\ =\leavevmode\nobreak\
0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ i=\left[1,\cdots,N\right]$ (15)
$\displaystyle{}_{i}^{c}{\Phi}\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{i=1}^{N}\left[\begin{array}[]{c}{}^{c}{\bm{w}}_{i}\\\
{}^{c}{\bm{z}}_{i}\end{array}\right]^{T}\left[\begin{array}[]{cc}{}_{i}^{c}{\bm{Y}}_{11}&{}_{i}^{c}{\bm{Y}}_{12}\\\
{}_{i}^{c}{\bm{Y}}_{21}&{}_{i}^{c}{\bm{Y}}_{22}\end{array}\right]\left[\begin{array}[]{c}{}^{c}{\bm{w}}_{i}\\\
{}^{c}{\bm{z}}_{i}\end{array}\right]$
$\displaystyle{}_{c}^{i}{\Phi}\leavevmode\nobreak\ =\leavevmode\nobreak\
\left[\begin{array}[]{c}{}^{i}{\bm{w}}\\\
{}^{i}{\bm{z}}\end{array}\right]^{T}\left[\begin{array}[]{cc}{}^{i}{\bm{Y}}_{11}&{}^{i}{\bm{Y}}_{12}\\\
{}^{i}{\bm{Y}}_{21}&{}^{i}{\bm{Y}}_{22}\end{array}\right]\left[\begin{array}[]{c}{}^{i}{\bm{w}}\\\
{}^{i}{\bm{z}}\end{array}\right]$

Here a clarification should be made that, ${}^{c}{\bm{G}}$ makes
interconnections with $N$ agents therefore it requires $N$ supply function
definitions. ${}^{c}{\bm{w}}_{i}$ and ${}^{c}{\bm{z}}_{i}$ are the partitions
of the input-output vectors ${}^{c}{\bm{w}}$ and ${}^{c}{\bm{z}}$ of
${}^{c}{\bm{G}}$ shared with ${}^{i}{\bm{G}}$. This dimensional complexity can
be eliminated by applying the extension described in (10) on systems
${}^{i}{\bm{G}}$ and ${}^{c}{\bm{G}}$. With ${}^{c}{\bm{G}}$ there will be
$N+1$ agents. Sizes of each system will be extended by $(N+1)N$ for the
synthesis such that new systems are denoted as
${}^{i}\hat{\bm{G}}\in\mathbb{R}^{N\cdot(N+1)\cdot(n_{z1i}+n_{zi})\times
N\cdot(N+1)\cdot(n_{w1i}+n_{wi})}$. System matrices of ${}^{i}\hat{\bm{G}}$ is
also extended to comply with the input-output sizes except for
${}^{i}\hat{\bm{A}}$, which is ${}^{i}\hat{\bm{A}}\leavevmode\nobreak\
=\leavevmode\nobreak\ ^{i}{\bm{A}}$. The extended system matrices are denoted
as ${}^{i}\hat{\bm{B}}_{1}\in\mathbb{R}^{n_{xi}\times N\cdot(N+1)\cdot
n_{w1i}}$, ${}^{i}\hat{\bm{B}}\in\mathbb{R}^{n_{xi}\times N\cdot(N+1)\cdot
n_{wi}}$, ${}^{i}\hat{\bm{C}}_{1}\in\mathbb{R}^{N\cdot(N+1)\cdot n_{z1i}\times
n_{xi}}$, ${}^{i}\hat{\bm{C}}\in\mathbb{R}^{N\cdot(N+1)\cdot n_{zi}\times
n_{xi}}$, ${}^{i}\hat{\bm{D}}_{1}\in\mathbb{R}^{N\cdot(N+1)\cdot n_{z1i}\times
N\cdot(N+1)\cdot n_{w1i}}$,
${}^{i}\hat{\bm{E}}_{1}\in\mathbb{R}^{N\cdot(N+1)\cdot n_{z1i}\times
N\cdot(N+1)\cdot n_{wi}}$ and
${}^{i}\hat{\bm{F}}_{1}\in\mathbb{R}^{N\cdot(N+1)\cdot n_{zi}\times
N\cdot(N+1)\cdot n_{w1i}}$. Since ${}^{c}\hat{\bm{G}}$ does not have
partitions like other agents do, it is convenient to write its system matrices
as ${}^{c}\hat{\bm{A}}\leavevmode\nobreak\ =\leavevmode\nobreak\
{\bm{0}}\in\mathbb{R}^{n_{xi}}$, ${}^{c}\hat{\bm{B}}\leavevmode\nobreak\
=\leavevmode\nobreak\ {\bm{0}}\in\mathbb{R}^{n_{xi}\times
N\cdot(N+1)\cdot(n_{w1i}+n_{wi})}$, ${}^{c}\hat{\bm{C}}\leavevmode\nobreak\
=\leavevmode\nobreak\
{\bm{0}}\in\mathbb{R}^{N\cdot(N+1)\cdot(n_{z1i}+n_{zi})\times n_{xi}}$ and
${}^{c}\hat{\bm{D}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\text{diag}(\text{diag}({\bm{0}},{\bm{\Upsilon}}),\text{diag}({\bm{0}},{\bm{\Upsilon}}))\in\mathbb{R}^{N\cdot(N+1)\cdot(n_{z1i}+n_{zi})\times
N\cdot(N+1)\cdot(n_{w1i}+n_{wi})}$.

Relationship in (15), has already been studied in the literature [13, 7] and
for completeness of the paper it will be shared here for this particular CS.
From (14) and (15) one can set the algebraic relationship given in (16).

$\begin{aligned}
\left[\begin{array}[]{cc}{}^{i}{\bm{Y}}_{11}&{}^{i}{\bm{Y}}_{12}\\\
{}^{i}{\bm{Y}}_{21}&{}^{i}{\bm{Y}}_{22}\end{array}\right]\leavevmode\nobreak\
=\leavevmode\nobreak\ -\left[\begin{array}[]{cc}{\bm{0}}&{\bm{I}}\\\
{\bm{I}}&{\bm{0}}\end{array}\right]&\left[\begin{array}[]{cc}{}_{i}^{c}{\bm{Y}}_{11}&{}_{i}^{c}{\bm{Y}}_{12}\\\
{}_{i}^{c}{\bm{Y}}_{21}&{}_{i}^{c}{\bm{Y}}_{22}\end{array}\right]\left[\begin{array}[]{cc}{\bm{0}}&{\bm{I}}\\\
{\bm{I}}&{\bm{0}}\end{array}\right]\end{aligned}$

(16)

This relationship yields
${}^{i}{\bm{Y}}_{11}={}^{i}{\bm{Y}}_{11}^{T}=-{}_{i}^{c}{\bm{Y}}_{22}$ and
${}^{i}{\bm{Y}}_{12}^{T}=-{}_{i}^{c}{\bm{Y}}_{12}$ for
$i=\left[1,\cdots,N\right]$.

Finally because of the communication topology for the modified system, a
supply function for agents including ${}^{c}\hat{\bm{G}}$ can be formulated as
in (17) with ${}^{d}\hat{\bm{Z}}$ for $d\in\\{i,\leavevmode\nobreak\ c\\}$.
Partitioning of these matrices are given as ${}^{d}\hat{\bm{Z}}_{*\\#}$, where
$*,\leavevmode\nobreak\ \\#\in\\{1,\leavevmode\nobreak\ 2\\}$ indicate the
respective partition of ${}^{d}\hat{\bm{Z}}$. All zero matrices with
respective sizes are defined by ${\bm{0}}$.

$\displaystyle{}^{i}\hat{\bm{Z}}_{11}$ $\displaystyle\leavevmode\nobreak\
=\leavevmode\nobreak\
\left[\begin{array}[]{c|c}\text{diag}\\{{\bm{0}}\\}_{i=1}^{N}&\text{col}\\{{\bm{0}}\\}_{i=1}^{N}\\\
\hline\cr\text{row}\\{{\bm{0}}\\}_{i=1}^{N}&{}^{i}{\bm{Y}}_{11}\end{array}\right]$
(17) $\displaystyle{}^{i}\hat{\bm{Z}}_{12}$ $\displaystyle\leavevmode\nobreak\
=\leavevmode\nobreak\
\left[\begin{array}[]{c|c}\text{diag}\\{{\bm{0}}\\}_{i=1}^{N}&\text{col}\\{{\bm{0}}\\}_{i=1}^{N}\\\
\hline\cr\text{row}\\{{\bm{0}}\\}_{i=1}^{N}&{}_{i}^{c}{\bm{Y}}_{12}\end{array}\right]$
$\displaystyle{}^{i}\hat{\bm{Z}}_{22}$ $\displaystyle\leavevmode\nobreak\
=\leavevmode\nobreak\
\left[\begin{array}[]{c|c}\text{diag}\\{{\bm{0}}\\}_{i=1}^{N}&\text{col}\\{{\bm{0}}\\}_{i=1}^{N}\\\
\hline\cr\text{row}\\{{\bm{0}}\\}_{i=1}^{N}&{}_{i}^{c}{\bm{Y}}_{11}\end{array}\right]$
$\displaystyle{}^{c}\hat{\bm{Z}}_{11}$ $\displaystyle\leavevmode\nobreak\
=\leavevmode\nobreak\
\left[\begin{array}[]{c|c}-\text{diag}\\{{}_{i}^{c}{\bm{Y}}_{11}\\}_{i=1}^{N}&\text{col}\\{{\bm{0}}\\}_{i=1}^{N}\\\
\hline\cr\text{row}\\{{\bm{0}}\\}_{i=1}^{N}&{\bm{0}}\end{array}\right]$
$\displaystyle{}^{c}\hat{\bm{Z}}_{12}$ $\displaystyle\leavevmode\nobreak\
=\leavevmode\nobreak\
\left[\begin{array}[]{c|c}-\text{diag}\\{{}^{i}{\bm{Y}}_{12}\\}_{i=1}^{N}&\text{col}\\{{\bm{0}}\\}_{i=1}^{N}\\\
\hline\cr\text{row}\\{{\bm{0}}\\}_{i=1}^{N}&{\bm{0}}\end{array}\right]$
$\displaystyle{}^{c}\hat{\bm{Z}}_{22}$ $\displaystyle\leavevmode\nobreak\
=\leavevmode\nobreak\
\left[\begin{array}[]{c|c}\text{diag}\\{{}^{i}{\bm{Y}}_{11}\\}_{i=1}^{N}&\text{col}\\{{\bm{0}}\\}_{i=1}^{N}\\\
\hline\cr\text{row}\\{{\bm{0}}\\}_{i=1}^{N}&{\bm{0}}\end{array}\right]$

Eventually, with supply functions of the form (7) and (8) described for the
modified cooperative system following lemma can be given.

###### Lemma IV.1

Let ${}^{i}\hat{\bm{G}}$ be a dissipative agent with state space realization
is derived from (2) using the extension (10). CS consisting of
${}^{i}\hat{\bm{G}}$ have a robust performance of $\gamma$ for input
${}^{i}\hat{w}_{1}\in\ell_{2}^{N\cdot(N+1)\cdot n_{w1i}}$ for
$i=\left[1,\cdots,N\right]$ if there exists
${}^{i}{\bm{X}}\in\mathbb{S}^{n_{xi}}$,
${}^{i}{\bm{Y}}_{11}\in\mathbb{S}^{(N+1)\cdot n_{wi}}$,
${}_{i}^{c}{\bm{Y}}_{11}\in\mathbb{S}^{(N+1)\cdot n_{wi}}$,
${}^{i}{\bm{Y}}_{12}\in\mathbb{S}_{s}^{(N+1)\cdot n_{wi}}$ and
${}_{i}^{c}{\bm{Y}}_{12}\in\mathbb{S}_{s}^{(N+1)\cdot n_{wi}}$ such that

$\displaystyle{}^{i}\hat{\bm{X}}$ $\displaystyle\leavevmode\nobreak\
\geq\leavevmode\nobreak\ {\bm{0}}$ (18)
$\displaystyle\left[\begin{array}[]{ccc}{}^{i}\hat{\bm{{X}}}\leavevmode\nobreak\
{}^{i}\hat{\bm{{A}}}+{}^{i}\hat{\bm{{A}}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{{X}}}&{}^{i}\hat{\bm{{X}}}\leavevmode\nobreak\
{}^{i}\hat{\bm{{B}}}_{1}&{}^{i}\hat{\bm{{X}}}\leavevmode\nobreak\
{}^{i}\hat{\bm{{B}}}\\\ {}^{i}\hat{\bm{{B}}}_{1}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{{X}}}&-{\bm{I}}&{\bm{0}}\\\
{}^{i}\hat{\bm{{B}}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{{X}}}&{\bm{0}}&{\bm{0}}\end{array}\right]\leavevmode\nobreak\
+\leavevmode\nobreak$
$\displaystyle\frac{1}{\gamma^{2}}\left[\begin{array}[]{c}{}^{i}\hat{\bm{{C}}}_{1}^{T}\\\
{}^{i}\hat{\bm{{D}}}_{1}^{T}\\\
{}^{i}\hat{\bm{{E}}}_{1}^{T}\end{array}\right]\left[\begin{array}[]{ccc}{}^{i}\hat{\bm{{C}}}_{1}&{}^{i}\hat{\bm{{D}}}_{1}&{}^{i}\hat{\bm{{E}}}_{1}\end{array}\right]\leavevmode\nobreak\
+\leavevmode\nobreak$
$\displaystyle\left[\begin{array}[]{cc}{}^{i}\hat{\bm{{C}}}^{T}&{\bm{0}}\\\
{}^{i}\hat{\bm{{F}}}_{1}^{T}&{\bm{0}}\\\
{\bm{0}}&{\bm{I}}\end{array}\right]{}^{i}\hat{\bm{Z}}\left[\begin{array}[]{ccc}{}^{i}\hat{\bm{{C}}}&{}^{i}\hat{\bm{{F}}}_{1}&{\bm{0}}\\\
{\bm{0}}&{\bm{0}}&{\bm{I}}\end{array}\right]$
$\displaystyle\leavevmode\nobreak\ <\leavevmode\nobreak\ {\bm{0}}$
$\displaystyle\left[\begin{array}[]{cc}{}^{c}\hat{\bm{C}}^{T}&{\bm{0}}\\\
{}^{c}\hat{\bm{D}}^{T}&{\bm{I}}\end{array}\right]\left[\begin{array}[]{c|c}\frac{1}{\gamma^{2}}{\bm{I}}&{\bm{0}}\\\
\hline\cr{\bm{0}}&-{\bm{I}}\end{array}\right]\left[\begin{array}[]{cc}{}^{c}\hat{\bm{C}}&{}^{c}\hat{\bm{D}}\\\
{\bm{0}}&{\bm{I}}\end{array}\right]\leavevmode\nobreak\ +\leavevmode\nobreak$
$\displaystyle\left[\begin{array}[]{cc}{}^{c}\hat{\bm{C}}^{T}&{\bm{0}}\\\
{}^{c}\hat{\bm{D}}^{T}&{\bm{I}}\end{array}\right]{}^{c}\hat{\bm{Z}}\left[\begin{array}[]{cc}{}^{c}\hat{\bm{C}}&{}^{c}\hat{\bm{D}}\\\
{\bm{0}}&{\bm{I}}\end{array}\right]$ $\displaystyle\leavevmode\nobreak\
<\leavevmode\nobreak\ {\bm{0}}$

It should be noted that, first two inequalities in (18) are LMIs. Because of
the specific structure of the interconnections of the third inequality
${}^{i}\hat{\bm{Z}}_{12}\leavevmode\nobreak\ =\leavevmode\nobreak\ {\bm{0}}$.
Eventually ${}^{i}\hat{\bm{Z}}$ and ${}^{c}\hat{\bm{Z}}$ are diagonal. By
investigating (17) and composite matrix
$\left[{}^{c}\hat{\bm{C}}\leavevmode\nobreak\ {}^{c}\hat{\bm{D}}\right]$ it
can be deduced that

$\left[\begin{array}[]{cc}{}^{c}\hat{\bm{C}}&{}^{c}\hat{\bm{D}}\end{array}\right]^{T}\hat{\bm{Z}}_{11}\left[\begin{array}[]{cc}{}^{c}\hat{\bm{C}}&{}^{c}\hat{\bm{D}}\end{array}\right]\leavevmode\nobreak\
=\leavevmode\nobreak\ {\bm{0}}$ (19)

As a result, third inequality in (18) is a quadratic MI. This inequality is
linearised and solved about current solution as described in section 4.4 and
section 4.5 of work [17]. Proof of the lemma is given with the proof of
Theorem IV.2.

###### Theorem IV.2

Let ${}^{i}\bar{\bm{G}}$ and ${}^{i}\hat{\bm{G}}$ be dissipative agents with
state space realizations originates from (2) and extended as provided in (10).
The following two statements are valid:

  1. 1.

CS consisting of ${}^{i}\bar{\bm{G}}$ achieves induced $\mathcal{L}_{2}$ gain
performance of $\gamma^{*}$ for input ${}^{i}\bar{w}_{1}\in\ell_{2}^{N\cdot
n_{w1i}}$ for $i=\left[1,\cdots,N\right]$ if there exist matrices
${}^{i}{\bm{{X}}}\leavevmode\nobreak\ =\leavevmode\nobreak\
{}^{i}{\bm{{X}}}^{T}$, ${}^{i}\bar{\bm{Z}}_{11}$, ${}^{i}\bar{\bm{Z}}_{12}$,
${}^{i}\bar{\bm{Z}}_{22}$ and $\Upsilon$ such that LMIs in (12) is satisfied
for $i=\left[1,\cdots,N\right]$.

  2. 2.

CS consisting of ${}^{i}\hat{\bm{G}}$ have a robust performance of $\gamma$
such that $\gamma^{*}<\gamma$ for input
${}^{i}\hat{w}_{1}\in\ell_{2}^{N\cdot(N+1)\cdot n_{w1i}}$ for
$i=\left[1,\cdots,N\right]$

Proof. As denoted previously, states of ${}^{i}{\bm{G}}$ is not modified
during the extension to obtain ${}^{i}\bar{\bm{G}}$ and ${}^{i}\hat{\bm{G}}$.
Extended input vectors are denoted as
$\left[{}^{i}\bar{\bm{w}}_{1}^{T}\leavevmode\nobreak\
{}^{i}\bar{\bm{w}}^{T}\right]^{T}$ and
$\left[{}^{i}\hat{\bm{w}}_{1}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{w}}^{T}\right]^{T}$ for systems ${}^{i}\bar{\bm{G}}$ and
${}^{i}\hat{\bm{G}}$, respectively. This being said, second inequality in (12)
is pre- and post-multiplied with state-input vector of
$\left[{}^{i}{\bm{x}}^{T}\leavevmode\nobreak\
{}^{i}\bar{\bm{w}}_{1}^{T}\leavevmode\nobreak\
{}^{i}\bar{\bm{w}}^{T}\right]^{T}$ regarding to the rules defined in extension
given in (10). This yields the following results.

$\displaystyle{}^{i}{\bm{x}}^{T}({}^{i}{\bm{X}}\leavevmode\nobreak\
{}^{i}{\bm{A}}+{}^{i}{\bm{A}}^{T}\leavevmode\nobreak\
{}^{i}{\bm{X}}){}^{i}{\bm{x}}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{i}{\bm{x}}^{T}\leavevmode\nobreak\ {}^{i}{\bm{X}}\leavevmode\nobreak\
{}^{i}{\bm{B}}_{1}\leavevmode\nobreak\ {}^{i}{\bm{w}}_{1}\leavevmode\nobreak\
+\leavevmode\nobreak$ (20)
$\displaystyle{}^{i}{\bm{x}}^{T}\leavevmode\nobreak\
{}^{i}{\bm{X}}\leavevmode\nobreak\ {}^{i}{\bm{B}}\leavevmode\nobreak\
{}^{i}{\bm{w}}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{i}{\bm{w}}^{T}\leavevmode\nobreak\ {}^{i}{\bm{B}}^{T}\leavevmode\nobreak\
{}^{i}{\bm{X}}\leavevmode\nobreak\ {}^{i}{\bm{x}}\leavevmode\nobreak\
+\leavevmode\nobreak$ $\displaystyle{}^{i}{\bm{w}}_{1}^{T}\leavevmode\nobreak\
{}^{i}{\bm{B}}_{1}^{T}\leavevmode\nobreak\ {}^{i}{\bm{X}}\leavevmode\nobreak\
{}^{i}{\bm{x}}\leavevmode\nobreak\ -\leavevmode\nobreak\
{}^{i}{\bm{w}}_{1}^{T}{}^{i}{\bm{w}}_{1}\leavevmode\nobreak\
+\leavevmode\nobreak\
\frac{1}{\gamma^{2}}{}^{i}{\bm{z}}_{1}^{T}{}^{i}{\bm{z}}_{1}\leavevmode\nobreak\
+\leavevmode\nobreak$ $\displaystyle{}^{i}{\bm{z}}^{T}\leavevmode\nobreak\
{}^{i}\tilde{\bm{Z}}_{11}\leavevmode\nobreak\
{}^{i}{\bm{z}}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{i}{\bm{z}}^{T}\leavevmode\nobreak\
{}^{i}\tilde{\bm{Z}}_{12}\leavevmode\nobreak\
{}^{i}{\bm{w}}\leavevmode\nobreak\ +\leavevmode\nobreak$
$\displaystyle{}^{i}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{i}\tilde{\bm{Z}}_{21}\leavevmode\nobreak\
{}^{i}{\bm{z}}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{i}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{i}\tilde{\bm{Z}}_{22}\leavevmode\nobreak\
{}^{i}{\bm{w}}\leavevmode\nobreak\ \leq\leavevmode\nobreak\ 0$

$\displaystyle{}^{i}\dot{\bar{V}}\leavevmode\nobreak\ =\leavevmode\nobreak$
$\displaystyle{}^{i}{\bm{x}}^{T}({}^{i}{\bm{X}}\leavevmode\nobreak\
{}^{i}{\bm{A}}+{}^{i}{\bm{A}}^{T}\leavevmode\nobreak\
{}^{i}{\bm{X}}){}^{i}{\bm{x}}\leavevmode\nobreak\ +\leavevmode\nobreak\ $ (21)
$\displaystyle{}^{i}{\bm{x}}^{T}\leavevmode\nobreak\
{}^{i}{\bm{X}}\leavevmode\nobreak\ {}^{i}{\bm{B}}_{1}\leavevmode\nobreak\
{}^{i}{\bm{w}}_{1}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{i}{\bm{x}}^{T}\leavevmode\nobreak\ {}^{i}{\bm{X}}\leavevmode\nobreak\
{}^{i}{\bm{B}}\leavevmode\nobreak\ {}^{i}{\bm{w}}\leavevmode\nobreak\
+\leavevmode\nobreak\ $ $\displaystyle{}^{i}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{i}{\bm{B}}^{T}\leavevmode\nobreak\ {}^{i}{\bm{X}}\leavevmode\nobreak\
{}^{i}{\bm{x}}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{i}{\bm{w}}_{1}^{T}\leavevmode\nobreak\
{}^{i}{\bm{B}}_{1}^{T}\leavevmode\nobreak\ {}^{i}{\bm{X}}\leavevmode\nobreak\
{}^{i}{\bm{x}}$ $\displaystyle{}^{i}\bar{\Phi}\leavevmode\nobreak\
=\leavevmode\nobreak$ $\displaystyle{}^{i}{\bm{z}}^{T}\leavevmode\nobreak\
{}^{i}\tilde{\bm{Z}}_{11}\leavevmode\nobreak\
{}^{i}{\bm{z}}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{i}{\bm{z}}^{T}\leavevmode\nobreak\
{}^{i}\tilde{\bm{Z}}_{12}\leavevmode\nobreak\
{}^{i}{\bm{w}}\leavevmode\nobreak\ +\leavevmode\nobreak\ $
$\displaystyle{}^{i}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{i}\tilde{\bm{Z}}_{21}\leavevmode\nobreak\
{}^{i}{\bm{z}}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{i}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{i}\tilde{\bm{Z}}_{22}\leavevmode\nobreak\ {}^{i}{\bm{w}}$

Redefining (20) with ${}^{i}\dot{\bar{V}}$ and ${}^{i}\bar{\Phi}$ and summing
over $i$ yields

$\displaystyle\sum_{i=1}^{N}{}^{i}\dot{\bar{V}}\leavevmode\nobreak\
+\leavevmode\nobreak\
\sum_{i=1}^{N}(\frac{1}{\gamma^{2}}{}^{i}{\bm{z}}_{1}^{T}{}^{i}{\bm{z}}_{1}\leavevmode\nobreak\
-\leavevmode\nobreak\
{}^{i}{\bm{w}}_{1}^{T}{}^{i}{\bm{w}}_{1})\leavevmode\nobreak\
+\leavevmode\nobreak\ \sum_{i=1}^{N}{}^{i}\bar{\Phi}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 0.$ (22)

Integrating (22) along the admissible trajectory for
$\left[{}^{i}{\bm{x}}\leavevmode\nobreak\
{}^{i}\bar{\bm{w}}_{1}^{T}\leavevmode\nobreak\
{}^{i}\bar{\bm{w}}^{T}\right]^{T}$ from time $t=0$ to $t=T$ yields
$\bar{V}(\\{{}^{i}{\bm{x}}(T)\\}_{i=1}^{N})-\bar{V}(\\{{}^{i}{\bm{x}}(0)\\}_{i=1}^{N})+\frac{1}{\gamma^{2}}\int_{0}^{T}\sum_{i=1}^{N}{}^{i}{\bm{z}}_{1}^{T}{}^{i}{\bm{z}}_{1}dt-\int_{0}^{T}\sum_{i=1}^{N}{}^{i}{\bm{w}}_{1}^{T}{}^{i}{\bm{w}}_{1}dt+\int_{0}^{T}\sum_{i=1}^{N}{}^{i}\bar{\Phi}dt\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 0$.

Following the same procedure for second inequality for
$i=\left[1,\cdots,N\right]$ and third inequality for $i\leavevmode\nobreak\
=\leavevmode\nobreak\ c$ of (18), following results are obtained. This time
column matrix that is being pre- and post-multiplied is
$\left[{}^{i}{\bm{x}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{w}}_{1}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{w}}^{T}\right]^{T}$ recalling that rules of extension given in
(10) applies.

$\displaystyle{}^{i}{\bm{x}}^{T}({}^{i}\hat{\bm{X}}\leavevmode\nobreak\
{}^{i}{\bm{A}}+{}^{i}{\bm{A}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{X}}){}^{i}{\bm{x}}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{i}{\bm{x}}^{T}\leavevmode\nobreak\ {}^{i}\hat{\bm{X}}\leavevmode\nobreak\
{}^{i}{\bm{B}}_{1}\leavevmode\nobreak\ {}^{i}{\bm{w}}_{1}\leavevmode\nobreak\
+\leavevmode\nobreak$ (23)
$\displaystyle{}^{i}{\bm{x}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{X}}\leavevmode\nobreak\ {}^{i}{\bm{B}}\leavevmode\nobreak\
{}^{i}{\bm{w}}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{i}{\bm{w}}^{T}\leavevmode\nobreak\ {}^{i}{\bm{B}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{X}}\leavevmode\nobreak\ {}^{i}{\bm{x}}\leavevmode\nobreak\
+\leavevmode\nobreak$ $\displaystyle{}^{i}{\bm{w}}_{1}^{T}\leavevmode\nobreak\
{}^{i}{\bm{B}}_{1}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{X}}\leavevmode\nobreak\ {}^{i}{\bm{x}}\leavevmode\nobreak\
-\leavevmode\nobreak\
{}^{i}{\bm{w}}_{1}^{T}{}^{i}{\bm{w}}_{1}\leavevmode\nobreak\
+\leavevmode\nobreak\
\frac{1}{\gamma^{2}}{}^{i}{\bm{z}}_{1}^{T}{}^{i}{\bm{z}}_{1}\leavevmode\nobreak\
+\leavevmode\nobreak$ $\displaystyle{}^{i}{\bm{z}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{Z}}_{11}\leavevmode\nobreak\ {}^{i}{\bm{z}}\leavevmode\nobreak\
+\leavevmode\nobreak\ {}^{i}{\bm{z}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{Z}}_{12}\leavevmode\nobreak\ {}^{i}{\bm{w}}\leavevmode\nobreak\
+\leavevmode\nobreak$ $\displaystyle{}^{i}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{Z}}_{21}\leavevmode\nobreak\ {}^{i}{\bm{z}}\leavevmode\nobreak\
+\leavevmode\nobreak\ {}^{i}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{Z}}_{22}\leavevmode\nobreak\ {}^{i}{\bm{w}}\leavevmode\nobreak\
+\leavevmode\nobreak$ $\displaystyle\leavevmode\nobreak\ -\leavevmode\nobreak\
{}^{c}{\bm{w}}_{1}^{T}{}^{c}{\bm{w}}_{1}\leavevmode\nobreak\
+\leavevmode\nobreak\
\frac{1}{\gamma^{2}}{}^{c}{\bm{z}}_{1}^{T}{}^{c}{\bm{z}}_{1}\leavevmode\nobreak\
+\leavevmode\nobreak$ $\displaystyle{}^{c}{\bm{z}}^{T}\leavevmode\nobreak\
{}^{c}\hat{\bm{Z}}_{11}\leavevmode\nobreak\ {}^{c}{\bm{z}}\leavevmode\nobreak\
+\leavevmode\nobreak\ {}^{c}{\bm{z}}^{T}\leavevmode\nobreak\
{}^{c}\hat{\bm{Z}}_{12}\leavevmode\nobreak\ {}^{c}{\bm{w}}\leavevmode\nobreak\
+\leavevmode\nobreak$ $\displaystyle{}^{c}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{c}\hat{\bm{Z}}_{21}\leavevmode\nobreak\ {}^{c}{\bm{z}}\leavevmode\nobreak\
+\leavevmode\nobreak\ {}^{c}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{c}\hat{\bm{Z}}_{22}\leavevmode\nobreak\ {}^{c}{\bm{w}}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 0$

$\displaystyle{}^{i}\dot{\hat{V}}\leavevmode\nobreak\ =\leavevmode\nobreak$
$\displaystyle{}^{i}{\bm{x}}^{T}({}^{i}\hat{\bm{X}}\leavevmode\nobreak\
{}^{i}{\bm{A}}+{}^{i}{\bm{A}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{X}}){}^{i}{\bm{x}}\leavevmode\nobreak\ +\leavevmode\nobreak\ $
(24) $\displaystyle{}^{i}{\bm{x}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{X}}\leavevmode\nobreak\ {}^{i}{\bm{B}}_{1}\leavevmode\nobreak\
{}^{i}{\bm{w}}_{1}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{i}{\bm{x}}^{T}\leavevmode\nobreak\ {}^{i}\hat{\bm{X}}\leavevmode\nobreak\
{}^{i}{\bm{B}}\leavevmode\nobreak\ {}^{i}{\bm{w}}\leavevmode\nobreak\
+\leavevmode\nobreak\ $ $\displaystyle{}^{i}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{i}{\bm{B}}^{T}\leavevmode\nobreak\ {}^{i}\hat{\bm{X}}\leavevmode\nobreak\
{}^{i}{\bm{x}}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{i}{\bm{w}}_{1}^{T}\leavevmode\nobreak\
{}^{i}{\bm{B}}_{1}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{X}}\leavevmode\nobreak\ {}^{i}{\bm{x}}$
$\displaystyle{}^{i}\hat{\Phi}\leavevmode\nobreak\ =\leavevmode\nobreak$
$\displaystyle{}^{i}{\bm{z}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{Z}}_{11}\leavevmode\nobreak\ {}^{i}{\bm{z}}\leavevmode\nobreak\
+\leavevmode\nobreak\ {}^{i}{\bm{z}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{Z}}_{12}\leavevmode\nobreak\ {}^{i}{\bm{w}}\leavevmode\nobreak\
+\leavevmode\nobreak\ $ $\displaystyle{}^{i}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{Z}}_{21}\leavevmode\nobreak\ {}^{i}{\bm{z}}\leavevmode\nobreak\
+\leavevmode\nobreak\ {}^{i}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{Z}}_{22}\leavevmode\nobreak\ {}^{i}{\bm{w}}$
$\displaystyle{}^{c}\hat{\Phi}\leavevmode\nobreak\ =\leavevmode\nobreak$
$\displaystyle{}^{c}{\bm{z}}^{T}\leavevmode\nobreak\
{}^{c}\hat{\bm{Z}}_{11}\leavevmode\nobreak\ {}^{c}{\bm{z}}\leavevmode\nobreak\
+\leavevmode\nobreak\ {}^{c}{\bm{z}}^{T}\leavevmode\nobreak\
{}^{c}\hat{\bm{Z}}_{12}\leavevmode\nobreak\ {}^{c}{\bm{w}}\leavevmode\nobreak\
+\leavevmode\nobreak\ $ $\displaystyle{}^{c}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{c}\hat{\bm{Z}}_{21}\leavevmode\nobreak\ {}^{c}{\bm{z}}\leavevmode\nobreak\
+\leavevmode\nobreak\ {}^{c}{\bm{w}}^{T}\leavevmode\nobreak\
{}^{c}\hat{\bm{Z}}_{22}\leavevmode\nobreak\ {}^{c}{\bm{w}}$

$\displaystyle\sum_{i=1}^{N}{}^{i}\dot{\hat{V}}\leavevmode\nobreak\
+\leavevmode\nobreak\
\sum_{i=1}^{N}(\frac{1}{\gamma^{2}}{}^{i}{\bm{z}}_{1}^{T}{}^{i}{\bm{z}}_{1}\leavevmode\nobreak\
-\leavevmode\nobreak\
{}^{i}{\bm{w}}_{1}^{T}{}^{i}{\bm{w}}_{1})\leavevmode\nobreak\
+\leavevmode\nobreak$ (25)
$\displaystyle\sum_{i=1}^{N}{}^{i}\hat{\Phi}\leavevmode\nobreak\
+\leavevmode\nobreak\ {}^{c}\hat{\Phi}\leavevmode\nobreak\
+\leavevmode\nobreak\
(\frac{1}{\gamma^{2}}{}^{c}{\bm{z}}_{1}^{T}{}^{c}{\bm{z}}_{1}\leavevmode\nobreak\
-\leavevmode\nobreak\
{}^{c}{\bm{w}}_{1}^{T}{}^{c}{\bm{w}}_{1})\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 0$

$\sum_{i=1}^{N}{}^{i}\hat{\Phi}\leavevmode\nobreak\ +\leavevmode\nobreak\
{}^{c}\hat{\Phi}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0$ due to the fact
that interconnections are neutral, which is dictated to the inequality by
${}^{i}\hat{\bm{Z}}$ and ${}^{c}\hat{\bm{Z}}$ matrices.

$\displaystyle\sum_{i=1}^{N}{}^{i}\dot{\hat{V}}\leavevmode\nobreak\
+\leavevmode\nobreak\
\sum_{i=1}^{N}(\frac{1}{\gamma^{2}}{}^{i}{\bm{z}}_{1}^{T}{}^{i}{\bm{z}}_{1}\leavevmode\nobreak\
-\leavevmode\nobreak\
{}^{i}{\bm{w}}_{1}^{T}{}^{i}{\bm{w}}_{1})\leavevmode\nobreak\
+\leavevmode\nobreak$ (26)
$\displaystyle(\frac{1}{\gamma^{2}}{}^{c}{\bm{z}}_{1}^{T}{}^{c}{\bm{z}}_{1}\leavevmode\nobreak\
-\leavevmode\nobreak\
{}^{c}{\bm{w}}_{1}^{T}{}^{c}{\bm{w}}_{1})\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 0$

Integrating (26) along the admissible trajectory for
$\left[{}^{i}{\bm{x}}\leavevmode\nobreak\
{}^{i}\hat{\bm{w}}_{1}^{T}\leavevmode\nobreak\
{}^{i}\hat{\bm{w}}^{T}\right]^{T}$ from time $t=0$ to $t=T$ yields
$\hat{V}(\\{{}^{i}{\bm{x}}(T)\\}_{i=1}^{N})-\hat{V}(\\{{}^{i}{\bm{x}}(0)\\}_{i=1}^{N})+\frac{1}{\gamma^{2}}\int_{0}^{T}\sum_{i=1}^{N}{}^{i}{\bm{z}}_{1}^{T}{}^{i}{\bm{z}}_{1}dt-\int_{0}^{T}\sum_{i=1}^{N}{}^{i}{\bm{w}}_{1}^{T}{}^{i}{\bm{w}}_{1}dt+\frac{1}{\gamma^{2}}\int_{0}^{T}{}^{c}{\bm{z}}_{1}^{T}{}^{c}{\bm{z}}_{1}dt-\int_{0}^{T}{}^{c}{\bm{w}}_{1}^{T}{}^{c}{\bm{w}}_{1}dt\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 0$.

To prove $H_{\infty}$ performance, assume
${}^{i}{\bm{x}}(0)\leavevmode\nobreak\ =\leavevmode\nobreak\ {\bm{0}}$ and
${}^{i}{\bm{x}}(T)\leavevmode\nobreak\ \neq\leavevmode\nobreak\ {\bm{0}}$. Let
${}^{i}{\bm{w}}_{1}\leavevmode\nobreak\ \neq\leavevmode\nobreak\ {\bm{0}}$ and
${}^{c}{\bm{w}}_{1}\leavevmode\nobreak\ \neq\leavevmode\nobreak\ {\bm{0}}$
combined as
${\bm{w}}_{1}=\text{col}\\{\text{col}\\{{}^{i}{\bm{w}}_{1}\\}_{i=1}^{N},\leavevmode\nobreak\
{}^{c}{\bm{w}}_{1}\\}$. Let ${}^{i}{\bm{z}}_{1}$ and ${}^{c}{\bm{z}}_{1}$
combined as
${\bm{z}}_{1}=\text{col}\\{\text{col}\\{{}^{i}{\bm{z}}_{1}\\}_{i=1}^{N},\leavevmode\nobreak\
{}^{c}{\bm{z}}_{1}\\}$. Then from (26) we get

$\frac{\|{\bm{z}}_{1}\|_{2}^{2}}{\|{\bm{w}}_{1}\|_{2}^{2}}\leq\gamma^{2}$ (27)

for the CS an this completes the proof for Lemma IV.1.

As seen in inequality (26), modified CS definition replaces
$\int_{0}^{T}\sum_{i=1}^{N}{}^{i}\bar{\Phi}dt$ with
$\frac{1}{\gamma^{2}}\int_{0}^{T}{}^{c}{\bm{z}}_{1}^{T}{}^{c}{\bm{z}}_{1}dt-\int_{0}^{T}{}^{c}{\bm{w}}_{1}^{T}{}^{c}{\bm{w}}_{1}dt$,
which is bounded quantity that depends on ${\bm{\Upsilon}}$.

If one would seek neutrality condition (14) for the original CS (see Figure 2)
and able to solve non-linear MI given in (12), this means that each agent is
dissipative. On top of that, due to (14), (28) is true and this leads to (29).

$\int_{0}^{T}\sum_{i=1}^{N}{}^{i}\bar{\Phi}dt\leavevmode\nobreak\
=\leavevmode\nobreak\ 0$ (28)

$\displaystyle\bar{V}(\\{{}^{i}{\bm{x}}(T)\\}_{i=1}^{N})-\bar{V}(\\{{}^{i}{\bm{x}}(0)\\}_{i=1}^{N})+$
(29)
$\displaystyle\frac{1}{\gamma^{2}}\int_{0}^{T}\sum_{i=1}^{N}{}^{i}{\bm{z}}_{1}^{T}{}^{i}{\bm{z}}_{1}dt-\int_{0}^{T}\sum_{i=1}^{N}{}^{i}{\bm{w}}_{1}^{T}{}^{i}{\bm{w}}_{1}dt$
$\displaystyle\leavevmode\nobreak\ \leq\leavevmode\nobreak\ 0$

Equation (26) infer that

$\displaystyle\hat{V}(\\{{}^{i}{\bm{x}}(T)\\}_{i=1}^{N})-\hat{V}(\\{{}^{i}{\bm{x}}(0)\\}_{i=1}^{N})+$
(30)
$\displaystyle\frac{1}{\gamma^{2}}\int_{0}^{T}\sum_{i=1}^{N}{}^{i}{\bm{z}}_{1}^{T}{}^{i}{\bm{z}}_{1}dt-\int_{0}^{T}\sum_{i=1}^{N}{}^{i}{\bm{w}}_{1}^{T}{}^{i}{\bm{w}}_{1}dt+$
$\displaystyle\frac{1}{\gamma^{2}}\int_{0}^{T}{}^{c}{\bm{z}}_{1}^{T}{}^{c}{\bm{z}}_{1}dt-\int_{0}^{T}{}^{c}{\bm{w}}_{1}^{T}{}^{c}{\bm{w}}_{1}dt\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 0$

Let adjacency matrix be a weighted adjacency matrix and note that
$\hat{V}(\\{{}^{i}{\bm{x}}(0)\\}_{i=1}^{N})\leavevmode\nobreak\
=\leavevmode\nobreak\ \bar{V}(\\{{}^{i}{\bm{x}}(0)\\}_{i=1}^{N})$ for
${}^{i}{\bm{x}}(0)\leavevmode\nobreak\ =\leavevmode\nobreak\ {\bm{0}}$. Then
from the subtraction of (29) from (30), it can be calculated that
$\hat{V}(\\{{}^{i}{\bm{x}}(T)\\}_{i=1}^{N})\leavevmode\nobreak\
-\leavevmode\nobreak\
\bar{V}(\\{{}^{i}{\bm{x}}(T)\\}_{i=1}^{N})\leavevmode\nobreak\
\leq\leavevmode\nobreak\ \alpha$, where $0\leavevmode\nobreak\
\leq\leavevmode\nobreak\ \alpha\leavevmode\nobreak\ \leq\leavevmode\nobreak\
(\|{\bm{\Upsilon}}\|_{2}-1)\|{}^{c}{\bm{w}}_{1}\|_{2}^{2}$. Later inequality
shows that, induced $\mathcal{L}_{2}$ norm of of ${}^{i}\hat{\bm{G}}$ is
greater or equal than the one of ${}^{i}\bar{\bm{G}}$ and this completes the
proof of Theorem IV.2.

##  V NUMERICAL VERIFICATION

Methodology described under Section IV, is verified using a CS of four
vehicles. Adjacency matrix for lumped and distributed cases are compared in
terms of edge weights ${\upsilon}$ and the robust performance level given by
$\gamma$. Simulations are executed on a PC with Intel(R) Core(TM) i7-4720HQ
CPU @2.6GHz, 16GB RAM running on Windows 10 OS and MATLAB 2019b.

Dynamics of each agent is defined using state space representation given in
(31).

${}^{i}{\bm{G}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\left[\begin{array}[]{cc|c}0&1&0\\\ -2&-2&1\\\ \hline\cr
1&0&0\end{array}\right]$ (31)

Agents are connected to each other with an adjacency matrix,
${\bm{\Upsilon}}$, which is defined as a function of a free scalar variable
$\alpha$ and this is provided in (32). $\alpha$ is contained in an interval
defined as $\alpha\leavevmode\nobreak\ =\leavevmode\nobreak\
\left[0,\leavevmode\nobreak\ 0.2\right]$. This interval is selected
arbitrarily without violating the fact that edge weight ${\upsilon}$ is
$0\leq{\upsilon}\leq 1$.

${\bm{\Upsilon}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\left[\begin{array}[]{cccc}0&2\alpha&3\alpha&1-5\alpha\\\
\alpha&0&3\alpha&1-4\alpha\\\ \alpha&2\alpha&0&1-3\alpha\\\
\alpha&2\alpha&1-3\alpha&0\end{array}\right]$ (32)

Using (31), ${\bm{S}}$ is constructed for $N=4$. Then ${\bm{H}}$ is created as
described in (4). For this lumped model of the CS, $H_{\infty}$ performance is
given in (5), which is a BMI optimization problem and size of the decision
variable is ${\bm{\mathcal{X}}}\in\mathbb{S}^{8}$. On top of that, there is
$\alpha\in\mathbb{R}$. Thus, total number of variables to be solved for this
case is $(8\cdot 9)/2+1=37$. Although it is out of the scope of this paper,
for proper comparison the method used to solve this problem should be noted.
This problem is solved using sequential LMI method using convex-concave
decompositions in 16.8125 s. Optimal values for $\alpha$ and $\gamma$ are
given as $\alpha_{optimal}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.1698$
and $\gamma_{optimal}\leavevmode\nobreak\ =\leavevmode\nobreak\ 1.0153$ these
values are provided on Figure 4 with red diamond. Purple diamond represents
the initial guess while green squares provided the calculated values for
$\gamma$ and $\alpha$ in every iteration.

Figure 4: Synthesis results for lumped case

For distributed case, agent dynamics is taken from the same system definition,
which is given in (31). Decision matrices denoted as
${}^{i}{\bm{X}}\in\mathbb{S}^{2}$ has $4\cdot(2\cdot 3)/2=12$ independent
variables in total for $i=\left[1,\cdots,4\right]$. In addition to that, there
are matrix variables ${}^{i}{\bm{Y}}_{11}\in\mathbb{S}^{4}$, which results in
$((4\cdot 5)/2)\cdot 4=40$. Most of these variables are multiplied with zeros
due to the extension given in (10) and disappear. Eventually, from each agent
$4$ and in total $16$ independent variables are being calculated out of $40$.
From ${}^{c}\hat{\bm{G}}$, we have 16 variables du to
${}^{c}{\bm{Y}}_{11}\in\mathbb{S}^{4}$. Finally, $\alpha$ should be accounted
in and this gives an overall $45$ variables for the whole simulation. The
distributed edge weight synthesis problem is solved in 18.3750 s. Optimal
values for $\alpha$ and $\gamma$ are given as
$\alpha_{optimal}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.1602$ and
$\gamma_{optimal}\leavevmode\nobreak\ =\leavevmode\nobreak\ 1.0712$ and these
values are provided on Figure 5 with red diamond. Initially, solver starts
from the initial guess provided with the purple diamond, where $\alpha=0.05$.
Compared to results of lumped method given in Figure 4, results of distributed
case given in Figure 5 show greater curvature and a greater infimum for the
induced $\mathcal{L}_{2}$ norm for the CS.

Figure 5: Synthesis results for distributed case.

Finally, a computational time analysis is executed for lumped and distributed
cases, where number of agents for lumped and distributed cases are designated
as $N_{lumped}=\left[3,\leavevmode\nobreak\ 4,\leavevmode\nobreak\ 5\right]$
and $N_{dist}=\left[3,\leavevmode\nobreak\ 4,\leavevmode\nobreak\
5,\leavevmode\nobreak\ 6,\leavevmode\nobreak\ 7,\leavevmode\nobreak\
8,\leavevmode\nobreak\ 9,\leavevmode\nobreak\ 10,\leavevmode\nobreak\
11,\leavevmode\nobreak\ 12\right]$. In both set of experiments, topology of
the adjacency matrix is kept such that (32) for first four agents are fixed
and each added agent listens to Agents $1$ and $N-1$. In $N=3$ case, first row
and column is taken out. Under these conditions, Figure 6 shows that
computation time increases quadratically as the number of agents are
increasing for the lumped case. Computation time shows quadratic behaviour for
distributed case.

Figure 6: Computation time analysis of distributed and lumped methods.

##  VI CONCLUSIONS

This paper focuses on finding the optimal values for edge weights in terms of
enhancing $H_{\infty}$ performance criteria of a CS in a distributed fashion.
Originally, this is an optimization problem with a BMI constraint, however,
when the CS is modeled in a distributed fashion problem includes non-linear
MIs as constraints. This problem is solved by modeling communication media as
a synthetic agent denoted as ${}^{c}{\bm{G}}$ and making ideal
interconnections between original agents and this synthetic agent. By this
way, information on adjacency matrix is secured and distributed edge weight
synthesis using dissipative system theory with neutrality condition on
interconnections is executed.

Edge weight synthesis is illustrated on a cooperative system with four
vehicles. Given the topology, edge weights of these systems are synthesized
and results reveal following key conclusions.

  1. 1.

Distributed method proposed in this paper is able to reach the optimal
$\alpha$ value that is being calculated for lumped CS. The error between these
values are 0.0099, which is 5.83% error. Difference in curvature and infimum
of the induced $\mathcal{L}_{2}$ norm is due to $\|{\bm{\Upsilon}}\|_{2}$ as
explained in the proof of Theorem IV.2.

  2. 2.

In numerical verification, each agent has two states and this gives an
advantage to lumped synthesis as there are 37 independent variables to be
solved. In distributed case, there are 45 variables in total. It is clearly
seen that increasing number of agents results in a quadratic increase in
computation time for lumped case. The quadratic relationship can be given as
$(n_{xi}\cdot N)^{2}$. On the other hand, number of variables to be solved in
distributed case grow with a relationship of $n_{xi}^{2}\cdot{N}+n_{wi}\cdot
N\cdot(N+1)$.

  3. 3.

These relationships are visible in Figure 6 as complexity of the problem so
the calculation time of the optimization problem increases quadratically for
lumped method; However, computational time trend for distributed method shows
a gradual shift from linear to quadratic as the number of agents increase.
Dominance of these trends are decided by the size of state and spatial
input/output vector of single agent. In addition to that number of states and
number of spatial inputs/outputs to single agent define, when one method is
superior to the other. Specifically, a larger spatial input/output vector for
same state vector pushes the number of agents, when distributed method become
superior, to higher values. It should also be noted that, there is a certain
threshold for spatial input/output to state vector ratio that distributed
method is no more superior for any number of agents.

## ACKNOWLEDGMENT

This work is supported by the Office of Naval Research via award number
N00014-18-1-2215. The authors thank the anonymous reviewers for critically
reading the manuscript and suggesting substantial improvements. This paper is
accepted to CDC 2021 [21].

## References

  * [1] W. Yu, Y. Yao, and W. Chen, “Analytical cooperative entry guidance for rendezvous and formation flight,” _Acta Astronautica_ , vol. 171, pp. 118–138, 2020, doi: https://doi.org/10.1016/j.actaastro.2020.02.044.
  * [2] L. Bai, L. Zhu, X. Zhang, W. Zhang, and Q. Yu, “Multi-satellite relay transmission in 5g: Concepts, techniques, and challenges,” _IEEE Network_ , vol. 32, no. 5, pp. 38–44, 2018, doi: http://doi.org/10.1109/MNET.2018.1800038.
  * [3] F. de Ponte Müller, E. M. Diaz, and I. Rashdan, “Cooperative positioning and radar sensor fusion for relative localization of vehicles,” in _2016 IEEE Intelligent Vehicles Symposium (IV)_ , 2016, doi: http://doi.org/10.1109/IVS.2016.7535520, pp. 1060–1065.
  * [4] D. Ding, Q. Han, Z. Wang, and X. Ge, “A survey on model-based distributed control and filtering for industrial cyber-physical systems,” _IEEE Transactions on Industrial Informatics_ , vol. 15, no. 5, pp. 2483–2499, 2019, doi: http://doi.org/10.1109/TII.2019.2905295.
  * [5] L. Ding, Q. Han, L. Y. Wang, and E. Sindi, “Distributed cooperative optimal control of dc microgrids with communication delays,” _IEEE Transactions on Industrial Informatics_ , vol. 14, no. 9, pp. 3924–3935, 2018, doi: http://doi.org/10.1109/TII.2018.2799239.
  * [6] P. B. g. Dohmann and S. Hirche, “Distributed control for cooperative manipulation with event-triggered communication,” _IEEE Transactions on Robotics_ , vol. 36, no. 4, pp. 1038–1052, 2020, doi: http://doi.org/10.1109/TRO.2020.2973096.
  * [7] E. P. van Horssen and S. Weiland, “Synthesis of distributed robust h-infinity controllers for interconnected discrete time systems,” _IEEE Transactions on Control of Network Systems_ , vol. 3, no. 3, pp. 286–295, 2016, doi: http://doi.org/10.1109/TCNS.2015.2476199.
  * [8] A. Ghosh, S. Boyd, and A. Saberi, “Minimizing effective resistance of a graph,” _SIAM Rev._ , vol. 50, no. 1, p. 37–66, Feb. 2008, doi: https://doi.org/10.1137/050645452.
  * [9] S. Y. Shafi, M. Arcak, and L. El Ghaoui, “Designing node and edge weights of a graph to meet laplacian eigenvalue constraints,” pp. 1016–1023, 2010, doi: https://doi.org/10.1109/ALLERTON.2010.5707021.
  * [10] S. Hassan-Moghaddam and M. R. Jovanović, “Topology design for stochastically forced consensus networks,” _IEEE Transactions on Control of Network Systems_ , vol. 5, no. 3, pp. 1075–1086, 2018, doi: http://doi.org/10.1109/TCNS.2017.2674962.
  * [11] D. Yuan, D. W. C. Ho, and S. Xu, “Regularized primal–dual subgradient method for distributed constrained optimization,” _IEEE Transactions on Cybernetics_ , vol. 46, no. 9, pp. 2109–2118, 2016, doi: http://doi.org/10.1109/TCYB.2015.2464255.
  * [12] S. Huang, P. Li, Q. Wu, F. Li, and F. Rong, “Admm-based distributed optimal reactive power control for loss minimization of dfig-based wind farms,” _International Journal of Electrical Power & Energy Systems_, vol. 118, p. 105827, 2020, doi: https://doi.org/10.1016/j.ijepes.2020.105827.
  * [13] C. Langbort, R. S. Chandra, and R. D’Andrea, “Distributed control design for systems interconnected over an arbitrary graph,” _IEEE Transactions on Automatic Control_ , vol. 49, no. 9, pp. 1502–1519, 2004, doi: http://doi.org/10.1109/TAC.2004.834123.
  * [14] M. Babazadeh and A. Nobakhti, “Direct synthesis of fixed-order ${\cal h}_{\infty}$ controllers,” _IEEE Transactions on Automatic Control_ , vol. 60, no. 10, pp. 2704–2709, 2015, doi: http://doi.org/10.1109/TAC.2015.2461832.
  * [15] T. Anderson, C.-Y. Chang, and S. Martínez, “Distributed approximate newton algorithms and weight design for constrained optimization,” _Automatica_ , vol. 109, p. 108538, 2019, doi: https://doi.org/10.1016/j.automatica.2019.108538.
  * [16] C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective output-feedback control via lmi optimization,” _IEEE Transactions on Automatic Control_ , vol. 42, no. 7, pp. 896–911, 1997 doi: https://doi.org/10.1109/9.599969.
  * [17] Y. Wang, R. Rajamani, and A. Zemouche, “Sequential lmi approach for the design of a bmi-based robust observer state feedback controller with nonlinear uncertainties,” _International Journal of Robust and Nonlinear Control_ , vol. 28, no. 4, pp. 1246–1260, 2018, doi: http://doi.org/10.1002/rnc.3948.
  * [18] J. W. Simpson-Porco, “Equilibrium-independent dissipativity with quadratic supply rates,” _IEEE Transactions on Automatic Control_ , vol. 64, no. 4, pp. 1440–1455, 2019, doi: http://doi.org/10.1109/TAC.2018.2838664.
  * [19] C.-e. Park, N. K. Kwon, and P. Park, “New bounded real lemma for singular markovian jump systems: Application to $h_{\infty}$ control,” _International Journal of Robust and Nonlinear Control_ , vol. 31, no. 3, pp. 907–919, 2021, doi: https://doi.org/10.1002/rnc.5340.
  * [20] A. Zecevic and D. Siljak, _Control of Complex Systems. Structural Constraints and Uncertainty_. Springer Science & Business Media, 01 2010, doi: https://doi.org/10.1007/978-1-4419-1216-9, ch. 6.
  * [21] _Distributed $H_{\infty}$ Edge Weight Synthesis for Cooperative Systems_, Austin, TX, USA, December 2021 doi: not published yet.