#  On Structured Lyapunov Functions and Dissipativity in Interconnected LTI
Systems

Andrej Jokić and Ivica Nakić A. Jokić is with the Faculty of Mechanical
Engineering and Naval Architecture, University of Zagreb, Ivana Lučića 5,
10000 Zagreb, Croatia, tel: +385921013815, e-mail<EMAIL_ADDRESS>Nakić is with the Department of Mathematics, Faculty of Science, University of
Zagreb, e-mail<EMAIL_ADDRESS>address: A. Jokić, Faculty of
Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana
Lučića 5, 10000 Zagreb, Croatia.This research has been supported by the
Croatian Science Foundation under the projects 9354 “Control of Spatially
Distributed Systems”, 9345 “MMACACMS”, IP-2016-06-2468 “Control of Dynamical
Systems” , and by the European Regional Development Fund under the grant
KK.01.1.1.01.0009 (DATACROSS).

###### Abstract

In this paper we study connections between structured storage or Lyapunov
functions of a class of interconnected systems (dynamical networks) and
dissipativity properties of the individual systems. We prove that if a
dynamical network, composed as a set of linear time invariant (LTI) systems
interconnected over an acyclic graph, admits an additive quadratic Lyapunov
function, then the individual systems in the network are dissipative with
respect to a (nonempty) set of interconnection neutral supply functions. Each
supply function from this set is defined on a single interconnection link in
the network. Specific characterizations of neutral supply functions are
presented which imply robustness of network stability/dissiptivity to removal
of interconnection links.

##  1 INTRODUCTION

In this paper we consider dissipative dynamical systems as introduced by Jan
Willems in [12], [13]. The dynamical system $\dot{x}=f(x,d)$, $z=g(x,d)$, is
defined to be dissipative with respect to the supply function $s(\cdot,\cdot)$
if there exists a storage $V:X\rightarrow\mathbb{R}$ such that the dissipation
inequality

$V(x(t_{1}))\leq V(x(t_{0}))+\int_{t_{0}}^{t_{1}}s(d(t),z(t))dt$ (1)

holds for all trajectories of $x,d,z$ satisfying the system’s dynamics, for
all $t\in[t_{0},t_{1}]$ and for all $t_{0}<t_{1}$. The supply function
quantifies the power supplied to (or extracted from) the system, while a
storage function111In general definition we do not require a storage function
to be non-negative, in accordance with the definition in [14]. In the paper we
explicitly indicate when some particular storage function is assumed/required
to be non-negative. quantifies the energy stored within the system at any
given moment. The dissipation inequality implies that the difference in the
stored energy over any finite time interval cannot exceed the amount of energy
supplied to the system over the same time interval.

Being an extension of Lyapunov theory to open systems, i.e., systems with
exogenous inputs and outputs, with inflow/outflow of power, the dissipativity
theory is one of the major tools in both _i)_ robust control theory (see,
e.g., [3, 11]), where many of the problems can be formulated, solved or
interpreted in this framework; and _ii)_ stability analysis/control synthesis
for large scale systems, see e.g. [8] for classical results and, e.g., [2, 7]
for some more recent results.

Loosely speaking, in robust control, supply functions are used to model the
way uncertainty in the system processes power. In large scale systems, supply
functions are used to capture the power exchanges between the subsystems. In
both cases, the notion of interconnection neutral supply functions, introduced
in [12], often plays a central role.

One of the results from [12] (Theorem 5 from [12]) states that dissipative
systems which are interconnected via a neutral interconnection constraint
define a new dissipative system where the sum of storage functions of the
individual subsystems is a storage function for the overall interconnected
system. In this paper we are concerned with the converse statement: _if a set
of interconnected systems is disspative (stable) with a storage function
(Lyapunov function) characterized by an additive structure 222In this context
we say that a function has an _additive structure_ if it is represented as a
sum of local functions, where each local function is defined on a level of an
individual system in a network., does then also necessarily exist a set of
interconnection neutral supply functions with respect to which the individual
systems in the network are dissipative?_

In this paper we consider linear time invariant (LTI) systems and quadratic
supply and storage/Lyapunov functions. The main contributions are summarized
as follows:

  * •

In the case of two open interconnected LTI systems, existence of an additive
storage function for an external supply rate with an additive structure
implies existence of interconnection neutral supply functions. In the case of
an autonomous interconnected LTI system, existence of a quadratic Lyapunov
function with an additive structure implies existence of interconnection
neutral supply functions.

  * •

Generalization of the above results to acyclic networks333Precise definition
of what we mean by acyclic network is presented in Section 4. This definition
allows for two adjacent systems ($G_{i}$, $G_{j}$) in the network to be
interconnected in feedback loops (there are outputs from $G_{i}$ acting as
inputs to $G_{j}$ and vice-versa), but places restrictions on having more
general cycles (loops) in the interconnections. of interconnected LTI systems.

  * •

Specific characterizations of neutral supply functions are presented which
imply robustness of network stability/dissiptivity to a removal of
interconnection links. Based on these results, we present sufficient
conditions under which acyclic networks are robustly stable with respect to
removal/disconnection of an interconnection link; as well as sufficient
conditions under which networks with cycles are robustly stable with respect
to removal/disconnection of a system.

To the best of our knowledge, and surprisingly, such (and similar) converse
statements to [12, Theorem 5] have not been presented yet, with exception of
our conference paper [5] which reported preliminary results on this topic. In
[5] we have presented a proof of existence of neutral supply rates for two
interconnected systems with no direct feed-through matrices (no algebraic
loop) and indicated that the generalization to acyclic networks is possible,
but with no detailed proof presented. In contrast, this paper contains
results, with detailed proofs, for systems with algebraic loops,
generalization to open systems and acyclic networks.

The remainder of the paper is organized as follows. In Section 2 we define the
notions of dissipativity and interconnection neutral supply functions and
present several results from [12] to set the stage for the main results of the
paper. The main results of the paper are presented in Sections 3, 4 and 5. In
Section 3 we consider interconnection of two systems only. These results are
further extended in Section 4 to accommodate arbitrary acyclic networks.
Characteristics of interconnection neutral supply rates and certain robustness
properties of a network are mutually related in Section 5. A numerical example
is presented in Section 6. To make the presentation more clear, most of the
proofs are collected in a single section, Section 7. Conclusions are
summarized in Section 8. There are two appendices included which contain
either well known (Appendix Appendix A: Robust stability and robust
dissipativity) or novel isolated results (Appendix Appendix B) which are used
in the proofs in Section 7.

##  2 PRELIMINARIES

###  2.1 Notation and terminology

Let $\mathbb{R}$ denote the field of real numbers and let $\mathbb{R}^{m\times n}$ denote $m$ by $n$ matrices with elements in $\mathbb{R}$. $I_{n}$ is the identity matrix with dimension $n$. Index $n$ will be omitted when the dimension is clear from the context. The transpose of a matrix $A$ is denoted by $A^{\top}$. $\operatorname{Ker}{A}$ and $\operatorname{Im}{A}$ are used to denote the kernel and the image space of $A$, respectively. The operator $\operatorname{col}(\cdot,\ldots,\cdot)$ stacks its operands into a column vector. For a set of (not necessarily square) matrices $\\{M_{1},M_{2},\ldots,M_{n}\\}$ we use $\operatorname{diag}(M_{1},\ldots,M_{n})$ to denote the matrix $\begin{pmatrix}M_{1}&0&\cdots&0\\\ 0&M_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&M_{n}\end{pmatrix}$.  The matrix inequalities $A\succ B$ ($A\prec B$) and $A\succeq B$ ($A\preceq B$) mean $A$ and $B$ are symmetric and $A-B$ is positive definite (negative definite) and positive semi-definite (negative semi-definite), respectively. For a transfer matrix $G$ with realization $G(s)=C(sI-A)^{-1}B+D$ we write $G=\left[\begin{array}[]{c | c}A&B\\\ \hline\cr C&D\end{array}\right]$. Blocks in matrices that can be inferred by symmetry are sometimes denoted by $\star$ to save space. For a finite set $\Omega$ we use $|\Omega|$ to denote its cardinality.

Throughout the paper, when we refer to stability of a system, we will mean
_asymptotic stability_. The term _stable_ should be interpreted in that way.

###  2.2 Dissipativity of LTI systems with quadratic supply functions

Here we recall characterization of dissipativity for linear time invariant
(LTI) systems in terms of linear matrix inequalities (LMI). For more details
we refer to, e.g., [11].

The dynamical system $\dot{x}=f(x,d)$, $z=g(x,d)$ is said to be strictly
dissipative if there exists $\epsilon>0$ so that the dissipation inequality
(1) holds when $s(d(t),z(t))$ is replaced with
$s(d(t),z(t))-\epsilon\|d(t)\|^{2}$. An LTI system $G$, given by

$G:\quad\begin{pmatrix}\dot{x}\\\ z\end{pmatrix}=\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\begin{pmatrix}x\\\ d\end{pmatrix},$ (2)

is dissipative with respect to the quadratic supply function

$s(d,z)=\left(\begin{matrix}d\\\ z\\\
\end{matrix}\right)^{\top}\left(\begin{matrix}Q&S\\\ S^{\top}&R\\\
\end{matrix}\right)\left(\begin{matrix}d\\\ z\\\ \end{matrix}\right),$ (3)

where $Q$ and $R$ are symmetric matrices of appropriate dimensions and $S$ is
a real matrix, if there exists a quadratic storage function

$V(x)=x^{\top}Px,$ (4)

such that the time derivative of $V(x(t))$ along the system’s trajectories
satisfies the _differential dissipation inequality_

$\dot{V}(x(t))\leq s(d(t),z(t))$ (5)

at any time $t$ and for all $(x,d,z)$ related via (2). The system (2) is
strictly dissipative with quadratic supply and quadratic storage if for all
$\operatorname{col}(x(t),d(t))\neq 0$ the inequality (5) holds with $\leq$
replaced by $<$. The differential strict dissipativity condition is equivalent
to the existence of a symmetric $P$ such that the following linear matrix
inequality (LMI) is feasible

$\left(\begin{array}[]{cc}I&0\\\ A&B\\\ \hdashline 0&I\\\
C&D\end{array}\right)^{\top}\left(\begin{array}[]{cc:cc}0&P&0&0\\\ P&0&0&0\\\
\hdashline 0&0&-Q&-S\\\
0&0&-S^{\top}&-R\end{array}\right)\left(\begin{array}[]{cc}I&0\\\ A&B\\\
\hdashline 0&I\\\ C&D\end{array}\right)\prec 0.$ (6)

Note that the we do not assume that the pair $(A,B)$ is controllable. Since in
this paper we will be exclusively dealing with _strict_ dissipativity, no
corresponding controllability assumptions are needed for the results presented
in the paper, see e.g. Chapter 2 in [11] for details.

Several well-known results regarding robust dissipativity and robust stability
are further presented in Appendix Appendix A: Robust stability and robust
dissipativity, as they are not required for presentation of the main results
of the paper (Sections 3, 4 and 5), but are however used in the proofs of
these results (Section 7).

###  2.3 Interconnection neutral supply rates

Consider two LTI systems $G_{i}$, $i=1,2$, with inputs $(v_{i},d_{i})$,
outputs $(w_{i},z_{i})$ and state vectors $x_{i}$, as presented in Figure 1.
Suppose the systems are interconnected with the interconnection constraint
$v_{1}=w_{2}$ and $v_{2}=w_{1}$.

$z_{1}$$d_{1}$$G_{1}$$v_{1}$$w_{1}$$w_{2}$$v_{2}$$G_{2}$$d_{2}$$z_{2}$ Figure
1: Open systems with external and interconnection signals.

Note that throughout the paper we will use symbols $v$ and $w$ (possibly with
indexes, e.g., $v_{i}$, $w_{i}$) to refer respectively to input and outputs of
a system, which are used to form interconnections with other systems. These
signals therefore become internal signals for the connected system, when the
interconnections are made. On the other side, we will use symbols $d$ and $z$
(possibly with indices, e.g., $d_{i}$, $z_{i}$) to refer respectively to
exogenous inputs and outputs (exogenous with respect to the set of
interconnected systems, that is, with respect to a network).

For $i=1,2$, let the system $G_{i}$ be dissipative with respect to supply
function $s_{i}(v_{i},d_{i},w_{i},z_{i})$ with a storage function
$V_{i}(x_{i})$ and suppose that the supply functions have the following
additive structure

$s_{i}(v_{i},d_{i},w_{i},z_{i})=s_{i,ext}(d_{i},z_{i})+s_{i,int}(v_{i},w_{i}),\quad\quad
i=1,2.$ (7)

The interconnection is said to be _neutral_ with respect to the supply
functions $s_{1,int}$, $s_{2,int}$ if

$s_{1,int}(v_{1},w_{1})+s_{2,int}(v_{2},w_{2})=0,$ (8)

for all $v_{1}$, $w_{1}$, $v_{2}$, $w_{2}$ such that $v_{1}=w_{2}$,
$v_{2}=w_{1}$. We will use the term _interconnection neutral supply functions_
to refer to supply functions $s_{1,int}(v_{1},w_{1})$ and
$s_{2,int}(v_{2},w_{2})$ which satisfy the above property.

We will use the symbol $G$ to refer to the system obtained by interconnecting
$G_{1}$ and $G_{2}$. The following proposition originates from [12].

######  Proposition 2.1.

Let $G_{1}$ and $G_{2}$ be strictly dissipative with respect to
$s_{1}(v_{1},d_{1},w_{1},z_{1})$ and $s_{2}(v_{2},d_{2},w_{2},z_{2})$, which
both have an additive structure as in (7). Let some corresponding storage
functions be $V_{1}(x_{1})$ and $V_{2}(x_{2})$. Suppose the interconnected
system $G$ is well-posed and $s_{1,int}(v_{1},w_{1})$ and
$s_{2,int}(v_{2},w_{2})$ are interconnection neutral supply functions. Then
the system $G$ is strictly dissipative with respect to the supply
$s_{ext}(d_{1},d_{2},z_{1},z_{2}):=s_{1,ext}(d_{1},z_{1})+s_{2,ext}(d_{2},z_{2})$.

###### Proof.

Adding the two dissipation inequalities
$\dot{V}_{1}(x_{1})<s_{1}(v_{1},d_{1},w_{1},z_{1})$ and
$\dot{V}_{2}(x_{2})<s_{2}(v_{2},d_{2},w_{2},z_{2})$, the desired result
follows directly from the neutrality condition (8), as it turns that
$V(x_{1},x_{2})=V_{1}(x_{1})+V_{2}(x_{2})$ acts as a storage function for $G$
with respect to $s_{ext}(d_{1},d_{2},z_{1},z_{2})$. ∎

$\tilde{G}_{1}$$v_{1}$$w_{1}$$w_{2}$$v_{2}$$\tilde{G}_{2}$ Figure 2:
Interconnected system as an autonomous system.

Let us now consider an autonomous system $\tilde{G}$ which consists of two
interconnected systems $\tilde{G}_{1}$ and $\tilde{G}_{2}$, as presented in
Figure 2. Let $x_{1}$ and $x_{2}$ denote state vectors of $\tilde{G}_{1}$ and
$\tilde{G}_{2}$, respectively.

######  Proposition 2.2.

Let $\tilde{G}_{1}$ and $\tilde{G}_{2}$ be _strictly_ dissipative with respect
to $s_{1,int}(v_{1},w_{1})$ and $s_{2,int}(v_{2},w_{2})$ with some
corresponding storage functions $V_{1}(x_{1})$ and $V_{2}(x_{2})$,
respectively. Suppose $V_{1}(\cdot)$ and $V_{2}(\cdot)$ are positive definite
functions, the interconnected system $\tilde{G}$ is well-posed and that
$s_{1,int}(v_{1},w_{1})$ and $s_{2,int}(v_{2},w_{2})$ are interconnection
neutral supply rates. Then the system $\tilde{G}$ is stable.

###### Proof.

Summing the strict dissipation inequalities
$\dot{V}_{1}(x_{1})<s_{1,int}(v_{1},w_{1})$ and
$\dot{V}_{2}(x_{2})<s_{2,int}(v_{2},w_{2})$, which hold for
$\operatorname{col}(x_{1},v_{1})\neq 0$ and
$\operatorname{col}(x_{2},v_{2})\neq 0$, with (8), we obtain
$\dot{V}_{1}(x_{1})+\dot{V}_{2}(x_{2})<0$ for $x_{1}\neq 0$ and/or $x_{2}\neq
0$. Therefore the positive definite function
$V(x_{1},x_{2}):=V_{1}(x_{1})+V_{2}(x_{2})$ is a Lyapunov function for the
interconnected system $G$. ∎

Extension of Propositions 2.1 and 2.2 to a larger number of interconnected
systems is straightforward, see [12].

##  3 TWO INTERCONNECTED SYSTEMS

In this section we present the main results of the paper for the case of two
interconnected systems. We prove suitably defined converse statements to those
of Proposition 2.1 and Proposition 2.2. While Propositions 2.1 and 2.2 hold
for general dissipative systems with general supply and storage functions, in
this section, and in the remainder of the paper, we consider LTI systems with
quadratic supply functions, i.e., supply functions in the form of (3) and
quadratic storage functions, i.e., storage functions in the form of (4).

###  3.1 Interconnection with an algebraic loop

Consider two interconnected systems $G_{1}$ and $G_{2}$, as depicted in Figure
3, where

$G_{i}:\,\,\,\begin{pmatrix}\dot{x}_{i}\\\ w_{i}\\\
z_{i}\end{pmatrix}=\begin{pmatrix}A_{i}&B_{i}&E_{i}\\\ C_{i}&D_{i}&0\\\
F_{i}&K_{i}&L_{i}\end{pmatrix}\begin{pmatrix}x_{i}\\\ v_{i}\\\
d_{i}\end{pmatrix},\quad i=1,2,$ (9)

where $A_{1}\in\mathbb{R}^{n_{1}\times n_{1}}$,
$D_{1}\in\mathbb{R}^{n_{w1}\times n_{v1}}$, $L_{1}\in\mathbb{R}^{n_{z1}\times
n_{d1}}$, $A_{2}\in\mathbb{R}^{n_{2}\times n_{2}}$,
$D_{2}\in\mathbb{R}^{n_{w2}\times n_{v2}}$, $L_{2}\in\mathbb{R}^{n_{z2}\times
n_{d2}}$. With $n_{v1}=n_{w2}$, $n_{v2}=n_{w1}$, the systems are
interconnected so that

$v_{1}=w_{2},\quad\quad v_{2}=w_{1}.$ (10)

Note that the existence of nonzero matrices $D_{1}$ and $D_{2}$ implies that
there is a direct feed-through of the signal $v_{1}$ to the signal $w_{2}$
(see Figure 3). This is what we mean by the term that there exists an
_algebraic loop_ in the system. Furthermore, note that there is a direct feed-
through from $d_{1}$ to $z_{1}$ and from $v_{1}$ to $z_{1}$, but there is no
direct feed-through from $d_{1}$ to $w_{1}$. Analogous situation is with the
system $G_{2}$. These feed-through channels are illustrated with dashed lines
in Figure 3. A distinguishing feature of the interconnected system (9), (10)
is that there is no direct feed-through path that goes from an external input
of $G_{1}$ to an external output of $G_{2}$, that is, there is no direct feed-
through path from $d_{1}$ to $z_{2}$. Analogously, there is no direct feed-
through path from $d_{2}$ to $z_{1}$.

$z_{1}$$d_{1}$$G_{1}$$v_{1}$$w_{1}$$w_{2}$$v_{2}$$G_{2}$$d_{2}$$z_{2}$ Figure
3: Two interconnected open systems.

We use the symbol $G$ to denote the overall interconnected system presented in
Figure 3 and defined by (9) and (10). The system $G$ has exogenous inputs
$(d_{1},d_{2})$ and outputs $(z_{1},z_{2})$.

######  Theorem 3.1.

Let $C_{1}$ and $C_{2}$ be full row rank matrices and suppose the
interconnected system $G$ is well-posed. Suppose that the system $G$ is
strictly dissipative with respect to a quadratic supply function
$s_{ext}(d_{1},d_{2},z_{1},z_{2})$, which is structured as follows (has an
additive structure)

$s_{\text{ext}}(d_{1},d_{2},z_{1},z_{2})=s_{1,ext}(d_{1},z_{1})+s_{2,ext}(d_{2},z_{2}),$
(11)

and suppose there exists a storage function $V(x_{1},x_{2})$ which is an
additive quadratic function as well, i.e., we have

$V(x_{1},x_{2})=V_{1}(x_{1})+V_{2}(x_{2}),$ (12)

with $V_{1}$ and $V_{2}$ quadratic functions. Then there exist quadratic
interconnection neutral supply functions $s_{1,int}(v_{1},w_{1})$ and
$s_{2,int}(v_{2},w_{2})$ such that

$s_{1,int}(v_{1},w_{1})+s_{2,int}(v_{2},w_{2})=0\quad\text{for}\,\,v_{1}=w_{2},v_{2}=w_{1},$
(13)

and

  * i)

$G_{1}$ is strictly dissipative with respect to the supply

$s_{1}(d_{1},v_{1},z_{1},w_{1}):=s_{1,ext}(d_{1},z_{1})+s_{1,int}(v_{1},w_{1})$

with the storage function $V_{1}(x_{1})$;

  * ii)

$G_{2}$ is strictly dissipative with respect to the supply

$s_{2}(d_{2},v_{2},z_{2},w_{2}):=s_{2,ext}(d_{2},z_{2})+s_{2,int}(v_{2},w_{2})$

with the storage function $V_{2}(x_{2})$.

We emphasize that in the above theorem the storage functions $V_{1}$, $V_{2}$
from the statements (_i_) and (_ii_) are the same $V_{1}$, $V_{2}$ from (12).

Theorem 3.1 is a converse result to Proposition 2.1 and its proof is given in
Section 7. The presented proof indeed exploits the assumption that both
$C_{1}$ and $C_{2}$ are full row rank matrices and it is still an open
question can this assumption be relaxed in this rather general setting. In
Section 3.2 we show that when $D_{1}=0$ and $D_{2}=0$ the rank assumption can
be omitted.

Let us now consider an autonomous system which consists of two interconnected
systems $\tilde{G}_{1}$ and $\tilde{G_{2}}$, as presented in Figure 2 and
given by

$\tilde{G}_{i}:\,\,\,\begin{pmatrix}\dot{x}_{i}\\\
w_{i}\end{pmatrix}=\begin{pmatrix}A_{i}&B_{i}\\\
C_{i}&D_{i}\end{pmatrix}\begin{pmatrix}x_{i}\\\ v_{i}\end{pmatrix},\quad
i=1,2.$ (14)

Let $\tilde{G}$ denote the interconnected system obtained by imposing
interconnection constraints $v_{1}=w_{2}$, $v_{2}=w_{1}$ on (14). The
following Corollary follows from Theorem 3.1.

######  Corollary 3.2.

Let $C_{1}$ and $C_{2}$ be full row rank matrices. Suppose $\tilde{G}$ is
well-posed, stable and admits an additive quadratic Lyapunov function of the
form $V(x_{1},x_{2})=V_{1}(x_{1})+V_{2}(x_{2})$. Then there exist quadratic
interconnection neutral supply functions $s_{1,int}(v_{1},w_{1})$ and
$s_{2,int}(v_{2},w_{2})$ such that
$s_{1,int}(v_{1},w_{1})+s_{2,int}(v_{2},w_{2})=0$ for $v_{1}=w_{2}$,
$v_{2}=w_{1}$ and

  1. (i)

$G_{1}$ is strictly dissipative with respect to $s_{1,int}(v_{1},w_{1})$ with
the storage function $V_{1}(x_{1})$,

  2. (ii)

$G_{2}$ is strictly dissipative with respect to $s_{2,int}(v_{2},w_{2})$ with
the storage function $V_{2}(x_{2})$.

Proof of Corollary 3.2 follows directly from the proof of Theorem 3.1
presented in Section 7.1 when in the proof we omit all the terms related to
exogenous inputs/outputs $d_{1},d_{2},z_{1},z_{2}$ (or equivalently, by
setting $E_{i}=0$, $F_{i}=0$, $K_{i}=0$, $L_{i}=0$) and the external supply
functions $s_{1,ext}(\cdot,\cdot)$ and $s_{2,ext}(\cdot,\cdot)$.

###  3.2 Interconnection without an algebraic loop

In many real-life cases systems are interconnected with no direct feed-through
matrices, i.e., matrices $D_{1}$ and $D_{2}$ from (9) are zero matrices. It
turns out that such interconnections have some specific features. Some of them
are summarized in the following proposition and in Section 7.

######  Proposition 3.3.

Consider LTI systems given by (9) and suppose $D_{1}=0$ and $D_{2}=0$. Then
Theorem 3.1 and Corollary 3.2 remain to hold even when $C_{1}$ and $C_{2}$ are
not necessarily full row rank matrices.

Note that with $D_{1}=0$ and $D_{2}=0$, the systems $G$ and $\tilde{G}$ are
necessarily well-posed.

Proof of Proposition 3.3 is given in Section 7.2.

##  4 DYNAMICAL NETWORKS

In this section we first extend the results from Section 3 from two
interconnected systems to more general dynamical networks. After introducing
necessary definitions, we study acyclic networks, where, due to space
limitations, we present results for _autonomous_ acyclic dynamical networks
only (i.e., generalization of Corollary 3.2). The corresponding results for
_open_ acyclic networks which are dissipative with respect to exogenous supply
functions with additive structure (i.e., generalization of Theorem 3.1)
straightforwardly follow along the same lines. At the end of the section, we
summarize several results regarding neutral supply rates in networks with
cycles.

###  4.1 Dynamical networks and additive Lyapunov functions

We define a _dynamical network_ as a finite set of dynamical systems
interconnected over some graph. More precisely, we use a directed graph
$\Gamma:=(\Omega,E)$ in which each vertex $G_{i}\in\Omega$ is identified with
a dynamical system, while a directed edge $(G_{i},G_{j})\in E$ means that
there is an output signal of $G_{i}$ that is input to $G_{j}$. With an edge
$(G_{i},G_{j})\in E$ we make the following input/output definitions: $w_{ij}$
is an output from the system $G_{i}$ and $v_{ji}$ is the input to the system
$G_{j}$. The interconnection constraint related to the edge $(G_{i},G_{j})$ is
given by $v_{ji}=w_{ij}$. We use $x_{i}$ to denote the state vector of system
$G_{i}$.

From the directed interconnection graph $\Gamma$, we define an undirected
graph $\hat{\Gamma}=(\Omega,\hat{E})$, as follows

$\Big{(}(G_{i},G_{j})\in E\Big{)}\,\,\text{or}\,\,\Big{(}(G_{j},G_{i})\in
E\Big{)}\Longleftrightarrow(G_{i},G_{j})\in\hat{E}.$

Graph $\hat{\Gamma}$ will be used below in this section to define what we mean
by the term acyclic network.

Let $\mathcal{N}_{i}$ denote the set of indices of systems adjacent to $G_{i}$
in the graph $\hat{\Gamma}$, that is,
$\mathcal{N}_{i}:=\\{j\,\,:\,\,(G_{i},G_{j})\in\hat{E}\\}$. Consider an
arbitrary system $G_{i}$ from the dynamical network and let
$\mathcal{N}_{i}=\\{N_{1}^{i},N_{2}^{i},\ldots,N_{r(i)}^{i}\\}$ where
$r(i)=|\mathcal{N}_{i}|$. We assume the system $G_{i}$ has a state space
realization of the following form

$G_{i}:\,\,\begin{pmatrix}\dot{x}_{i}\\\ w_{iN_{1}^{i}}\\\ w_{iN_{2}^{i}}\\\
\vdots\\\
w_{iN_{r(i)}^{i}}\end{pmatrix}=\begin{pmatrix}A_{i}&B_{iN_{1}^{i}}&B_{iN_{2}^{i}}&\ldots&B_{iN_{r(i)}^{i}}\\\
C_{iN_{1}^{i}}&D_{iN_{1}^{i}}&0&\ldots&0\\\
C_{iN_{2}^{i}}&0&D_{iN_{2}^{i}}&\ldots&0\\\ \vdots&\vdots&\ddots&\vdots&0\\\
C_{iN_{r(i)}^{i}}&0&0&\ldots&D_{iN_{r(i)}^{i}}\end{pmatrix}\begin{pmatrix}x_{i}\\\
v_{iN_{1}^{i}}\\\ v_{iN_{2}^{i}}\\\ \vdots\\\ v_{iN_{r(i)}^{i}}\end{pmatrix}.$
(15)

A distinguishing feature of (15) is that there are no direct feed-through
paths from $v_{ik}$ to $w_{il}$ when $k\neq l$.

Let $N$ denote the number of systems in a dynamical network, i.e.,
$N=|\Omega|$, and suppose the overall dynamics of the network is described by
$\dot{x}=\mathcal{A}x$, where $x=\operatorname{col}(x_{1},\ldots,x_{N})$,
$x_{i}\in\mathbb{R}^{n_{i}}$. We say that the dynamical network admits an
additive quadratic Lyapunov function if there exists a block diagonal matrix
$P=\operatorname{diag}(P_{1},\ldots,P_{N})$ with symmetric matrices
$P_{i}\in\mathbb{R}^{n_{i}\times n_{i}}$, $P_{i}\succ 0$, such that
$\mathcal{A}^{\top}P+P\mathcal{A}\prec 0$. Indeed, the term _additive_ is used
since the Lyapunov function defined with $P$ is given by

$V(x)=\underbrace{x_{1}^{\top}P_{1}x_{1}}_{V_{1}(x_{1})}+\ldots\underbrace{x_{N}^{\top}P_{N}x_{N}}_{V_{N}(x_{N})},$
(16)

that is, $V(x)$ is a sum of local functions $V_{i}(x_{i})$, where each $V_{i}$
is _local_ to the system $i$ in a sense that it depends only on the states of
that system.

###  4.2 Acyclic dynamical networks

We say that a dynamical network defined with a directed graph $\Gamma$ is
_acyclic_ dynamical network if the corresponding undirected graph
$\hat{\Gamma}=(\Omega,\hat{E})$ is acyclic. For definition of an acyclic graph
we refer to, e.g., [1].

######  Theorem 4.1.

Suppose the graph $\hat{\Gamma}$ is acyclic, each $G_{i}$ is represented in
the form of (15) and either one of the following two cases holds:

  * 1)

the matrices $D_{ij}$ in (15) are in general non-zero matrices while each
matrix $C_{ij}$ in (15) has full row rank;

  * 2)

$D_{ij}=0$ for all $(i,j)$ such that $(G_{i},G_{j})\in E$.

Let the overall interconnected network be well-posed 444In the case (2) this
is not an assumption as then the network is necessarily well-posed.. Then the
following two statements are equivalent:

  * i)

The dynamical network admits an additive quadratic Lyapunov function of the
form (16).

  * ii)

For each $G_{i}\in\Omega$ and each $j\in\mathcal{N}_{i}$ there exists a
quadratic function $s_{ij}(v_{ij},w_{ij})$ such that

    * a)

$G_{i}$ is strictly dissipative with respect to the supply function
$\sum_{j\in\mathcal{N}_{i}}s_{ij}(v_{ij},w_{ij})$ with local function
$V_{i}(x_{i})=x_{i}^{\top}P_{i}x_{i}$, with $P_{i}\succ 0$, as a storage
function;

    * b)

$s_{ij}(v_{ij},w_{ij})+s_{ji}(v_{ji},w_{ji})=0$ for each $(i,j)$ such that
$(G_{i},G_{j})\in\hat{E}$.

Note that the condition in part (_b_) of the statement (_ii_) means that the
supplies $s_{ij}$ and $s_{ji}$ are interconnection neutral supply functions
for the interconnections between the systems $G_{i}$ and $G_{j}$. We also
emphasize that in the above theorem each local function $V_{i}(x_{i})$ from
(_i_) is indeed the same $V_{i}(x_{i})$ as in (_ii_).

###### Proof of Theorem 4.1.

The implication (_ii_) $\implies$ (_i_) is trivial and is a straightforward
generalization of Proposition 2.2. It remains to show (_i_) $\implies$ (_ii_).
The proof is based on a repeated applications of Theorem 3.1 and Corollary 3.2
for the case (1), or Proposition 3.3 for the case (2).

Suppose first that $\hat{\Gamma}$ is a connected graph. Consider an arbitrary
system $G_{i}$ from the network and let
$\mathcal{N}_{i}=\\{N_{1}^{i},N_{2}^{i},\ldots,N_{r(i)}^{i}\\}$ and $r(i)$ be
defined as above in Section 4.1. Let
$\mathcal{E}_{i}=\\{e_{1}^{i},e_{2}^{i},\ldots,e_{r(i)}^{i}\\}:=\\{(G_{i},G_{N_{1}^{i}}),\ldots,(G_{i},G_{N_{r(i)}^{i}})\\}$,
that is, $\mathcal{E}_{i}$ is the set of all undirected edges which have
$G_{i}$ as an end vertex. For each $G_{i}\in\Omega$ and
$j\in\\{1,\ldots,r(i)\\}$ we define the following two subgraphs of
$\hat{\Gamma}$:

  1. i)

$\hat{\Gamma}_{+}(\hat{\Gamma},e^{i}_{j})$ is the connected component of the
graph $\hat{\Gamma}-e_{j}^{i}$ (the graph obtained by removing the edge
$e_{j}^{i}$ from the graph $\hat{\Gamma}$) which contains $G_{i}$;

  2. ii)

$\hat{\Gamma}_{-}(\hat{\Gamma},e^{i}_{j})$ is the connected component of the
graph $\hat{\Gamma}-e_{j}^{i}$ which does not contain $G_{i}$.

Note that since $\hat{\Gamma}$ is an acyclic graph, for each edge $e^{i}_{j}$
we can view the overall dynamical network as an interconnection of two
systems: one as the dynamical network corresponding to
$\hat{\Gamma}_{+}(\hat{\Gamma},e^{i}_{j})$ and the other as the dynamical
network corresponding to $\hat{\Gamma}_{-}(\hat{\Gamma},e^{i}_{j})$. In the
overall network $\hat{\Gamma}$, the two systems are interconnected over the
edge $e^{i}_{j}$ only. In the remainder we will refer to a dynamical system
(dynamical network) simply by referring to its graph, i.e., we will use the
term “system $\hat{\Gamma}_{+}(\hat{\Gamma},e^{i}_{j})$”.

Next we infer existence of interconnection neutral supply functions $s_{ij}$,
$s_{ji}$ for an arbitrary $G_{i}\in\Omega$ and for all $j\in\mathcal{N}_{i}$,
which satisfy conditions (_ii_) from the theorem. In other words, we infer
existence of neutral supply functions defined on all edges in
$\mathcal{E}_{i}$ for an arbitrary $G_{i}\in\Omega$. This is an iterative
procedure described as follows, with $k$ as the iteration counter:
Let $\hat{\Gamma}_{0}:=\hat{\Gamma}$, $k=1$.
While $k<r(i)+1$:

  * •

if $k=1$ we apply Corollary 3.2 (in case (1)) or Proposition 3.3 (in case
(2)),

  * •

if $k>1$ we apply Theorem 3.1 (in case (1)), or Proposition 3.3 (in case (2)),

to infer the existence of interconnection neutral supply functions on the edge
$e^{i}_{k}$ between the systems
$\hat{\Gamma}_{+}(\hat{\Gamma}_{k-1},e^{i}_{k})$ and
$\hat{\Gamma}_{-}(\hat{\Gamma}_{k-1},e^{i}_{k})$. With $j=N_{k}^{i}$, denote
these interconnection neutral supply functions with $s_{ij}(v_{ij},w_{ij})$
and $s_{ji}(v_{ji},w_{ji})$ from the statement (ii) of the theorem.
Denote $\hat{\Gamma}_{k}=\hat{\Gamma}_{+}(\hat{\Gamma}_{k-1},e_{k}^{i})$.
Increase $k$ by one and continue the procedure.

At each step of the procedure, the system $\hat{\Gamma}_{k}$ is strictly
dissipative with respect to the external supply function
$\sum_{j\in\\{N_{1}^{i},\ldots,N_{k}^{i}\\}}s_{ij}(v_{ij},w_{ij})$ with the
corresponding storage function being $\sum_{p\in T_{k}}V_{p}(x_{p})$. Here
$T_{k}$ denotes the set of indices $p$ of systems $G_{p}$ which belong to the
vertex set of $\hat{\Gamma}_{k}$.

Finally, if the graph $\hat{\Gamma}$ is not connected, we can apply the
presented procedure to each connected component of $\hat{\Gamma}$ separately.
∎

The key feature in the procedure described in the above proof is that in each
step all the assumptions for applying Corollary 3.2 or Theorem 3.1 are
necessarily satisfied at each iteration.

######  Remark 4.2.

In the case of a graph $\hat{\Gamma}$ with a cycle, if $e\in E$ is an edge
from a cycle in $\hat{\Gamma}$, we cannot use the above stated definitions of
$\hat{\Gamma}_{+}(\hat{\Gamma},e)$ and $\hat{\Gamma}_{-}(\hat{\Gamma},e)$ to
divide the network into two systems and to subsequently apply Corollary 3.2 or
Theorem 3.1.

######  Example 4.3.

In this example we illustrate the iterative procedure (presented in the proof
of Theorem 4.1) for obtaining interconnection neutral supply rates for all
three edges in the graph $\hat{\Gamma}$ in Figure 4a. Suppose the considered
network belongs to the case (1) from Theorem 4.1. First the overall network is
seen as the interconnection of $G_{1}$ and $G_{\alpha}$, where $G_{\alpha}$
consists of interconnected $G_{2}$, $G_{3}$ and $G_{4}$, as presented in
Figure 4a. Corollary 3.2 applies and we infer existence of interconnection
neutral supply functions $s_{1\alpha}$, $s_{\alpha 1}$. Next, consider
interconnection of $G_{4}$ and $G_{\beta}$, where $G_{\beta}$ consists of
interconnected $G_{2}$ and $G_{3}$, as presented in Figure 4b. Neutral
supplies follow from Theorem 3.1. Finally, consider $G_{2}$ and $G_{3}$ and
apply again Theorem 3.1 to obtain $s_{23},s_{32}$, see Figure 4c. Note that
$s_{1\alpha}=s_{1\beta}=s_{12}$, $s_{\alpha 1}=s_{\beta 1}=s_{21}$,
$s_{4\beta}=s_{42}$, $s_{\beta 4}=s_{24}$.

$G_{1}$$G_{2}$$G_{3}$$G_{4}$$G_{\alpha}$a)$s_{1\alpha}$$s_{\alpha
1}$$G_{2}$$G_{3}$$G_{4}$$G_{\beta}$$s_{1\beta}$$s_{\beta
1}$$s_{4\beta}$$s_{\beta
4}$b)$G_{2}$$G_{3}$$s_{12}$$s_{21}$$s_{23}$$s_{32}$$s_{42}$$s_{24}$c) Figure
4: Illustration of the iterative procedure from the proof of Theorem 4.1.

###  4.3 Networks with cycles

We have already in Remark 4.2 pointed out to the main difficulty of
generalizing Theorem 4.1 to networks where $\hat{\Gamma}$ contains cycles.
However, Theorem 4.1 can be applied to infer certain dissipativiy properties
in networks with cycles as well, based on the following simple observation.

If we group subsets of connected systems from a dynamical network (i.e., from
$\hat{\Gamma}$) and consider each of these groups as a single dynamical
system, we obtain a graph of a network characterized with new set of vertices
(“new systems”) and edges (“new interconnections”). We assume this grouping of
systems is such that each system belongs to one and only one group. In this
way, from $\hat{\Gamma}$ we obtain new undirected graph $\hat{\Gamma}_{new}$.
Indeed, the underlying dynamical network behind $\Gamma_{new}$ remains the
same. This process is illustrated in Figure 5.

$G_{1}$$G_{3}$$G_{2}$$G_{4}$$G_{6}$$G_{5}$$G_{1}^{new}$$G_{2}^{new}$$G_{1}^{new}$$G_{2}^{new}$$\rightarrow$a)
Graph $\hat{\Gamma}$b) Graph $\hat{\Gamma}_{new}$ Figure 5: Example of
grouping of vertices in $\hat{\Gamma}$ to obtain $\hat{\Gamma}_{new}$.

If the system behind the original graph $\hat{\Gamma}$ is stable and admits an
additive Lyapunov function, then the system represented with
$\hat{\Gamma}_{new}$ admits an additive Laypunov function as well. Note that
we can always partition networks with cycles in $\hat{\Gamma}$ to obtain
$\hat{\Gamma}_{new}$ which is acyclic, and typically this partitioning can be
done in several ways. If a network admits an additive Lyapunov function and
$\hat{\Gamma}_{new}$ is an acyclic graph, we can use Theorem 4.1 to infer
existence of interconnection neutral supply functions in connection to the
edges of $\hat{\Gamma}_{new}$. A notable specific case of partitioning and the
corresponding dissipativity characterizations are summarised in the following
remark.

######  Remark 4.4.

Suppose $\hat{\Gamma}$ is an arbitrary graph, possibly containing cycles, and
let $G_{i}$ be an arbitrary system from the corresponding network. Suppose
that the considered dynamical network is stable and admits an additive
quadratic Lyapunov function. We can view the network as interconnection of two
systems $G_{1}^{new}$ and $G_{2}^{new}$, where $G_{1}^{new}:=G_{i}$ and
$G_{2}^{new}$ is composed of all the remaining systems in the network. The
corresponding graph $\hat{\Gamma}_{new}$ is acyclic, contains only two systems
and we can apply Theorem 3.1 and Corollary 3.2. In this way we infer existence
of interconnection neutral supply rate defined as quadratic function of all
interconneting signals between $G_{i}$ and the rest of the network.

##  5 NEUTRAL SUPPLY FUNCTIONS AND ROBUSTNESS

###  5.1 Stability robustness to connection/disconnection of an
interconnection link

Consider an LTI dynamical system $G$ with inputs $(v_{A},v_{B})$ and outputs
$(w_{A},w_{B})$, as depicted in Figure 6. Suppose that
$v_{A}(t)\in\mathbb{R}^{p}$, $v_{B}(t)\in\mathbb{R}^{q}$ and
$w_{A}(t)\in\mathbb{R}^{q}$, $w_{B}(t)\in\mathbb{R}^{p}$, that is, $v_{A}(t)$
and $w_{B}(t)$ are of the same dimension, while $v_{B}(t)$ and $w_{A}(t)$ are
of the same dimension. Let $v:=\operatorname{col}(v_{A},v_{B})$,
$w:=\operatorname{col}(w_{A},w_{B})$ and let $x$ denote the state vector of
$G$.

$\alpha I$$\alpha I$$G$$v_{A}$$w_{A}$$w_{B}$$v_{B}$ Figure 6: Dynamical system
with an isolated interconnection link.

######  Lemma 5.1.

Let the system $G$ be strictly dissipative with respect to a quadratic supply
function $s(v,w)$ with an additive structure

$s(v,w)=s_{A}(v_{A},w_{A})+s_{B}(v_{B},w_{B})$ (17)

and with some corresponding quadratic storage function $V(x)=x^{\top}Px$.
Suppose the following holds

  1. a)

$s_{A}(v_{A},w_{A})+s_{B}(v_{B},w_{B})=0$ for all $v_{A}=w_{B}$,
$v_{B}=w_{A}$;

  2. b)

$s_{A}(0,w_{A})\leq 0$ for all $w_{A}\neq 0$, $s_{B}(0,w_{B})\leq 0$ for all
$w_{B}\neq 0$;

  3. c)

$P\succ 0.$

Then the system $G$ is stable and it remains stable if the following
interconnection is made: $v_{A}=\alpha w_{B}$, $v_{B}=\alpha w_{A}$, for all
$\alpha\in[0,1]$.

###### Proof.

The conditions (a) and (b) are satisfied if and only if $R\succeq 0$,
$Q\preceq 0$ and the supply $s(v,w)$ is structured as follows

$s(v,w)=-\begin{pmatrix}v_{A}\\\ v_{B}\\\ \hdashline w_{A}\\\
w_{B}\end{pmatrix}^{\top}\underbrace{\left(\begin{array}[]{c c : c
c}Q&0&S&0\\\ 0&-R&0&-S^{\top}\\\ \hdashline S^{\top}&0&R&0\\\
0&-S&0&-Q\end{array}\right)}_{\Pi}\begin{pmatrix}v_{A}\\\ v_{B}\\\ \hdashline
w_{A}\\\ w_{B}\end{pmatrix}.$ (18)

Since for $\alpha\in[0,1]$ we have

$\left(\begin{array}[]{c
}\star\end{array}\right)^{\top}\Pi\left(\begin{array}[]{c c}0&\alpha I\\\
\alpha I&0\\\ \hdashline I&0\\\
0&I\end{array}\right)=\begin{pmatrix}(1-\alpha^{2})R&\\\
0&-(1-\alpha^{2})Q\end{pmatrix}\succeq 0$ (19)

the desired result follows directly from Theorem 8.1 in Appendix Appendix A:
Robust stability and robust dissipativity with

$H=\begin{pmatrix}0&\alpha I\\\ \alpha
I&0\end{pmatrix}\in\mathbf{H}=\Big{\\{}\begin{pmatrix}0&\alpha I\\\ \alpha
I&0\end{pmatrix}\,\,:\alpha\in[0,1]\Big{\\}}$

∎

In Lemma 5.1, the exposed interconnection link defined with an output-input
pair $(w,v)$ does not need to be an interconnection link from an acyclic
network. Indeed, the system $G$ is an arbitrary LTI system, with no specific
structure required. Also note that no specific structure on the Laypunov
function (i.e., on the matrix $P$) was imposed. In the following subsection,
we will use the following convenient interpretation of this result. Suppose
that some interconnection link in a system is characterized with an
interconnection neutral supply function which satisfies the property (b) from
Lemma 5.1 and with a positive definite storage function. Then the system is
stable irrespective of whether the interconnection is present or not.

###  5.2 Acyclic networks with no algebraic loops

The following proposition is proved in Section 7.3.

######  Proposition 5.2.

Consider LTI systems given by (9).

  1. (a)

Let $D_{i}=0$, $K_{i}=0$ and $L_{i}=0$, $i=1,2$. Then the statement of Theorem
3.1 holds with full row rank assumption on $C_{1}$ and $C_{2}$ omitted and
with some interconnection neutral supply functions $s_{1,int}$ and $s_{2,int}$
which satisfy the following condition

$s_{1,int}(0,w_{1})\leq 0,\,\,s_{2,int}(0,w_{2})\leq 0\,\,\,\text{for
all}\,\,\,w_{1}\neq 0,w_{2}\neq 0.$ (20)

  2. (b)

Let $D_{1}=0$, $D_{2}=0$. Then Corollary 3.2 holds with full row rank
assumption on $C_{1}$ and $C_{2}$ omitted and with some interconnection
neutral supply functions $s_{1,int}$ and $s_{1,int}$ which satisfy (20).

The following result follows directly from Proposition 5.2, Theorem 4.1 and
Lemma 5.1.

######  Corollary 5.3.

Consider a dynamical network which belongs to the case (2) from Theorem 4.1
and suppose it admits an additive Lyapunov function of the form (16). Then the
network is robustly stable with respect to removal (disconnection) of an
arbitrary edge from $\hat{\Gamma}$.

###  5.3 Networks with cycles and no algebraic loops

The following result follows directly from Remark 4.4, Proposition 5.2 and
Lemma 5.1.

######  Corollary 5.4.

Consider an arbitrary dynamical network defined in Section 4.1 (hence graph
$\hat{\Gamma}$ can contain cycles) in which dynamics of the system $G_{i}$ is
given by (15) with $D_{ij}=0$ for all $(i,j)$ such that $(G_{i},G_{j})\in E$.
Suppose that the network admits an additive Lyapunov function of the form
(16). Then the network is robustly stable with respect to removal
(disconnection) of the system $G_{i}$ from the network555By this we mean
removal of the system $G_{i}$ together with all edges $(G_{i},G_{j})\in E$
from the graph $\Gamma$..

Simpler, alternative approach to prove Corollary 5.3 and Corollary 5.4, which
avoids proving existence of interconnection neutral supply rates (Theorem 4.1)
and their characterizations (Proposition 5.2), is presented in [5]. Still we
believe that the characterizations and insights obtained from the proofs based
on existence of interconnection neutral supply rates are of independent
interest.

##  6 EXAMPLE

Numerical example presented in this section is based on the example from
Section VI in [4]. There a DC-grid is presented, which is here illustrated in
Figure 7. The grid consists of a DC voltage source $E$, two resistive loads
with resistances $R_{L1}$ and $R_{L3}$ and a battery. Each of three
transmission lines of the network has resistance $R$ and inductance $L$.

$R$$L$$E$$i_{3}$$d_{3}i_{3}$$d_{3}^{-1}u_{3}$$u_{3}$$R_{L3}$DC-DCload
3$R$$L$$i_{2}$$d_{2}i_{2}$$d_{2}^{-1}u_{2}$$u_{2}$DC-
DCbattery$R$$L$$i_{1}$$d_{1}i_{1}$$d_{1}^{-1}u_{1}$$u_{1}$$R_{L1}$DC-DCload 1
Figure 7: An example of a DC-grid, from [4].

The voltage and the current of load $1$, the battery, and the load $3$ are
denoted by $(u_{1},i_{1})$, $(u_{2},i_{2})$ and $(u_{3},i_{3})$, respectively.
Furthermore, load $1$, the battery and load $3$ are equipped with ideal DC-DC
converters with voltage gains $d_{1},d_{2}$ and $d_{3}$, respectively. The
overall network is seen as an interconnection of $3$ dynamical systems, as
presented in Figure 8 and as described next.

The first system $G_{1}$ consists of load $1$, its converter and the
transmission line. This is a first order system with a single state
$x_{1}:=i_{1}$, which is the current over the corresponding transmission
inductance. Input to $G_{1}$ is voltage $u_{2}$, and output of $G_{1}$ is
voltage $u_{1}$.

The second system $G_{2}$ consists of the battery, its converter and a
transmission line. The sate vector of this system is
$x_{2}:=\operatorname{col}(s,i_{2})$, where $s$ denotes the state of charge of
the battery and is assumed to be linear with respect to $u_{2}$, that is,
$s=Ku_{2}$ where $K$ is a given constant. The dynamics of the battery is
modelled as $\dot{s}=i_{2}$. Input to $G_{2}$ is
$\operatorname{col}(u_{1},u_{3})$, while the output is
$\operatorname{col}(u_{2},u_{2})$, as presented in Figure 8.

The third system $G_{3}$ consists of load $3$, its converter, a transmission
line and the voltage source $E$. Here the state vector is $x_{3}:=i_{3}$, the
input is $u_{2}$ and the output is $u_{3}$.

$u_{2}$$u_{1}$$u_{3}$$u_{2}$$G_{1}$$G_{2}$$G_{3}$ Figure 8: Interconnection of
systems in the DC-grid

With the above descriptions, the state-space realizations of the three systems
are given as follows:

$G_{1}=\left[\begin{array}[]{c | c}-\frac{R+d_{1}^{-2}R_{L1}}{L}&\frac{d_{1}^{-1}d_{2}^{-1}}{L}\\\ \hline\cr R_{L1}&0\end{array}\right],\,G_{3}=\left[\begin{array}[]{c | c}-\frac{R+2d_{3}^{-2}R_{L3}}{L}&\frac{d_{2}^{-1}d_{3}^{-1}}{L}\\\ \hline\cr R_{L3}&0\end{array}\right],$ $G_{2}=\left[\begin{array}[]{c c | c c}0&1&0&0\\\ -\frac{2d_{2}^{-2}}{KL}&-\frac{R}{L}&\frac{d_{1}^{-1}d_{2}^{-1}}{L}&\frac{d_{2}^{-1}d_{3}^{-1}}{L}\\\ \hline\cr\frac{1}{K}&0&0&0\\\ \frac{1}{K}&0&0&0\end{array}\right].$

Note that in the above models, and in Figure 8, we do not consider $E$ as an
exogenous input. Instead we assume $E$ to be a constant signal while the
values of the states and the interconnecting signals denote their deviations
from the corresponding equilibrium values. Numerical values of the parameters
are as well taken from [4] and are as follows: $R=1\,\Omega$,
$L=10^{-3}\,\text{H}$, $K=10\,\text{F}$, $R_{L1}=R_{L3}=20\,\Omega$.
$(d_{1},d_{2},d_{3})=(0.3953,0.1634,0.7785)$.

It can easily be verified that the considered dynamical network is stable and
admits an additive quadratic Lyapunov function
$V(x_{1},x_{2},x_{3})=V_{1}(x_{1})+V_{2}(x_{2})+V_{3}(x_{3})$, where

$\displaystyle V_{1}(x_{1})$
$\displaystyle=3.3282\,\,i_{1}^{2},\,\,V_{2}(x_{2})=\begin{pmatrix}s\\\
i_{2}\end{pmatrix}^{\top}\begin{pmatrix}14.3127&0.0261\\\
0.0261&0.0069\end{pmatrix}\begin{pmatrix}s\\\ i_{2}\end{pmatrix},$
$\displaystyle V_{3}(x_{3})$ $\displaystyle=2.3523\,\,i_{3}^{2}.$

Theorem 4.1 states that there exist interconnection neutral supply functions
defined on the interconnection links. We have applied the iterative procedure
described in the proof of Theorem 4.1 to construct a set of such supply
functions. We have that $G_{1}$ is strictly dissipative w.r.t. supply function
$s_{12}(u_{2},u_{1})$ with the storage $V_{1}(x_{1})$; the system $G_{2}$ is
strictly dissipative w.r.t. $s_{21}(u_{1},u_{2})+s_{23}(u_{3},u_{2})$ with the
storage $V_{2}(x_{2})$; and the system $G_{3}$ is strictly dissipative w.r.t.
$s_{32}(u_{2},u_{3})$ with the storage $V_{3}(x_{3})$, where

$\displaystyle s_{12}(u_{2},u_{1})$ $\displaystyle=\begin{pmatrix}u_{2}\\\
u_{1}\end{pmatrix}^{\top}\begin{pmatrix}4754.6&1543.5\\\
1543.5&-1637.6\end{pmatrix}\begin{pmatrix}u_{2}\\\ u_{1}\end{pmatrix},$ (21)
$\displaystyle s_{21}(u_{1},u_{2})$ $\displaystyle=\begin{pmatrix}u_{1}\\\
u_{2}\end{pmatrix}^{\top}\begin{pmatrix}1637.6&-1543.5\\\
-1543.5&-4754.6\end{pmatrix}\begin{pmatrix}u_{1}\\\ u_{2}\end{pmatrix},$ (22)
$\displaystyle s_{23}(u_{3},u_{2})$ $\displaystyle=\begin{pmatrix}u_{3}\\\
u_{2}\end{pmatrix}^{\top}\begin{pmatrix}608&-506.8\\\
-506.8&-2298.6\end{pmatrix}\begin{pmatrix}u_{3}\\\ u_{2}\end{pmatrix},$ (23)
$\displaystyle s_{32}(u_{2},u_{3})$ $\displaystyle=\begin{pmatrix}u_{2}\\\
u_{3}\end{pmatrix}^{\top}\begin{pmatrix}2298.6&506.8\\\
506.8&-608\end{pmatrix}\begin{pmatrix}u_{2}\\\ u_{3}\end{pmatrix}.$ (24)

Note that $s_{12}(u_{2},u_{1})+s_{21}(u_{1},u_{2})=0$ and
$s_{23}(u_{3},u_{2})+s_{32}(u_{2},u_{3})=0$. We remark that in the calculation
of the supply functions according to the constructive procedure from the proof
of Theorem 3.1, we have chosen $\alpha=0.5$ in (55).

_A posteriori_ tests, based on verification of the corresponding strict
dissipativity LMIs of the form (6), indeed confirm that all three systems are
strictly dissipative w.r.t. supply functions and with the storage functions as
descried above in the text. Note that $s_{12}(0,u_{1})<0$ for all $u_{1}\neq
0$; $s_{21}(0,u_{2})+s_{23}(0,u_{2})<0$ for all $u_{2}\neq 0$ and
$s_{32}(0,u_{3})<0$ for all $u_{3}\neq 0$, which is in accordance with the
statements of Proposition 5.2. Finally, it is easily verified that the network
stability is robust w.r.t. removal of the interconnection links, what is in
conformity with Corollary 5.3.

##  7 PROOFS

This section contains proofs of the novel results presented in this paper.
Some of the proofs make use of the non-conservative robust stability and
robust dissipativity characterizations based on the full block S-procedure
[10, 9], which are suitably summarized in Appendix Appendix A: Robust
stability and robust dissipativity.

###  7.1 Proof of Theorem 3.1

###### Step 0: _Starting point, main idea and overview_.

The system $G$ can be presented as system $G_{0}$ with an interconnection
matrix $H=\left(\begin{smallmatrix}0&I\\\ I&0\end{smallmatrix}\right)$ in a
feedback loop, as illustrated in Figure 12b in Appendix Appendix A: Robust
stability and robust dissipativity, where the system $G_{0}$ is presented in
Figure 9.

$G_{1}$$G_{2}$$G_{0}$$v_{1}$$v_{2}$$d_{1}$$d_{2}$$w_{1}$$w_{2}$$z_{1}$$z_{2}$$v$$w$$d$$z$
Figure 9: The system $G_{0}$.

The system $G_{0}$ from Figure 9 is given by (93) with $M=0$,
$x=\operatorname{col}(x_{1},x_{2})$, $w=\operatorname{col}(w_{1},w_{2})$,
$v=\operatorname{col}(v_{1},v_{2})$, $d=\operatorname{col}(d_{1},d_{2})$,
$z=\operatorname{col}(z_{1},z_{2})$; with $A=\operatorname{diag}(A_{1},A_{2})$
and with $B$, $C$, $D$, $E$, $F$, $K$, $L$ defined as block diagonal matrices
in the same way as $A$.

Let the external supply function be given by

$s_{ext}(d,z):=-\begin{pmatrix}d\\\
z\end{pmatrix}^{\top}\begin{pmatrix}Q^{P}&S^{P}\\\
(S^{P})^{\top}&R^{P}\end{pmatrix}\begin{pmatrix}d\\\ z\end{pmatrix}.\;$

The fact that $s_{ext}$ has an additive structure implies that the matrices in
the above definition of $s_{ext}$ are block diagonal, as follows:

$\Pi^{P}=\begin{pmatrix}Q^{P}&S^{P}\\\
(S^{P})^{\top}&R^{P}\end{pmatrix}:=\left(\begin{array}[]{c c : c
c}Q_{1}^{P}&0&S_{1}^{P}&0\\\ 0&Q_{2}^{P}&0&S_{2}^{P}\\\
\hdashline(S_{1}^{P})^{\top}&0&R_{1}^{P}&0\\\
0&(S_{2}^{P})^{\top}&0&R_{2}^{P}\end{array}\right),$ (25)

since then we have

$s_{ext}(d,z)=\sum_{i=1}^{2}\underbrace{\begin{pmatrix}d_{i}\\\
z_{i}\end{pmatrix}^{\top}\begin{pmatrix}-Q_{i}^{P}&-S_{i}^{P}\\\
-(S_{i}^{P})^{\top}&-R_{i}^{P}\end{pmatrix}\begin{pmatrix}d_{i}\\\
z_{i}\end{pmatrix}}_{s_{i,ext}(d_{i},z_{i})}.$ (26)

The dissipativity condition with respect to $s_{ext}(d,z)$ and with a storage
function $V(x)=x^{\top}Px$ is, according to statement (b) of Theorem 8.2 from
Appendix Appendix A: Robust stability and robust dissipativity, equivalent to
the existence of a multiplier

$\Pi=\begin{pmatrix}Q&S\\\ S^{\top}&R\end{pmatrix}$ (27)

such that

$\begin{pmatrix}H\\\ I\end{pmatrix}^{\top}\underbrace{\begin{pmatrix}Q&S\\\
S^{\top}&R\end{pmatrix}}_{\Pi}\begin{pmatrix}H\\\ I\end{pmatrix}\succeq 0$
(28)

and

$\left(\star\right)^{\top}\left(\begin{array}[]{cc:cc:cc}0&P&0&0&0&0\\\
P&0&0&0&0&0\\\ \hdashline 0&0&Q&S&0&0\\\ 0&0&S^{\top}&R&0&0\\\ \hdashline
0&0&0&0&Q^{P}&S^{P}\\\
0&0&0&0&(S^{P})^{\top}&R^{P}\end{array}\right)\left(\begin{array}[]{ccc}I&0&0\\\
A&B&E\\\ \hdashline 0&I&0\\\ C&D&0\\\ \hdashline 0&0&I\\\
F&K&L\end{array}\right)\prec 0.$ (29)

Note that $\Pi$ is in general a full-block multiplier (i.e., $\Pi$ is a _full
symmetric matrix_). Furthermore, note that we do not require any condition
regarding definiteness of the matrix $R^{P}$, as opposed to the case in
statement (a) of Theorem 8.2.

Since the statement of Theorem 3.1 assumes a storage function $V(x)$ with an
additive structure, the matrix $P$ in (29) is block diagonal matrix, that is,
$P=\operatorname{diag}(P_{1},P_{2})$.

Next we present the main idea of the proof.

_The starting point:_ The LMIs (28) and (29) are necessarily satisfied for
some full block $\Pi$. The LMI (29) implies (see Section 2.2) that the system
$G_{0}$ is dissipative with respect to the supply function
$s(v,w,d,z):=s_{H}(v,w)+s_{ext}(d,z)$, where

$s_{H}(v,w)=-\begin{pmatrix}v\\\
w\end{pmatrix}^{\top}\underbrace{\begin{pmatrix}Q&S\\\
S^{\top}&R\end{pmatrix}}_{\Pi}\begin{pmatrix}v\\\ w\end{pmatrix}.$ (30)

_The goal:_ Recall that our goal is to show that for $i=1,2$, the system
$G_{i}$ is strictly dissipative with
$s_{i}(d_{i},v_{i},z_{i},w_{i})=s_{i,ext}(d_{i},z_{i})+s_{i,int}(v_{i},w_{i})$
with respect to the storage function $V_{i}(x_{i})=x_{i}^{\top}P_{i}x_{i}$. We
will call these conditions _local dissipativity conditions_. Furthermore, we
require $s_{1,int}(v_{1},w_{1})+s_{2,int}(v_{2},w_{2})=0$, for all
$v_{1}=w_{2}$, $v_{2}=w_{1}$. We will call this condition the _neutrality
condition_. Next we show how we can “translate” these desired properties into
equivalent conditions imposed on $G_{0}$ and $\Pi$, via (28), (29) and the
dissipativity interpretation of (29).

Suppose that the matrices $Q$, $S$ and $R$ from (29), i.e., from (30), are
block diagonal, that is, $Q=Q^{D}:=\operatorname{diag}(Q_{1}^{D},Q_{2}^{D})$,
$R=R^{D}:=\operatorname{diag}(R_{1}^{D},R_{2}^{D})$ and
$S=S^{D}:=\operatorname{diag}(S_{1}^{D},S_{2}^{D})$, with the dimensions of
the blocks in conformity with dimensions of $v_{1}$, $v_{2}$, $w_{1}$, $w_{2}$
that appear in $v$ and $w$ from (30). In that case, and since _all_ the
matrices (e.g., $P$, $A$, $B$, etc.) that appear in (29) are block diagonal
(now including also $Q=Q^{D}$, $R=R^{D}$, $S=S^{D}$), the LMI (29) can be
decomposed into the following two independent LMIs:

$\left(\star\right)^{\top}\left(\begin{array}[]{cc:cc:cc}0&P_{i}&0&0&0&0\\\
P_{i}&0&0&0&0&0\\\ \hdashline 0&0&Q^{D}_{i}&S^{D}_{i}&0&0\\\
0&0&(S^{D}_{i})^{\top}&R^{D}_{i}&0&0\\\ \hdashline
0&0&0&0&Q^{P}_{i}&S^{P}_{i}\\\
0&0&0&0&(S^{P}_{i})^{\top}&R^{P}_{i}\end{array}\right)\left(\begin{array}[]{ccc}I&0&0\\\
A_{i}&B_{i}&E_{i}\\\ \hdashline 0&I&0\\\ C_{i}&D_{i}&0\\\ \hdashline 0&0&I\\\
F_{i}&K_{i}&L_{i}\end{array}\right)\prec 0,$ (31)

$i=1,2$. These two LMIs in turn imply (see Section 2.2) that the _local
dissipativity conditions_ are satisfied with
$s_{i,int}:=-\begin{pmatrix}v_{i}\\\
w_{i}\end{pmatrix}^{\top}\begin{pmatrix}Q_{i}^{D}&S_{i}^{D}\\\
(S_{i}^{D})^{\top}&R_{i}^{D}\end{pmatrix}\begin{pmatrix}v_{i}\\\
w_{i}\end{pmatrix}$, for $i=1,2$. Furthermore, with such definitions of
$s_{1,int}$ and $s_{2,int}$, it is easy to verify that the _neutrality_
condition is satisfied if and only if

$\begin{pmatrix}Q_{2}^{D}&S_{2}^{D}\\\
(S_{2}^{D})^{\top}&R_{2}^{D}\end{pmatrix}=\begin{pmatrix}-R_{1}^{D}&-(S_{1}^{D})^{\top}\\\
-S_{1}^{D}&-Q_{1}^{D}\end{pmatrix}.$ (32)

With the abbreviation $\Pi_{D}:=\begin{pmatrix}Q^{D}&S^{D}\\\
(S^{D})^{\top}&R^{D}\end{pmatrix}$, it is crucial to note that for $\Pi_{D}$
which satisfies the neutrality condition (32), the LMI (28) is necessarily
satisfied with equality sign when $\Pi$ is replaced by $\Pi_{D}$.

To summarize, the goal we want to reach is equivalent to showing that the
dissipativity LMI (29) is feasible when the full block multiplier $\Pi$ is
replaced with some structured multiplier $\Pi_{D}$, while $\Pi_{D}$ also
satisfies the condition (32).

_The main idea and proof overview_ : The main idea is to _construct_ a
structured multiplier $\Pi_{D}$ which satisfies all the above described
conditions, starting from existence of a full-block multiplier $\Pi$ which
satisfies (28) and (29).

The remaining part of the proof is divided in 3 steps. In _Step 1_ we derive
an alternative dissipativity condition, which is equivalent to feasibility of
(29). Note that in this step we exploit the assumption that both $C_{1}$ and
$C_{2}$ are full row rank matrices. In _Step 2_ , this alternative condition
is combined with (28) to derive an auxiliary result which will be used in
_Step 3_ to show that the proposed $\Pi_{D}$ satisfies the alternative
dissipativity condition. In _Step 3_ we construct $\Pi_{D}$ which satisfies
all required conditions.

_Step 1_ : Alternative characterization of dissipativity.
We first consider the case when $n_{w1}<n_{1}$ and $n_{w2}<n_{2}$. At the end
of the proof we remark on the case when $n_{w1}=n_{1}$ and $n_{w2}=n_{2}$, or
when we have some combination of the above equalities/inequalities.

Let $V_{1}$ span the kernel of $C_{1}$ and $V_{2}$ span the kernel of $C_{2}$.
Furthermore, let $W_{1}$ and $W_{2}$ be the matrices whose columns span the
orthogonal subspaces to $V_{1}$ and $V_{2}$, respectively. Let

$T:=\begin{pmatrix}\Xi&0&0\\\ 0&C_{W}&0\\\ 0&0&I\end{pmatrix},$ (33)

where $\Xi=\begin{pmatrix}V&W\end{pmatrix}$ with
$V=\operatorname{diag}(V_{1},V_{2})$, $W=\operatorname{diag}(W_{1},W_{2})$,
while $C_{W}=\operatorname{diag}(C_{2}W_{2},C_{1}W_{1})$. Note that $T$ is a
nonsingular square matrix (which would not be the case if $C_{1}$ and/or
$C_{2}$ would not have full row rank). After congruence transformation with
$T$ on (29) we obtain the inequality (34),

$\begin{pmatrix}V^{\top}MV&V^{\top}MW&V^{\top}N_{A}C_{W}&V^{\top}N_{B}\\\
W^{\top}MV&W^{\top}(M+C^{\top}RC)W&W^{\top}(N_{A}+C^{\top}S^{\top}+C^{\top}RD)C_{W}&W^{\top}N_{B}\\\
C_{W}^{\top}N_{A}^{\top}V&C_{W}^{\top}(N_{A}^{\top}+SC+D^{\top}RC)W&C_{W}^{\top}(K^{\top}R^{P}K+Q+SD+D^{\top}S^{\top}+D^{\top}RD)C_{W}&C_{W}^{\top}N_{C}\\\
N_{B}^{\top}V&N_{B}^{\top}W&N_{C}^{\top}C_{W}&N_{D}\end{pmatrix}\prec 0$ (34)

where we have used the abbreviations

$\displaystyle M$ $\displaystyle:=A^{\top}P+PA+F^{\top}R^{P}F,$ (35)
$\displaystyle N_{A}$ $\displaystyle:=PB+F^{\top}R^{P}K,$ (36) $\displaystyle
N_{B}$ $\displaystyle:=PE+F^{\top}(S^{P})^{\top}+F^{\top}R^{P}L,$ (37)
$\displaystyle N_{C}$ $\displaystyle:=K^{\top}(S^{P})^{\top}+K^{\top}R^{P}L,$
(38) $\displaystyle N_{D}$
$\displaystyle:=Q^{P}+S^{P}L+L^{\top}(S^{P})^{\top}+L^{\top}R^{P}L.$ (39)

After applying Schur complement rule twice on the inequality (34), first time
with the diagonal block $V^{\top}MV$ (upper left block in (34)) to be
inverted, and the second time with the diagonal block
$N_{D}-N_{B}^{\top}V(V^{\top}MV)^{-1}V^{\top}N_{B}$ (which appears on diagonal
in one of the matrices after the first Schur complement has been applied) to
be inverted, we obtain the following inequalities which are equivalent to
(34), hence also to (29):

$\displaystyle V^{\top}MV\prec 0$ (40a) $\displaystyle
N_{D}-N_{B}^{\top}V(V^{\top}MV)^{-1}V^{\top}N_{B}\prec 0,$ (40b)
$\displaystyle\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}Q&S\\\
S^{\top}&R\end{pmatrix}\begin{pmatrix}0&I\\\
I&D\end{pmatrix}\begin{pmatrix}CW&0\\\ 0&C_{W}\end{pmatrix}+$
$\displaystyle+\underbrace{\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}\breve{R}_{I}+\breve{R}_{II}&\breve{S}_{I}^{\top}+\breve{S}_{II}^{\top}\\\
\breve{S}_{I}+\breve{S}_{II}&\breve{Q}_{I}+\breve{Q}_{II}\end{pmatrix}\begin{pmatrix}W&0\\\
0&C_{W}\end{pmatrix}}_{Y}\prec 0.$ (40c)

In the above inequalities where we have used the abbreviations

$\displaystyle\breve{R}_{I}=M-MV(V^{\top}MV)^{-1}V^{\top}M,$ (41a)
$\displaystyle\breve{S}_{I}=N_{A}^{\top}-N_{A}^{\top}V(V^{\top}MV)^{-1}V^{\top}M,$
(41b)
$\displaystyle\breve{Q}_{I}=K^{\top}R^{P}K-N_{A}^{\top}V(V^{\top}MV)^{-1}V^{\top}N_{A},$
(41c)
$\displaystyle\breve{R}_{II}=-\Big{(}\star\Big{)}^{\top}Z^{-1}\Big{(}N_{B}^{\top}-N_{B}^{\top}V(V^{\top}MV)^{-1}V^{\top}M\Big{)},$
(41d) $\displaystyle\breve{S}_{II}=-\tilde{Z}\,Z^{-1}\,\tilde{\tilde{Z}},$
(41e)
$\displaystyle\breve{Q}_{II}=-\Big{(}\star\Big{)}^{\top}Z^{-1}\Big{(}N_{C}^{\top}-N_{B}^{\top}V(V^{\top}MV)^{-1}V^{\top}N_{A}\Big{)}.$
(41f)

with $Z:=N_{D}-N_{B}^{\top}V(V^{\top}MV)^{-1}V^{\top}N_{B}$,
$\tilde{Z}:=N_{C}-N_{A}^{\top}V(V^{\top}MV)^{-1}V^{\top}N_{B}$,
$\tilde{\tilde{Z}}:=N_{B}^{\top}-N_{B}^{\top}V(V^{\top}MV)^{-1}V^{\top}M$. Let

$\hat{R}:=\breve{R}_{I}+\breve{R}_{II},\,\,\,\hat{S}:=\breve{S}_{I}+\breve{S}_{II}\,\,\,\text{and}\,\,\,\hat{Q}:=\breve{Q}_{I}+\breve{Q}_{II}.$
(42)

Note that $\hat{R},\hat{S}$ and $\hat{Q}$ are by construction block diagonal
matrices, i.e., we can write

$\hat{R}=\begin{pmatrix}\hat{R}_{1}&0\\\
0&\hat{R}_{2}\end{pmatrix},\quad\hat{S}=\begin{pmatrix}\hat{S}_{1}&0\\\
0&\hat{S}_{2}\end{pmatrix},\quad\hat{Q}=\begin{pmatrix}\hat{Q}_{1}&0\\\
0&\hat{Q}_{2}\end{pmatrix},$

where $\hat{R}_{i}\in\mathbb{R}^{n_{i}\times n_{i}}$ for $i=1,2$,
$\hat{Q}_{1}\in\mathbb{R}^{n_{w}\times n_{w}}$,
$\hat{Q}_{2}\in\mathbb{R}^{n_{z}\times n_{z}}$,
$\hat{S}_{1}\in\mathbb{R}^{n_{w}\times n_{1}}$ and
$\hat{S}_{2}\in\mathbb{R}^{n_{z}\times n_{2}}$.

Let us define $L_{1}=\left(\begin{smallmatrix}C_{1}\\\
V_{1}^{\top}\end{smallmatrix}\right)$,
$L_{2}=\left(\begin{smallmatrix}C_{2}\\\
V_{2}^{\top}\end{smallmatrix}\right)$. Note that
$L_{1}\in\mathbb{R}^{n_{1}\times n_{1}}$ and $L_{2}\in\mathbb{R}^{n_{2}\times
n_{2}}$ are nonsingular square matrices and that

$L_{1}W_{1}=\begin{pmatrix}C_{1}W_{1}\\\ 0\end{pmatrix},\quad\quad
L_{2}W_{2}=\begin{pmatrix}C_{2}W_{2}\\\ 0\end{pmatrix}.$

With $L=\operatorname{diag}(L_{1},L_{2})$ the matrix $Y$ from (40c) can be
presented as

$Y=\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}L^{-\top}\hat{R}L^{-1}&L^{-\top}\hat{S}^{\top}\\\
\hat{S}L^{-1}&\hat{Q}\end{pmatrix}\begin{pmatrix}LW&0\\\
0&C_{W}\end{pmatrix}.$ (43)

Note that

$LW=\left(\begin{array}[]{c : c}C_{1}W_{1}&0\\\ 0&0\\\ \hdashline
0&C_{2}W_{2}\\\ 0&0\end{array}\right)$ (44)

and that we can, in conformity with the above partition of $LW$, partition
$L^{-\top}\hat{R}L^{-1}$ and $\hat{S}L^{-1}$ into blocks

$L^{-\top}\hat{R}L^{-1}=\left(\begin{array}[]{c c : c
c}\tilde{R}^{1}_{11}&\tilde{R}^{1}_{12}&0&0\\\
(\tilde{R}^{1}_{12})^{\top}&\tilde{R}^{1}_{22}&0&0\\\ \hdashline
0&0&\tilde{R}^{2}_{11}&\tilde{R}^{2}_{12}\\\
0&0&(\tilde{R}^{2}_{12})^{\top}&\tilde{R}^{2}_{22}\end{array}\right),$ (45)
$\hat{S}L^{-1}=\left(\begin{array}[]{c c : c
c}\tilde{S}^{1}_{11}&\tilde{S}^{1}_{12}&0&0\\\ \hdashline
0&0&\tilde{S}^{2}_{11}&\tilde{S}^{2}_{12}\end{array}\right).$

The matrix $Y$ from (43), after multiplications and with
$C_{W}=\operatorname{diag}(C_{2}W_{2},C_{1}W_{1})$, becomes

$Y=\begin{pmatrix}CW&0\\\
0&C_{W}\end{pmatrix}^{\top}\begin{pmatrix}\mathcal{R}&\mathcal{S}^{\top}\\\
\mathcal{S}&\mathcal{Q}\end{pmatrix}\begin{pmatrix}CW&0\\\
0&C_{W}\end{pmatrix},$ (46)

where

$\displaystyle\mathcal{R}$ $\displaystyle=\begin{pmatrix}\mathcal{R}_{1}&0\\\
0&\mathcal{R}_{2}\end{pmatrix}:=\begin{pmatrix}\tilde{R}^{1}_{11}&0\\\
0&\tilde{R}^{2}_{11}\end{pmatrix},$ (47a) $\displaystyle\mathcal{S}$
$\displaystyle=\begin{pmatrix}\mathcal{S}_{1}&0\\\
0&\mathcal{S}_{2}\end{pmatrix}:=\begin{pmatrix}\tilde{S}^{1}_{11}&0\\\
0&\tilde{S}^{2}_{11}\end{pmatrix},$ (47b) $\displaystyle\mathcal{Q}$
$\displaystyle=\begin{pmatrix}\mathcal{Q}_{1}&0\\\
0&\mathcal{Q}_{2}\end{pmatrix}:=\begin{pmatrix}\hat{Q}_{1}&0\\\
0&\hat{Q}_{2}\end{pmatrix}=\hat{Q}.$ (47c)

The inequality (40c) now reads as

$\begin{pmatrix}\star\end{pmatrix}^{\top}\left(\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}Q&S\\\
S^{\top}&R\end{pmatrix}\begin{pmatrix}0&I\\\
I&D\end{pmatrix}+\begin{pmatrix}\mathcal{R}&\mathcal{S}^{\top}\\\
\mathcal{S}&\mathcal{Q}\end{pmatrix}\right)\begin{pmatrix}CW&0\\\
0&C_{W}\end{pmatrix}\prec 0.$ (48)

Since $CW$ and $C_{W}$ are nonsigular square matrices, (40c) is equivalent to

$\begin{pmatrix}I&0\\\ D&I\end{pmatrix}^{\top}\begin{pmatrix}Q&S\\\
S^{\top}&R\end{pmatrix}\begin{pmatrix}I&0\\\
D&I\end{pmatrix}+\begin{pmatrix}\mathcal{Q}&\mathcal{S}\\\
\mathcal{S}^{\top}&\mathcal{R}\end{pmatrix}\prec 0.$ (49)

Recall that $Q$, $S$ and $R$ are full matrices, while $\mathcal{Q}$,
$\mathcal{S}$ and $\mathcal{R}$ are block diagonal matrices, derived from the
parameters of the systems ($A_{i},B_{i},C_{i}),i=1,2$ and the Laypunov
matrices $P_{1},P_{2}$. The derived results up to now can be summarized in the
following equivalence

$\text{\eqref{DisCompact0}, \eqref{DisCompactA},
\eqref{QSRinequality_53}}\quad\iff\quad\text{\eqref{NN2_53}}.$ (50)

The inequalities (40a), (40b) do not depend on $Q$, $S$ and $R$, and since our
goal is to devise new multipliers $Q$, $S$ and $R$, which define the
interconnection neutral supply rates, in the remainder we focus on the
inequality (49).

_Step 2_ : Auxiliary result.
After congruence transformation of (49) with $\left(\begin{smallmatrix}I&0\\\
-D&I\end{smallmatrix}\right)$ we have

$\begin{pmatrix}Q&S\\\ S^{\top}&R\end{pmatrix}+\begin{pmatrix}I&0\\\
-D&I\end{pmatrix}^{\top}\begin{pmatrix}\mathcal{Q}&\mathcal{S}\\\
\mathcal{S}^{\top}&\mathcal{R}\end{pmatrix}\begin{pmatrix}I&0\\\
-D&I\end{pmatrix}\prec 0.$ (51)

Pre-multiplying (51) with $\left(\begin{smallmatrix}H\\\
I\end{smallmatrix}\right)^{\top}$ and post-multiplying with
$\left(\begin{smallmatrix}H\\\ I\end{smallmatrix}\right)$, and using (28), we
obtain

$\displaystyle\mathcal{M}=\begin{pmatrix}\mathcal{M}_{1}&\mathcal{M}_{2}\\\
\mathcal{M}_{2}^{\top}&\mathcal{M}_{3}\end{pmatrix}\prec 0,$ (52)
$\displaystyle\mathcal{M}_{1}=\mathcal{R}_{1}+\mathcal{Q}_{2}-\mathcal{S}_{2}D_{2}-D_{2}^{\top}\mathcal{S}_{2}^{\top}+D_{2}^{\top}\mathcal{R}_{2}D_{2},$
$\displaystyle\mathcal{M}_{2}=\mathcal{S}_{1}^{\top}+\mathcal{S}_{2}-\mathcal{R}_{1}D_{1}-D_{2}^{\top}\mathcal{R}_{2},$
$\displaystyle\mathcal{M}_{3}=\mathcal{Q}_{1}+\mathcal{R}_{2}-\mathcal{S}_{1}D_{1}-D_{1}^{\top}\mathcal{S}_{1}^{\top}+D_{1}^{\top}\mathcal{R}_{1}D_{1}.$

_Step 3_ : Construction of structured multipliers.
Our goal is now to find a new multiplier

$\Pi_{D}=\begin{pmatrix}Q^{D}&S^{D}\\\ (S^{D})^{\top}&R^{D}\end{pmatrix}$ (53)

with block diagonal matrices $Q^{D}=\operatorname{diag}(Q^{D}_{1},Q^{D}_{2})$,
$S^{D}=\operatorname{diag}(S^{D}_{1},S^{D}_{2})$ and
$R^{D}=\operatorname{diag}(R^{D}_{1},R^{D}_{2})$, which can replace $\Pi$ from
(27) in the inequality (29) so that this inequality remains to hold, while in
addition the multiplier $\Pi_{D}$ satisfies the neutrality condition (32). Due
to (50), the multiplier $\Pi_{D}$ satisfies the inequality (29) if and only if

$\begin{pmatrix}I&0\\\ D&I\end{pmatrix}^{\top}\begin{pmatrix}Q^{D}&S^{D}\\\
(S^{D})^{\top}&R^{D}\end{pmatrix}\begin{pmatrix}I&0\\\
D&I\end{pmatrix}+\begin{pmatrix}\mathcal{Q}&\mathcal{S}\\\
\mathcal{S}^{\top}&\mathcal{R}\end{pmatrix}\prec 0.$ (54)

Consider the multiplier $\Pi_{D}$ defined with the following matrices

$\displaystyle Q^{D}_{1}$
$\displaystyle=\alpha(-\mathcal{Q}_{1}+D_{1}^{\top}\mathcal{S}_{1}^{\top}+\mathcal{S}_{1}D_{1}-D_{1}^{\top}\mathcal{R}_{1}D_{1})+(1-\alpha)\mathcal{R}_{2},$
(55a) $\displaystyle S^{D}_{1}$
$\displaystyle=\alpha(-\mathcal{S}_{1}+D_{1}^{\top}\mathcal{R}_{1})+(1-\alpha)(\mathcal{S}_{2}^{\top}-\mathcal{R}_{2}D_{2}),$
(55b) $\displaystyle R^{D}_{1}$
$\displaystyle=\alpha(-\mathcal{R}_{1})+(1-\alpha)(\mathcal{Q}_{2}-\mathcal{S}_{2}D_{2}-D_{2}^{\top}\mathcal{S}_{2}^{\top}+D_{2}^{\top}\mathcal{R}_{2}D_{2}),$
(55c) $\displaystyle Q^{D}_{2}$ $\displaystyle=-R^{D}_{1},\quad
S^{D}_{2}=-(S^{D}_{1})^{\top},\quad R^{D}_{2}=-Q^{D}_{1},$ (55d)

where $\alpha$ is a real number from the interval $(0,1)$. We have

$\displaystyle\begin{pmatrix}I&0\\\
D&I\end{pmatrix}^{\top}\begin{pmatrix}Q^{D}&S^{D}\\\
(S^{D})^{\top}&R^{D}\end{pmatrix}\begin{pmatrix}I&0\\\
D&I\end{pmatrix}+\begin{pmatrix}\mathcal{Q}&\mathcal{S}\\\
\mathcal{S}^{\top}&\mathcal{R}\end{pmatrix}=$
$\displaystyle\left(\star\right)^{\top}\left(\begin{array}[]{c c : c
c}(1-\alpha)\mathcal{M}_{3}&0&(1-\alpha)\mathcal{M}_{2}^{\top}&0\\\
0&\alpha\mathcal{M}_{1}&0&\alpha\mathcal{M}_{2}\\\
\hdashline(1-\alpha)\mathcal{M}_{2}&0&(1-\alpha)\mathcal{M}_{1}&0\\\
0&\alpha\mathcal{M}_{2}^{\top}&0&\alpha\mathcal{M}_{3}\end{array}\right)\begin{pmatrix}I&0\\\
D&I\end{pmatrix}\prec 0,$ (60)

that is, the condition (54) is satisfied. The inequality in (60) follows from
(52). Finally, note that due to (55d), $\Pi_{D}$ by construction also
satisfies the neutrality condition (32).

This concludes the proof for the case when $n_{w1}<n_{1}$ and $n_{w2}<n_{2}$.
In the case when either $n_{w1}=n_{1}$ or $n_{w2}=n_{2}$ (or both), the proof
follows along the similar lines, except that the congruence transformation
with the matrix $T$, defined in (33), is either completely omitted (when
$n_{w1}=n_{1}$, $n_{w2}=n_{2}$) or $T$ is suitably modified. ∎

###  7.2 Proof of Proposition 3.3

We present proof of the extension of Theorem 3.1 in the case when $C_{1}$ and
$C_{2}$ are not full row rank. From there, proof of the analogous extension of
Corollary 3.2 follows directly by omitting the matrices related to exogenous
inputs/outputs $d_{1},d_{2},z_{1},z_{2}$, i.e., the matrices
$E_{i},F_{i},K_{i},L_{i}$, $i=1,2$, and indeed by omitting the terms related
to the external supply functions $s_{1,ext}$ and $s_{2,ext}$.

###### Step 0: The main idea and overview.

Suppose that $C_{1}\in\mathbb{R}^{n_{w1}\times n_{1}}$ and
$C_{2}\in\mathbb{R}^{n_{w2}\times n_{2}}$ are row rank deficient. Let
$\tilde{n}_{w1}$ ($\tilde{n}_{w1}<n_{w1}$) and $\tilde{n}_{w2}$
$(\tilde{n}_{w2}<n_{w2})$ denote respectively rank of $C_{1}$ and rank of
$C_{2}$. Then, without loss of generality, we can take

$C_{1}=\begin{pmatrix}J_{1}\\\
I_{\tilde{n}_{w1}}\end{pmatrix}\tilde{C}_{1},\quad\quad
C_{2}=\begin{pmatrix}J_{2}\\\ I_{\tilde{n}_{w2}}\end{pmatrix}\tilde{C}_{2},$
(61)

where $\tilde{C}_{1}\in\mathbb{R}^{\tilde{n}_{w1}\times n_{1}}$ and
$\tilde{C}_{2}\in\mathbb{R}^{\tilde{n}_{w2}\times n_{2}}$ are full row rank
matrices. The matrix $J_{1}$ defines the first $(n_{w1}-\tilde{n}_{w1})$ rows
of $C_{1}$ as linear combinations of rows of $\tilde{C}_{1}$, while the matrix
$J_{2}$ does the same for $C_{2}$.

$G$$z_{1}$$d_{1}$$G_{1}$$v_{1}$$w_{1}$$w_{2}$$v_{2}$$G_{2}$$d_{2}$$z_{2}$$\downarrow$$\tilde{G}$$z_{1}$$d_{1}$$\tilde{G}_{1}$$\tilde{v}_{1}$$\tilde{w}_{1}$$\tilde{w}_{2}$$\tilde{v}_{2}$$\tilde{G}_{2}$$d_{2}$$z_{2}$
Figure 10: Redefinition of local systems and interconnecting signals.

Instead of considering interconnection of systems $G_{1}$ and $G_{2}$ given by
(9) with $D_{1}=0$ and $D_{2}=0$ (let us denote this interconnection with $G$,
as done in Figure 10), we consider dissipativity properties of the system
obtained by interconnecting $\tilde{G}_{1}$ and $\tilde{G}_{2}$ with

$\tilde{G}_{i}:\,\begin{pmatrix}\dot{x}_{i}\\\ \tilde{w}_{i}\\\
z_{i}\end{pmatrix}=\begin{pmatrix}A_{i}&\tilde{B}_{i}&E_{i}\\\
\tilde{C}_{i}&0&0\\\
F_{i}&\tilde{K}_{i}&L_{i}\end{pmatrix}\begin{pmatrix}x_{i}\\\ \tilde{v}_{i}\\\
d_{i}\end{pmatrix},\quad i=1,2,$ (62)

where

$\tilde{B}_{1}=B_{1}\begin{pmatrix}J_{2}\\\
I\end{pmatrix},\,\,\tilde{K}_{1}=K_{1}\begin{pmatrix}J_{2}\\\
I\end{pmatrix},\,\,\tilde{B}_{2}=B_{2}\begin{pmatrix}J_{1}\\\
I\end{pmatrix},\,\,\tilde{K}_{2}=K_{2}\begin{pmatrix}J_{1}\\\ I\end{pmatrix}.$
(63)

Let $\tilde{G}$ denote the interconnection of $\tilde{G}_{1}$ and
$\tilde{G}_{2}$ obtained with constraints $\tilde{v}_{1}=\tilde{w}_{2}$ and
$\tilde{v}_{2}=\tilde{w}_{1}$. Then $\tilde{G}$ and $G$ are the same system
when observed from the input-output behaviour of the exogenous signals
$d:=\operatorname{col}(d_{1},d_{2})\rightarrow
z=\operatorname{col}(z_{1},z_{2})$. The only difference is in the definitions
of internal “subsystems” ($G_{1}$ and $G_{2}$ in $G$; $\tilde{G}_{1}$ and
$\tilde{G}_{2}$ in $\tilde{G}$) and the corresponding interconnection signals:
$w_{1}$, $v_{1}$ in $G$ and $\tilde{w}_{1}$, $\tilde{v}_{1}$ in $\tilde{G}$,
as presented in Figure 10.

$B_{1}$$\begin{pmatrix}J_{2}\\\
I\end{pmatrix}$$\tilde{C}_{2}$$\tilde{B}_{1}$$C_{2}$$x_{2}$$\tilde{w}_{2}$$\tilde{v}_{1}$$w_{2}$$v_{1}$
Figure 11: Interconnection between $G_{1}$ and $G_{2}$, versus interconnection
between $\tilde{G}_{1}$ and $\tilde{G}_{2}$.

The main principle behind the above described redefinition of signals and
system matrices is illustrated in Figure 11, which presents relation of the
interconnection signal between $G_{1}$ and $G_{2}$ (signal $v_{1}=w_{2}$), and
the signal between $\tilde{G_{1}}$ and $\tilde{G}_{2}$ (signal
$\tilde{v}_{1}=\tilde{w}_{2}$). The same figure illustrates the relations
between $\tilde{B}_{1}$ and $B_{1}$ as well as $\tilde{C}_{2}$ and $C_{2}$. By
altering all the indices in the figure from $1$ to $2$ and from $2$ to $1$, we
obtain illustration of the relations between $\tilde{B}_{2}$ and $B_{2}$,
between $\tilde{C}_{1}$ and $C_{1}$, and between signals pars $(v_{2},w_{1})$
and $(\tilde{v}_{2},\tilde{w}_{1})$.

We emphasize that the above redefinition of signals, matrices and subsystems
cannot be done in case when $D_{1}\neq 0$ and/or $D_{2}\neq 0$.

_The main idea_ : Note that in $\tilde{G}$, both $\tilde{C}_{1}$ and
$\tilde{C}_{2}$ are full row rank matrices and therefore we can apply Theorem
3.1 to infer existence of interconnection neutral supply functions
$\tilde{s}_{1,int}(\tilde{v}_{1},\tilde{w}_{1})$ and
$\tilde{s}_{2,int}(\tilde{v}_{2},\tilde{w}_{2})$ on the interconnecting
signals of $\tilde{G}$. The main idea of the proof is to devise
interconnection neutral supply functions $s_{1,int}(v_{1},w_{1})$ and
$s_{2,int}(v_{2},w_{2})$ on the interconnecting signals of $G$, starting from
existence of $\tilde{s}_{1,int}$ and $\tilde{s}_{2,int}$.

_Overview_ :The remainder of the proof is divided in 4 steps. In _Step 1_ we
define $\tilde{s}_{1,int}$ and $\tilde{s}_{2,int}$ using a triplet of matrices
$(\tilde{Q},\tilde{R},\tilde{S})$. The goal is to find a new triplet $(Q,R,S)$
which defines $s_{1,int}(v_{1},w_{1})$ and $s_{2,int}(v_{2},w_{2})$. Since the
interconnection signals $\tilde{v}_{1}$, $\tilde{w_{1}}$, $\tilde{v}_{2}$,
$\tilde{w_{2}}$ have lower spatial dimension than the corresponding signals
$v_{1}$, $w_{1}$, $v_{2}$, $w_{2}$, i.e.,
$\tilde{v}_{1}(t)\in\mathbb{R}^{\tilde{n}_{w1}}$,
$v_{1}(t)\in\mathbb{R}^{n_{w1}}$ with $\tilde{n}_{w1}<n_{w1}$, etc., the
matrices $(Q,R,S)$ are characterized with larger number of rows/columns than
the corresponding matrices $(\tilde{Q},\tilde{R},\tilde{S})$. We say that
$(Q,R,S)$ are constructed by an appropriate _extension_ of
$(\tilde{Q},\tilde{R},\tilde{S})$. In _Step 1_ we present a set of algebraic
conditions for this extension. In _Step 2_ we present a procedure how to
construct $Q$ by an appropriate extension from $\tilde{Q}$. In _Step 3_ and
_Step 4_ we present similar extensions to construct $R$ and $S$, respectively.

_Step 1_ : Conditions for extension.
Let

$\tilde{s}_{1,int}(\tilde{v}_{1},\tilde{w}_{1})=\begin{pmatrix}\tilde{v}_{1}\\\
\tilde{w}_{1}\end{pmatrix}^{\top}\begin{pmatrix}\tilde{Q}&\tilde{S}\\\
\tilde{S}^{\top}&\tilde{R}\end{pmatrix}\begin{pmatrix}\tilde{v}_{1}\\\
\tilde{w}_{1}\end{pmatrix}.$ (64)

Then, with the neutrality condition
$(\tilde{v}_{1}=\tilde{w}_{2},\tilde{v}_{2}=\tilde{w}_{1})\implies\tilde{s}_{1,int}(\tilde{v}_{1},\tilde{w}_{1})+\tilde{s}_{2,int}(\tilde{v}_{2},\tilde{w}_{2})=0$
accounted for, we have

$\tilde{s}_{2,int}(\tilde{v}_{2},\tilde{w}_{2})=\begin{pmatrix}\tilde{v}_{2}\\\
\tilde{w}_{2}\end{pmatrix}^{\top}\begin{pmatrix}-\tilde{R}&-\tilde{S}^{\top}\\\
-\tilde{S}&-\tilde{Q}\end{pmatrix}\begin{pmatrix}\tilde{v}_{2}\\\
\tilde{w}_{2}\end{pmatrix}.$ (65)

Furthermore, let

$s_{i,ext}(d_{i},z_{i})=\left(\begin{smallmatrix}d_{i}\\\
z_{i}\end{smallmatrix}\right)^{\top}\left(\begin{smallmatrix}Q^{P}_{i}&S^{P}_{i}\\\
(S^{P}_{i})^{\top}&R^{P}_{i}\end{smallmatrix}\right)\left(\begin{smallmatrix}d_{i}\\\
z_{i}\end{smallmatrix}\right),\,\,\,\,i=1,2.$

With LMI characterization from Section 2.2, strict dissipativity of
$\tilde{G}_{1}$ with respect to
$s_{1,ext}(d_{1},z_{1})+\tilde{s}_{1,int}(\tilde{v}_{1},\tilde{w}_{1})$ and
strict dissipativity of $\tilde{G}_{2}$ with respect to
$s_{2,ext}(d_{2},z_{2})+\tilde{s}_{2,int}(\tilde{v}_{2},\tilde{w}_{2})$ are
respectively given by the following two inequalities

$\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}0&P_{1}\\\
P_{1}&0\end{pmatrix}\begin{pmatrix}I&0&0\\\
A_{1}&\tilde{B}_{1}&E_{1}\end{pmatrix}\prec\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}\tilde{Q}&\tilde{S}\\\
\tilde{S}^{\top}&\tilde{R}\end{pmatrix}\begin{pmatrix}0&I&0\\\
\tilde{C}_{1}&0&0\end{pmatrix}\\\
+\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}Q^{P}_{1}&S^{P}_{1}\\\
(S^{P}_{1})^{\top}&R^{P}_{1}\end{pmatrix}\begin{pmatrix}0&0&I\\\
F_{1}&\tilde{K}_{1}&L_{1}\end{pmatrix},$ (66)
$\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}0&P_{2}\\\
P_{2}&0\end{pmatrix}\begin{pmatrix}I&0&0\\\
A_{2}&\tilde{B}_{2}&E_{2}\end{pmatrix}\prec\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}-\tilde{R}&-\tilde{S}^{\top}\\\
-\tilde{S}&-\tilde{Q}\end{pmatrix}\begin{pmatrix}0&I&0\\\
\tilde{C}_{2}&0&0\end{pmatrix}\\\
+\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}Q^{P}_{2}&S^{P}_{2}\\\
(S^{P}_{2})^{\top}&R^{P}_{2}\end{pmatrix}\begin{pmatrix}0&0&I\\\
F_{2}&\tilde{K}_{2}&L_{2}\end{pmatrix}.$ (67)

Consider the equality

$\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}Q&S\\\
S^{\top}&R\end{pmatrix}\begin{pmatrix}\begin{pmatrix}J_{2}\\\
I\end{pmatrix}&0\\\ 0&\begin{pmatrix}J_{1}\\\
I\end{pmatrix}\end{pmatrix}=\begin{pmatrix}\tilde{Q}&\tilde{S}\\\
\tilde{S}^{\top}&\tilde{R}\end{pmatrix},$ (68)

which is a linear equation in $Q,S,R$ for some known
$\tilde{Q},\tilde{S},\tilde{R}$. Substituting (68) into (66), together with
(61) and (63), we obtain

$\tilde{W}_{1}^{\top}X_{1}\underbrace{\begin{pmatrix}I&0&0\\\
0&\begin{pmatrix}J_{2}\\\ I\end{pmatrix}&0\\\
0&0&I\end{pmatrix}}_{\tilde{W}_{1}}\prec\tilde{W}_{1}^{\top}(Y_{1}^{int}+Y_{1}^{ext})\underbrace{\begin{pmatrix}I&0&0\\\
0&\begin{pmatrix}J_{2}\\\ I\end{pmatrix}&0\\\
0&0&I\end{pmatrix}}_{\tilde{W}_{1}},$ (69)

where

$\displaystyle X_{1}=\begin{pmatrix}I&0&0\\\
A_{1}&B_{1}&E_{1}\end{pmatrix}^{\top}\begin{pmatrix}0&P_{1}\\\
P_{1}&0\end{pmatrix}\begin{pmatrix}I&0&0\\\ A_{1}&B_{1}&E_{1}\end{pmatrix},$
(70) $\displaystyle Y_{1}^{int}=\begin{pmatrix}0&I&0\\\
C_{1}&0&0\end{pmatrix}^{\top}\begin{pmatrix}Q&S\\\
S^{\top}&R\end{pmatrix}\begin{pmatrix}0&I&0\\\ C_{1}&0&0\end{pmatrix},$ (71)
$\displaystyle Y_{1}^{ext}=\begin{pmatrix}0&0&I\\\
F_{1}&K_{1}&L_{1}\end{pmatrix}^{\top}\begin{pmatrix}Q^{P}_{1}&S^{P}_{1}\\\
(S^{P}_{1})^{\top}&R^{P}_{1}\end{pmatrix}\begin{pmatrix}0&0&I\\\
F_{1}&K_{1}&L_{1}\end{pmatrix}.$ (72)

Similarly, substituting (68) into (67), together with (61) and (63), we obtain

$\tilde{W}_{2}^{\top}X_{2}\underbrace{\begin{pmatrix}I&0&0\\\
0&\begin{pmatrix}J_{1}\\\ I\end{pmatrix}&0\\\
0&0&I\end{pmatrix}}_{\tilde{W}_{2}}\prec\tilde{W}_{2}^{\top}(Y_{2}^{int}+Y_{2}^{ext})\underbrace{\begin{pmatrix}I&0&0\\\
0&\begin{pmatrix}J_{1}\\\ I\end{pmatrix}&0\\\
0&0&I\end{pmatrix}}_{\tilde{W}_{2}},$ (73)

where

$\displaystyle X_{2}=\begin{pmatrix}I&0&0\\\
A_{2}&B_{2}&E_{2}\end{pmatrix}^{\top}\begin{pmatrix}0&P_{2}\\\
P_{2}&0\end{pmatrix}\begin{pmatrix}I&0&0\\\ A_{2}&B_{2}&E_{2}\end{pmatrix},$
(74) $\displaystyle Y_{2}^{int}=\begin{pmatrix}0&I&0\\\
C_{2}&0&0\end{pmatrix}^{\top}\begin{pmatrix}-R&-S^{\top}\\\
-S&-Q\end{pmatrix}\begin{pmatrix}0&I&0\\\ C_{2}&0&0\end{pmatrix},$ (75)
$\displaystyle Y_{2}^{ext}=\begin{pmatrix}0&0&I\\\
F_{2}&K_{2}&L_{2}\end{pmatrix}^{\top}\begin{pmatrix}Q^{P}_{2}&S^{P}_{2}\\\
(S^{P}_{2})^{\top}&R^{P}_{2}\end{pmatrix}\begin{pmatrix}0&0&I\\\
F_{2}&K_{2}&L_{2}\end{pmatrix}.$ (76)

Our aim is to show that we can always select $Q,S$ and $R$ with (68) such that
$X_{1}\prec Y_{1}^{int}+Y_{1}^{ext}$ and $X_{2}\prec Y_{2}^{int}+Y_{2}^{ext}$.
Indeed, the latter two inequalities are precisely the dissipation inequalities
stating that $Q,S$ and $R$ define the interconnection neutral supply functions
$s_{1,int}(v_{1},w_{1})$ and $s_{2,int}(v_{2},w_{2})$ on the interconnecting
signals of $G$.

_Step 2_ : Constructing $Q$.
Consider first the inequality $X_{1}\prec Y_{1}^{int}+Y_{1}^{ext}$. The
inequality (69) implies that $X_{1}\prec Y_{1}^{int}+Y_{1}^{ext}$ on
$\operatorname{Im}\tilde{W}_{1}$, that is,
$x^{\top}X_{1}x<x^{\top}(Y_{1}^{int}+Y_{1}^{ext})x$ for all
$x\in\operatorname{Im}\tilde{W}_{1}$, $x\neq 0$, but not necessarily also for
an arbitrary $x\neq 0$. Note that
$\operatorname{Ker}\tilde{W}_{1}^{\top}=\operatorname{Im}\tilde{\tilde{W}}_{1}$
where
$\tilde{\tilde{W}}_{1}=\left(\begin{smallmatrix}0&\left(\begin{smallmatrix}I&-J_{2}\end{smallmatrix}\right)&0\end{smallmatrix}\right)^{\top}$,
and
$W_{1}:=\left(\begin{smallmatrix}\tilde{W}_{1}&\tilde{\tilde{W}}_{1}\end{smallmatrix}\right)$
is a nonsingular square matrix.

Next, we show that we can always select $Q,S$ and $R$ in (68) so that

$W_{1}^{\top}X_{1}W_{1}\prec W_{1}^{\top}(Y_{1}^{int}+Y_{1}^{ext})W_{1}.$ (77)

Since $W_{1}$ is a nonsingular square matrix, the latter inequality indeed
implies $X_{1}\prec Y_{1}^{int}+Y_{1}^{ext}$.

In addition to (68), we further constrain $Q$ by adding the following relation
between $Q$ and $\tilde{Q}$

$\begin{pmatrix}\star\end{pmatrix}^{\top}Q\begin{pmatrix}J_{2}&I\\\
I&-J_{2}^{\top}\end{pmatrix}=\begin{pmatrix}\tilde{Q}&0\\\
0&\gamma_{Q}I-\begin{pmatrix}I\\\
-J_{2}^{\top}\end{pmatrix}^{\top}K_{1}^{\top}R_{1}^{P}K_{1}\begin{pmatrix}I\\\
-J_{2}^{\top}\end{pmatrix}\end{pmatrix}$ (78)

for some fixed real number $\gamma_{Q}$. For given $\tilde{Q}$ and
$\gamma_{Q}$, the above equation uniquely defines $Q$. Note that the only
constraint on $Q$ from (68) is given by $\left(\begin{smallmatrix}J_{2}\\\
I\end{smallmatrix}\right)^{\top}Q\left(\begin{smallmatrix}J_{2}\\\
I\end{smallmatrix}\right)=\tilde{Q}$ and is also present in (78). In that
sense, uniquely defined $Q$ from (78) necessarily satisfies the constraint on
$Q$ imposed by (68).

With $Z_{1}:=X_{1}-Y_{1}^{int}-Y_{1}^{ext}$, the inequality (69) reads as
$\tilde{W}_{1}^{\top}Z_{1}\tilde{W}_{1}\prec 0$, while the inequality (77),
which we want to obtain, reads as

$W_{1}^{\top}Z_{1}W_{1}=\begin{pmatrix}\tilde{W}_{1}^{\top}Z_{1}\tilde{W}_{1}&\tilde{W}_{1}^{\top}Z_{1}\tilde{\tilde{W}}_{1}\\\
\tilde{\tilde{W}}_{1}^{\top}Z_{1}\tilde{W}_{1}&\tilde{\tilde{W}}_{1}^{\top}Z_{1}\tilde{\tilde{W}}_{1}\end{pmatrix}\prec
0.$ (79)

Note that
$\tilde{\tilde{W}}_{1}^{\top}Z_{1}\tilde{\tilde{W}}_{1}=\left(\begin{smallmatrix}I\\\
-J_{2}^{\top}\end{smallmatrix}\right)^{\top}(-Q-K_{1}^{\top}R_{1}^{P}K_{1})\left(\begin{smallmatrix}I\\\
-J_{2}^{\top}\end{smallmatrix}\right)$, which with (78) implies
$\tilde{\tilde{W}}_{1}^{\top}Z_{1}\tilde{\tilde{W}}_{1}=-\gamma_{Q}I$. Due to
this fact, and since $\tilde{W}_{1}^{\top}Z_{1}\tilde{W}_{1}\prec 0$, the
inequality (79) can always be rendered feasible by taking sufficiently large
positive real number $\gamma_{Q}$. To summarize, with sufficiently large
$\gamma_{Q}$, the equation (78) gives us the parameter matrix $Q$ for the
neutral supply rate within $G$, starting from the parameter matrix $\tilde{Q}$
of the system $\tilde{G}$ with modified interconnections.

_Step 3_ : Constructing $R$.
The inequality $X_{2}\prec Y_{2}^{int}+Y_{2}^{ext}$ follows by symmetry and as
a result gives us the following conditions which relate $R$ with $\tilde{R}$:

$\begin{pmatrix}\star\end{pmatrix}^{\top}R\begin{pmatrix}J_{1}&I\\\
I&-J_{1}^{\top}\end{pmatrix}=\begin{pmatrix}\tilde{R}&0\\\
0&-\gamma_{R}I+\begin{pmatrix}I\\\
-J_{1}^{\top}\end{pmatrix}^{\top}K_{2}^{\top}R_{2}^{P}K_{2}\begin{pmatrix}I\\\
-J_{1}^{\top}\end{pmatrix}\end{pmatrix}$ (80)

for some sufficiently large positive real number $\gamma_{R}$. The procedure
is completely analogous to the one for $Q$, i.e., with definitions
$W_{2}:=\left(\begin{smallmatrix}\tilde{W}_{2}&\tilde{\tilde{W}}_{2}\end{smallmatrix}\right)$,$\tilde{\tilde{W}}_{2}=\left(\begin{smallmatrix}0&\left(\begin{smallmatrix}I&-J_{1}\end{smallmatrix}\right)&0\end{smallmatrix}\right)^{\top}$,
we can always select $\gamma_{R}$ in (80) to impose the inequality
$W_{2}^{\top}X_{2}W_{2}\prec W_{2}^{\top}(Y_{2}^{int}+Y_{2}^{ext})W_{2}$.

_Step 4_ : Constructing $S$.
Finally, to complete the proof, note that conditions
$W_{1}^{\top}X_{1}W_{1}\prec W_{1}^{\top}(Y_{1}^{int}+Y_{1}^{ext})W_{1}$ and
$W_{2}^{\top}X_{2}W_{2}\prec W_{2}^{\top}(Y_{2}^{int}+Y_{2}^{ext})W_{2}$, with
(78) and (80), do not impose any additional constraints on $S$, that is, the
only constraints on $S$ that we consider are the ones imposed by (68), and it
is easy to see that they always have a solution. More precisely, (68) gives
$\left(\begin{smallmatrix}J_{2}^{\top}&I\end{smallmatrix}\right)S\left(\begin{smallmatrix}J_{1}\\\
I\end{smallmatrix}\right)=\tilde{S}$ as the relation between $S$ and
$\tilde{S}$, which always has a solution in $S$ for any given $\tilde{S}$. ∎

###  7.3 Proof of Proposition 5.2

###### Proof of the part (a).

_Step 0_ : The main idea and overview.
_The main idea_ : The presented proof is based on the proof of Theorem 3.1.
The proof of Theorem 3.1 is constructive in a sense that starting from
existence of a general (full-block) multiplier $\Pi$ in (27), we are able to
construct a structured multiplier
$\Pi_{D}=\left(\begin{smallmatrix}Q^{D}&S^{D}\\\
(S^{D})^{\top}&R^{D}\end{smallmatrix}\right)$ with
$Q^{D}=\operatorname{diag}(Q^{D}_{1},Q^{D}_{2})$,
$S^{D}=\operatorname{diag}(S^{D}_{1},S^{D}_{2})$,
$R^{D}=\operatorname{diag}(R^{D}_{1},R^{D}_{2})$ (see (53)), which defines the
interconnection neutral supply rates as $Q^{D}_{2}=-R^{D}_{1}$,
$S^{D}_{2}=-(S^{D}_{1})^{\top}$, $R^{D}_{2}=-Q^{D}_{1}$. Our goal here is to
show that with $D=\operatorname{diag}(D_{1},D_{2})=0$,
$K=\operatorname{diag}(K_{1},K_{2})=0$ and
$L=\operatorname{diag}(L_{1},L_{2})=0$, there exists a multiplier $\Pi_{D}$
which, in addition to the above properties, also satisfies the condition

$R^{D}=\begin{pmatrix}R^{D}_{1}&0\\\ 0&R^{D}_{2}\end{pmatrix}\succeq 0.$ (81)

Indeed, then the supply functions

$s_{i,int}:=-\left(\begin{smallmatrix}v_{i}\\\
w_{i}\end{smallmatrix}\right)^{\top}\left(\begin{smallmatrix}Q^{D}_{i}&S^{D}_{i}\\\
(S^{D}_{i})^{\top}&R^{D}_{i}\end{smallmatrix}\right)\left(\begin{smallmatrix}v_{i}\\\
w_{i}\end{smallmatrix}\right),\quad i=1,2,$ (82)

are interconnection neutral supply functions, while (81) implies (20).

_Starting point and overview_ : Note that with $D=0$, from (55) we have

$R^{D}_{1}=-\alpha\mathcal{R}_{1}+(1-\alpha)\mathcal{Q}_{2},\quad
R^{D}_{2}=\alpha\mathcal{Q}_{1}-(1-\alpha)\mathcal{R}_{2},$ (83)

where $0<\alpha<1$ and
$\mathcal{R}_{1},\mathcal{R}_{2},\mathcal{Q}_{1},\mathcal{Q}_{2}$ are defined
in (47). In the remainder of the proof we show that

$\mathcal{R}:=\begin{pmatrix}\mathcal{R}_{1}&0\\\
0&\mathcal{R}_{2}\end{pmatrix}\preceq
0,\quad\mathcal{Q}:=\begin{pmatrix}\mathcal{Q}_{1}&0\\\
0&\mathcal{Q}_{2}\end{pmatrix}\succeq 0,$ (84)

and therefore by (83) we have (81). This is done in 3 steps. For the first two
steps, we assume that $C_{1}$ and $C_{2}$ are full row rank matrices. In _Step
1_ we prove the inequality $\mathcal{Q}\succeq 0$, while in _Step 2_ we prove
the inequality $\mathcal{R}\preceq 0$. In _Step 3_ we relax the rank
assumptions on $C_{1}$ and $C_{2}$.

_Step 1_ : Proving $\mathcal{Q}\succeq 0$.
From (47) we have $\mathcal{Q}=\hat{Q}$, while from (42) we have
$\hat{Q}=\breve{Q}_{I}+\breve{Q}_{II}$. With $D=0,K=0,L=0$, from (41) we get

$\displaystyle\breve{Q}_{I}=-N_{A}^{\top}V(V^{\top}MV)^{-1}V^{\top}N_{A},$
(85a)
$\displaystyle\breve{Q}_{II}=-\Big{(}\star\Big{)}^{\top}Z^{-1}\Big{(}N_{B}^{\top}V(V^{\top}MV)^{-1}V^{\top}N_{A}\Big{)},$
(85b)

where $Z=N_{D}-N_{B}^{\top}V(V^{\top}MV)^{-1}V^{\top}N_{B}$,
$M:=A^{\top}P+PA+F^{\top}R^{P}F$, $N_{A}:=PB$,
$N_{B}:=PE+F^{\top}(S^{P})^{\top}$ and $N_{D}:=Q^{P}$.

Next we show that $M\prec 0$ and $Z\prec 0$. From there we conclude that
$\breve{Q}_{I}\succeq 0$ (from (85a) with $M\prec 0$) and
$\breve{Q}_{II}\succeq 0$ (from (85b) with $Z\preceq 0$), and therefore
$\mathcal{Q}=\hat{Q}\succeq 0$.

The inequality $M\prec 0$ follows directly from (29), since after the
multiplication of the matrices in (29), the matrix $M$ appears as a block on
diagonal of the matrix on the left hand side of the inequality (29). The
inequality $Z\prec 0$ has already been inferred in the proof of Theorem 3.1,
see (40b).

_Step 2_ : Proving $\mathcal{R}\preceq 0$.
This part of the proof is somewhat less straightforward and relies on Lemma
8.3 presented in Appendix Appendix B. This lemma implies that, with $D=0$,
$K=0$, $L=0$, there exists a multiplier $\Pi$ from (27) such that (28) and
(29) hold not only for $H=\left(\begin{smallmatrix}0&I\\\
I&0\end{smallmatrix}\right)$, but also for all

$H\in\mathbf{H}:=\Big{\\{}\begin{pmatrix}0&\alpha I\\\ \alpha
I&0\end{pmatrix}\,\,:\,\,\alpha\in[0,1]\Big{\\}}.$

In that case from (28) we have $R\succeq 0$. To see this, take the element
from $\mathbf{H}$ with $\alpha=0$ and apply (28). We conclude that there
necessarily exists a multiplier $\Pi$ from (27) such that (28) and (29) hold,
while $R\succeq 0$.

With $D=0$ the condition (49) becomes

$\begin{pmatrix}Q&S\\\
S^{\top}&R\end{pmatrix}+\begin{pmatrix}\mathcal{Q}&\mathcal{S}\\\
\mathcal{S}^{\top}&\mathcal{R}\end{pmatrix}\prec 0.$

Since $R\succeq 0$, we have $\mathcal{R}\preceq 0$.

_Step 3_ : Relaxing the full row rank assumption for $C_{1}$ and $C_{2}$.
We make use of the proof of Proposition 3.3. Recall that in the proof of
Proposition 3.3 we have first concluded existence of interconnection supply
rates on the modified system which had full row rank matrices $\tilde{C}_{1}$
and $\tilde{C}_{2}$. The interconnection signals in the modified system had
lower dimension than the corresponding interconnection signals of the original
systems. Then we showed that starting from the matrices $\tilde{Q}$,
$\tilde{R}$ and $\tilde{S}$, which define the interconnection neutral supply
functions of the modified system, as presented in (64) and (65), we can
construct interconnection neutral supply functions for the original systems.
These supply functions are defined through matrices $Q$, $R$ and $S$, which
are respectively “extensions” of the matrices $\tilde{Q}$, $\tilde{R}$ and
$\tilde{S}$. The extensions for $Q$ and $R$ are given by (78), and (80),
respectively. Comparing (64) and (65) with (82), we have that $R_{1}^{D}$ and
$R_{2}^{D}$ from (81) correspond respectively to $-\tilde{R}$ and $\tilde{Q}$
from (64) and (65). To finalize the proof for relaxing the rank conditions of
$C_{i}$ it is therefore sufficient to show that _i_) using extension (78) on
$\tilde{Q}\succeq 0$ we can obtain $Q$ from (78) such that $Q\succeq 0$; and
_ii_) using extension (80) on $\tilde{R}\preceq 0$ we can obtain $R$ from (80)
such that $R\preceq 0$. Indeed, both (i) and (ii) can be achieved when
selecting sufficiently large $\gamma_{Q}$ and $\gamma_{R}$, since the matrices
in (78) and (80) are related with congruence transformations.

_Proof of the part (b):_

_Step 0_ : The main idea and overview.
We present proof in the case of full row rank matrices $C_{1}$ and $C_{2}$.
The proof for relaxation of this assumption is fully analogous to the proof
presented for the part (a) above.

_The main idea and overview_ : Following the same path as in the proof for the
part (a) presented above, the proof again boils down to proving inequality
(81). Note that (81) is satisfied if $\mathcal{R}\preceq 0$ and
$\mathcal{Q}\succeq 0$, due to (83).

The remaining part of the proof is divided in 3 steps. In _Step 1_ we give an
auxiliary result which will be instrumental in both _Step 2_ and _Step 3_ , in
which the inequalities $\mathcal{R}\preceq 0$ and $\mathcal{Q}\succeq 0$ are
proven, respectively.

_Step 1_ : Preliminaries and auxiliary results
Recall that the proof of Corollary 3.2 follows directly from the proof of
Theorem 3.1 when we omit all the terms related to exogenous inputs/outputs
$d_{1},d_{2},z_{1},z_{2}$ and the external supply functions
$s_{1,ext}(\cdot,\cdot)$ and $s_{2,ext}(\cdot,\cdot)$, that is, by taking

$E=0,\,F=0,\,K=0,\,L=0,\,Q^{P}=0,\,S^{P}=0,\,R^{P}=0$ (86)

in (29) and in the remainder of the proof of Theorem 3.1. Additionally, the
statement of Corollary 3.2 requires $P=\operatorname{diag}(P_{1},P_{2})\succ
0$, where $P_{1}$ and $P_{2}$ are the matrices defining $V_{1}(\cdot)$ and
$V_{2}(\cdot)$, i.e., $V_{1}(x_{1})=x_{1}^{\top}P_{1}x_{1}$,
$V_{2}(x_{2})=x_{2}^{\top}P_{2}x_{2}$. Finally, we consider the case when
$D=\operatorname{diag}(D_{1},D_{2})=0$.

From (35), with (86), we have

$M=A^{\top}P+PA=\begin{pmatrix}A_{1}^{\top}P_{1}+P_{1}A_{1}&0\\\
0&A_{2}^{\top}P_{2}+P_{2}A_{2}\end{pmatrix}=:\begin{pmatrix}M_{1}&0\\\
0&M_{2}\end{pmatrix}.$ (87)

Next we show that $M\prec 0$. Let $x=\operatorname{col}(x_{1},x_{2})$ denote
the state vector of the interconnected system, which can be represented as
$\dot{x}=\mathcal{A}x$ with
$\mathcal{A}=\left(\begin{smallmatrix}A_{1}&B_{1}C_{2}\\\
B_{2}C_{1}&A_{2}\end{smallmatrix}\right)$. Since $V(x)=x^{\top}Px$ is a
Lyapunov function for the interconnected system, we have
$\mathcal{A}^{\top}P+P\mathcal{A}\prec 0$. It remains to note that
$\mathcal{A}^{\top}P+P\mathcal{A}=\left(\begin{smallmatrix}M_{1}&*\\\
*&M_{2}\\\ \end{smallmatrix}\right)$ (here $*$ denote submatrices which are
not of interest) and therefore we conclude that $M_{1}\prec 0$, $M_{2}\prec
0$, which implies $M\prec 0$.

_Step 2_ : Proving $\mathcal{Q}\succeq 0$.
From (47), (42), (41) we have $\mathcal{Q}=\breve{Q}_{I}+\breve{Q}_{II}$,
which with (86) reduces to

$\mathcal{Q}=-N_{A}^{\top}V(V^{\top}MV)^{-1}V^{\top}N_{A}$ (88)

with $N_{A}=PB$. We have $M\prec 0\,\,\implies(V^{\top}MV)^{-1}\prec
0\implies\mathcal{Q}\succeq 0$.

_Step 3_ : Proving $\mathcal{R}\preceq 0$.
It is possible to prove this inequality following a similar path as done in
the proof for the part (a). Here we present an alternative and somewhat more
direct proof. From (47a) and (45) we have that $\mathcal{R}\preceq 0$ is
implied if $\hat{R}\preceq 0$. Note that $\hat{R}$ is defined in (42). With
(86), from (42) and (41) we have

$\hat{R}=M-MV(V^{\top}MV)^{-1}V^{\top}M,$ (89)

where $V$ is a tall matrix666The number of rows in larger than the number of
columns. with full column rank. Indeed, recall from _Step 1_ in the proof of
Theorem 3.1 that $V:=\operatorname{diag}(V_{1},V_{2})$, where $V_{1}$ and
$V_{2}$ are defined as full column rank matrices whose columns span the kernel
spaces of $C_{1}$ and $C_{2}$, respectively.

Since $M\prec 0$, we have

$X:=\begin{pmatrix}M&MV\\\
V^{\top}M&V^{\top}MV\end{pmatrix}=\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}M&0\\\
0&M\end{pmatrix}\begin{pmatrix}I&0\\\ 0&V\end{pmatrix}\prec 0.$ (90)

The above inequality implies, via the Schur complement rule, that
$M-MV(V^{\top}MV)^{-1}V^{\top}M\prec 0$, that is, $\hat{R}\prec 0$. Therefore,
we also have the desired (weaker) inequality $\hat{R}\preceq 0$. ∎

##  8 CONCLUSIONS

The main results of this paper give insights into interplay between structured
storage/Lyapunov functions for a class of interconnected systems and
dissipativity properties of the individual systems. These results complement
some of the results from the seminal papers [12, 13] of Jan Willems, making
suitable converse statements to Theorem 5 in [12]. More precisely, we have
proven that if a dynamical network, composed as a set of linear time invariant
systems interconnected over an acyclic graph, admits an additive Lyapunov
function, then the individual systems in the network are dissipative with
respect to a (nonempty) set of interconnection neutral supply functions. Each
supply function from this set is defined on a single interconnection link in
the network.

From a more practical/application oriented point of view, the presented
results can be used to relate certain analysis or controller synthesis
approaches for uncertain and/or large scale systems, precisely by relating
structural properties of Lyapunov functions with dissipativity properties.
Indeed, while some control synthesis approaches build on employment of
additive Lyapunov functions (see, e.g., [15, 16]), others directly use
interconnection neutral supply functions, e.g., [7]. Furthermore, the relation
between interconnection neutral supply functions and robustness properties,
which have also be presented in this paper, has a potential to become a
constructive element in both analysis and control synthesis.

## Appendix A: Robust stability and robust dissipativity

Let the system from Figure 12 a) be defined by

$\tilde{G}_{0}:\quad\begin{pmatrix}\dot{x}\\\
w\end{pmatrix}=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\begin{pmatrix}x\\\
v\end{pmatrix},$ (91)

and

$v=Hw,\quad H\in\mathbf{H},$ (92)

where $A\in\mathbb{R}^{n\times n},D\in\mathbb{R}^{n_{w}\times n_{v}}$, and
$\mathbf{H}$ is a set of matrices in $\mathbb{R}^{n_{v}\times n_{w}}$. Let
$\tilde{G}$ denote the overall interconnected system given by (91), (92).
Furthermore, let the system from Figure 12 b) be defined by

$G_{0}:\quad\begin{pmatrix}\dot{x}\\\ w\\\
z\end{pmatrix}=\begin{pmatrix}A&B&E\\\ C&D&M\\\
K&F&L\end{pmatrix}\begin{pmatrix}x\\\ v\\\ d\end{pmatrix},$ (93)

and

$v=Hw,\quad H\in\mathbf{H},$ (94)

where $A\in\mathbb{R}^{n\times n},D\in\mathbb{R}^{n_{w}\times
n_{v}},L\in\mathbb{R}^{n_{z}\times n_{d}}$ and $\mathbf{H}$ is a set of
matrices in $\mathbb{R}^{n_{v}\times n_{w}}$. Let $G$ denote the overall
interconnected system given by (93), (94), with $d$ and $z$ as input and
output, respectively.

$\tilde{G}_{0}$$H$$w$$v$a)$G_{0}$$H$$d$$z$$w$$v$b) Figure 12: a) Autonomous
uncertain system; b) Uncertain open system.

The system $\tilde{G}$ is robustly exponentially stable if it is well-posed
and if there exists a quadratic function $V(x)=x^{\top}Px$ with $P\succ 0$
such that $\dot{V}(x(t))<0$ for all $x(t)\neq 0$ and for all $H\in\mathbf{H}$.
The system $G$ is robustly strictly dissipative with respect to the supply
function

$s(d,z)=\begin{pmatrix}d\\\ z\\\
\end{pmatrix}^{\top}\begin{pmatrix}Q_{P}&S_{P}\\\ S_{P}^{\top}&R_{P}\\\
\end{pmatrix}\begin{pmatrix}d\\\ z\\\ \end{pmatrix}$ (95)

if it is well-posed and if there exists a quadratic storage function
$V(x)=x^{\top}Px$, such that $\dot{V}(x(t))<s(d(t),z(t))$ at each time $t$,
for all $H\in\mathbf{H}$ and for all $\operatorname{col}(x(t),d(t))\neq 0$.
The system $G$ is robustly exponentially stable if it is well-posed and if
there exists a quadratic function $V(x)=x^{\top}Px$ with $P\succ 0$ such that
$\dot{V}(x(t))<0$ for all $x(t)\neq 0$, for $d(t)=0$ and for all
$H\in\mathbf{H}$.

######  Theorem 8.1.

Suppose that $\mathbf{H}$ is a compact set. Then the system $\tilde{G}$ is
robustly exponentially stable if and only if there exist symmetric matrices
$P\succ 0$, $Q$, $R$ and a real matrix $S$ such that the following
inequalities are satisfied

$\displaystyle\quad\quad\quad\,\,\,\begin{pmatrix}H\\\
I\end{pmatrix}^{\top}\begin{pmatrix}Q&S\\\
S^{\top}&R\end{pmatrix}\begin{pmatrix}H\\\ I\end{pmatrix}\succeq
0,,\quad\text{for all}\,\,H\in\mathbf{H},$ (96a)
$\displaystyle\left(\begin{array}[]{c}\star\end{array}\right)^{\top}\left(\begin{array}[]{cc:cc}0&P&0&0\\\
P&0&0&0\\\ \hdashline 0&0&Q&S\\\
0&0&S^{\top}&R\end{array}\right)\left(\begin{array}[]{cc}I&0\\\ A&B\\\
\hdashline 0&I\\\ C&D\end{array}\right)\prec 0.$ (96k)

######  Theorem 8.2.

Let $Q_{P},S_{P},R_{P}$ be given real matrices, where $Q_{P}$ and $R_{P}$ are
symmetric, and suppose $\mathbf{H}$ is a compact set. Consider the following
inequalities

$\displaystyle\quad\quad\quad\,\,\,\begin{pmatrix}H\\\
I\end{pmatrix}^{\top}\begin{pmatrix}Q&S\\\
S^{\top}&R\end{pmatrix}\begin{pmatrix}H\\\ I\end{pmatrix}\succeq
0,,\quad\text{for all}\,\,H\in\mathbf{H},$ (97a)
$\displaystyle\left(\begin{array}[]{ccc}\star\end{array}\right)^{\top}\left(\begin{array}[]{cc:cc:cc}0&P&0&0&0&0\\\
P&0&0&0&0&0\\\ \hdashline 0&0&Q&S&0&0\\\ 0&0&S^{\top}&R&0&0\\\ \hdashline
0&0&0&0&-Q_{P}&-S_{P}\\\
0&0&0&0&-S_{P}^{\top}&-R_{P}\end{array}\right)\left(\begin{array}[]{ccc}I&0&0\\\
A&B&E\\\ \hdashline 0&I&0\\\ C&D&M\\\ \hdashline 0&0&I\\\
K&F&L\end{array}\right)\prec 0.$ (97o)

  1. (a)

Suppose $-R_{P}\succeq 0$. Then the system $G$ is well-posed, robustly
exponentially stable and robustly strictly dissipative with respect to supply
(95), if and only if there exist symmetric matrices $P\succ 0$, $Q$ and $R$
and a real matrix $S$ such that the inequalities (97) hold.

  2. (b)

Suppose $G$ is well-posed. Then the system $G$ is robustly strictly
dissipative with respect to supply (95) if and only if there exist symmetric
matrices $P$, $Q$ and $R$ and a real matrix $S$ such that the inequalities
(97) hold.

We will refer to the matrix $\begin{pmatrix}Q&S\\\ S^{\top}&R\end{pmatrix}$ in
Theorem 8.1 and Theorem 8.2 as the _multiplier_ , as commonly done in the
literature. The following remarks are in order.

When (97) holds, the matrix $P$ defines the function $V(x)=x^{\top}Px$ as a
storage function with respect to the supply (95). The assumption
$-R_{P}\succeq 0$ in (_a_) is instrumental to infer well-posedness from (97).
Both assumptions $-R_{P}\succeq 0$ and $P\succ 0$ in (_a_) are instrumental to
infer robust stability from (97). Also note that adding the condition $P\succ
0$ alone to (_b_) does not necessarily imply robust stability.

Proofs of Theorem 8.1 and Theorem 8.2 are based on the full block S-procedure
and can be directly found or deduced (statement (b) from Theorem 8.2) from
[10, 9]. The condition on compactness of $\mathbf{H}$ can be suitably relaxed,
as shown recently in [6], but this is not used in this paper.

## Appendix B

Consider interconnected system $G$ given by (9)-(10) with $D_{i}=0$,
$K_{i}=0$, $L_{i}=0$, $i=1,2$, and as presented in Figure 1. With
$x:=\operatorname{col}(x_{1},x_{2})$, $d:=\operatorname{col}(d_{1},d_{2})$,
$z=\operatorname{col}(z_{1},z_{2})$, we have that $G$ is given by

$\dot{x}=Ax+Ed,\quad z=Fx,$ (98)

and $E=\operatorname{diag}(E_{1},E_{2})$,
$F=\operatorname{diag}(F_{1},F_{2})$,
$A=\left(\begin{smallmatrix}A_{1}&B_{1}C_{2}\\\
B_{2}C_{1}&A_{2}\end{smallmatrix}\right)$.

######  Lemma 8.3.

Consider the interconnected system $G$ given by (98) and suppose it is
strictly dissipative with an additive quadratic supply function
$s_{\text{ext}}(d_{1},d_{2},z_{1},z_{2})=s_{1,ext}(d_{1},z_{1})+s_{2,ext}(d_{2},z_{2})$
with an additive quadratic storage function
$V(x)=x_{1}^{\top}P_{1}x_{1}+x_{2}^{\top}P_{2}x_{2}$. Let the system
$G_{\alpha}$ be defined from $G$ when the matrix $A$ from (98) is replaced by
$A(\alpha):=\left(\begin{smallmatrix}A_{1}&\alpha B_{1}C_{2}\\\ \alpha
B_{2}C_{1}&A_{2}\end{smallmatrix}\right)$ for $\alpha$ an arbitrary real
constant such that $0\leq\alpha\leq 1$. Then $G_{\alpha}$ is strictly
dissipative with respect to the same additive supply function
$s_{\text{ext}}(d_{1},d_{2},z_{1},z_{2})$ and with the same additive storage
function $V(x)$.

###### Proof.

Let $s_{ext}(d,z):=\left(\begin{smallmatrix}d\\\
z\end{smallmatrix}\right)^{\top}\left(\begin{smallmatrix}-Q^{P}&-S^{P}\\\
-(S^{P})^{\top}&-R^{P}\end{smallmatrix}\right)\left(\begin{smallmatrix}d\\\
z\end{smallmatrix}\right)$. Strict dissipativity of $G$ with respect to
$s_{ext}(d,z)$ and with storage function $V(x)=x^{\top}Px$ implies that the
following matrix inequality holds

$\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}0&P\\\
P&0\end{pmatrix}\begin{pmatrix}I&0\\\
A&E\end{pmatrix}+\begin{pmatrix}\star\end{pmatrix}^{\top}\begin{pmatrix}Q^{P}&S^{P}\\\
(S^{P})^{\top}&R^{P}\end{pmatrix}\begin{pmatrix}0&I\\\ F&0\end{pmatrix}\prec
0.$ (99)

Note that the hypothesis that both supply and storage functions are additive
implies that the matrices $P$, $Q^{P}$, $S^{P}$ and $R^{P}$ are block diagonal
($P:=\operatorname{diag}(P_{1},P_{2})$). The dissipativity condition (99),
after multiplications, reads as
$\left(\begin{smallmatrix}A^{\top}P+PA+F^{\top}R^{P}F&PE+F^{\top}(S^{P})^{\top}\\\
E^{\top}P+S^{P}F&Q^{P}\end{smallmatrix}\right)\prec 0.$ Applying the Schur
complement rule on the above inequality we obtain $A^{\top}P+PA+Z\prec 0$,
where
$Z=F^{\top}R^{P}F-(PE+F^{\top}(S^{P})^{\top})(Q^{P})^{-1}(E^{\top}P+S^{P}F)$.
Note that $Z$ is by construction block diagonal matrix, i.e., we have
$Z=\operatorname{diag}(Z_{1},Z_{2})$. We have

$A(\alpha)^{\top}P+PA(\alpha)=\begin{pmatrix}A_{1}^{\top}P_{1}+P_{1}A_{1}&\alpha(C_{1}^{\top}B_{2}^{\top}P_{2}+P_{1}B_{1}C_{2})\\\
\star&A_{2}^{\top}P_{2}+P_{2}A_{2}\end{pmatrix}.$

Next we show that $A(\alpha)^{\top}P+PA(\alpha)+Z\prec 0$ for $\alpha=1$
implies that this inequality holds also for all $\alpha\in[0,1)$. After
applying the Schur complement rule on the inequality
$A(\alpha)^{\top}P+PA(\alpha)+Z\prec 0$, we have
$A_{1}^{\top}P_{1}+P_{1}A_{1}+Z_{1}-\alpha^{2}\Lambda\prec 0$ with
$\Lambda=(\star)^{\top}(A_{2}^{\top}P_{2}+P_{2}A_{2}+Z_{2})^{-1}(C_{2}^{\top}B_{1}^{\top}P_{1}+P_{2}B_{2}C_{1})$.
Therefore $A_{1}^{\top}P_{1}+P_{1}A_{1}+Z_{1}\prec\alpha^{2}\Lambda\preceq 0$.
Since $\Lambda\prec 0$ and the above inequality holds for $\alpha=1$, it also
holds for any $\alpha\in[0,1)$. ∎

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