remarkRemark hypothesisHypothesis claimClaim assumptionAssumption Sparse
Representations of Dynamical NetworksŞ. Sabău, A. Sperilă, C. Oară, and A.
Jadbabaie [][nocite]ex_supplement

# Sparse Representations of Dynamical Networks:
A Coprime Factorization Approach††thanks: This work was supported by a grant
of the Ministry of Research, Innovation and Digitization, CCCDI - UEFISCDI,
project no. PN-III-P2-2.1-PED-2021-1626, within PNCDI III.

Şerban Sabău Electrical and Computer Engineering Department, Stevens Institute
of Technology, Hoboken, NJ 07030 USA<EMAIL_ADDRESS>Andrei Sperilă
Faculty of Automatic Control and Computers, “Politehnica” University of
Bucharest, Sector 6, 060042 Romania<EMAIL_ADDRESS><EMAIL_ADDRESS>Cristian Oară33footnotemark: 3 Ali Jadbabaie Institute
for Data, Systems and Society, Massachusetts Institute of Technology (MIT),
Cambridge, MA 02139 USA<EMAIL_ADDRESS>
###### Abstract

We study a class of dynamical networks modeled by linear and time-invariant
systems which are described by state-space realizations. For these networks,
we investigate the relations between various types of factorizations which
preserve the structure of their component subsystems’ interconnection. In
doing so, we provide tractable means of shifting between different types of
sparsity-preserving representations and we show how to employ these
factorizations to obtain distributed implementations for stabilizing and
possibly stable controllers. By formulating all these results for both
discrete- and continuous-time systems, we develop specialized distributed
implementations that, up to this point, were only available for networks
modeled as discrete-time systems.

###### keywords:

Sparse representations, distributed controllers, coprime factors, networked
systems.

93A14, 93B99, 93C05

##  1 Introduction

With the growing trend of modern technical applications becoming more
interconnected and spatially widespread, distributed process control has
become an area of intense research in the specialized literature catering to
dynamical systems. However, with its convexity refuted and its generic
intractability revealed (see, for example, [3]), recent efforts in the area of
distributed control for Linear and Time-Invariant (LTI) systems have shifted
from the classical formulation of imposing sparsity upon the Transfer Function
Matrix (TFM) of the controller [32, 33, 22, 26] towards the implementation of
decentralized state-space-based control laws.

###  1.1 Context

Among the first attempts to formalize a state-space-oriented approach to
distributed control was the work from [17], in which structured realizations
are employed in order to obtain control laws that inherit the sparsity
structure of the controlled network. A key concept pioneered in [17], which
turns out to be crucial for the results presented in this manuscript, is the
fact that the distributed state-space implementation of the sought-after
control laws must possess certain structural properties, such as
stabilizability and detectability, in order to certify closed-loop stability.

Yet, arguably, the most celebrated results in this new paradigm of distributed
control came with the advent of the System-Level Synthesis (SLS) design
framework from [35]. Based upon the classical work in [1], the SLS approach
produces decentralized control laws for discrete-time systems, by employing an
affine parametrization of all the achievable closed-loop maps. Additionally,
the work in [13] has shown that, in the SLS framework, it is possible to
decouple controller synthesis from closed-loop design. Closely following the
developments in [35], the body of work presented in [10] aims to provide a
continuous-time parallel to the SLS framework, with important contributions to
the vehicular platoon problem, which was also tackled in [25]. Notably, the
approaches from [10] and [25] are linked via the implicit use of coprime
factorizations, and an underlying dependence upon the Youla parametrization.

In the same vein as the techniques from [10] and [25], a new design paradigm,
based upon coprime factorizations and heavily inspired by the work from [37],
was recently formalized in [24], with the aim of proposing a unitary
continuous- and discrete-time approach to optimal distributed control. The
theoretical developments presented in [24] are also supported by the convex
relaxation-based procedures from [27], which are part of a larger class of
design methods currently gaining traction in the distributed control
literature (see [5, 14, 16, 34, 23]). Fortunately, the intrinsic connections
between the SLS-focused methods and their Youla-based counterparts has been
elucidated in [29] and [39], while a system-theoretical middle ground between
these two approaches was proposed in [7] and [38], via the Input-Output
Parametrization.

###  1.2 Motivation

Although the various techniques mentioned in the previous subsection clearly
show that significant progress has been made in moving away from the
traditional sparse TFM-based approach to distributed control, a number of
important limitations still remain unaddressed in literature. While certain
procedures, such as the ones in [17], build upon highly structured
realizations for the investigated networks, obtaining these exceptional
representations becomes somewhat unwieldy for large-scale systems. Except for
those networks endowed with particularly sparse or with recursive inter-nodal
connections (see the numerical example in [27]), the effort needed to compose
the interconnected plant’s model out of its nodes’ realizations does not scale
gracefully with respect to the network’s complexity.

Similar to the results from [17], other methods proposed in literature focus
exclusively on networks modeled in either continuous- or discrete-time, with
[10] and [35] serving as respective examples. Oftentimes, this exclusivity is
owed to the manner in which stability guarantees are offered for the proposed
control schemes. We point out that this particular limitation, which will be
further commented upon in the sequel, has been lifted following the
introduction of the design framework from [24]. In doing so, however, the
control laws from [24] take on a distinct input-output interpretation,
regardless of their distributed and stabilizing state-space-based
implementations, as in Section III.C in [24]. Thus, the cost paid for this
increased flexibility is a degree of opaqueness with respect to network state
dynamics, at the implementation level.

###  1.3 Contribution

The aim of this paper is to propose a factorized representation, inspired by
the system response TFMs from [35], which is suitable to both network analysis
and distributed controller synthesis. This representation, dubbed a System
Response-Type Realization (SRTR), captures both the dynamics and the structure
of the network, regardless of whether it is modeled as either a continuous- or
a discrete-time system. This factorization offers a notably keener insight
into the state dynamics of the plant than the Network Realization Function
(NRF) representation from [24], while ensuring the same guarantees of closed-
loop stability as NRF-based control laws, when implementing controllers via
SRTR-based feedback schemes.

In developing the theory associated with this new representation, we
implicitly investigate the subtle connections between the classical left
coprime factorizations discussed in [31], the system response TFMs from [35]
and the NRF pairs introduced in [24]. Additionally, we propose numerically
tractable means of shifting between the aforementioned types of
representations, all while placing a particular emphasis on the preservation
of three key properties: stability, coprimeness and sparsity structure.
Finally, we show how the first property allows us to elegantly tackle the
problem of strong stabilization in the decentralized setting, how the second
property implicitly allows for the commutation between representations, and
how the third property directly promotes the implementation of distributed
control laws.

###  1.4 Outline of the Paper

In the second section, we include a series of preliminary notions which are
standard to control theory literature. The third section of the paper contains
a brief outline of the theoretical concepts behind various representations for
LTI systems. In the fourth section, we show that, while the NRF representation
of a given LTI system is not guaranteed to be coprime, this property can be
ensured for a closely related representation associated with our system,
heavily inspired by those in [35]. We also provide state-space formulas for
the class of all such coprime representations, related to a particular type of
realization for our LTI system.

The fourth section also contains the main results of the paper: an explanation
of the intrinsic connections between NRF pairs, the coprime representations
inspired by [35], and the classical left coprime factorizations associated
with a given TFM, along with the means of employing them to obtain distributed
and stabilizing control laws. The fifth section showcases the direct
application of these results through a numerical example. Finally, the sixth
section contains a number of concluding remarks.

In order to ensure that the paper is as self-contained as possible, some more
specialized notions and technical results have been included in Appendices A
and B, while Appendix C contains a series of auxiliary statements, employed in
the proofs of the main results. Moreover, the aforementioned proofs have been
placed in Appendix D, in order to promote the flow and readability of the
paper.

##  2 Preliminaries

###  2.1 Notation

We denote by $\mathbb{R}$ the set of real numbers and by $\mathbb{N}$ the set
of natural ones, while $\mathbb{C}$ stands for the complex plane and
$\mathbb{C}_{\infty}:=\mathbb{C}\cup\\{\infty\\}$ denotes the extended complex
plane. Let $\mathbb{R}^{p\times m}$ be the set of $p\times m$ real matrices,
for whose elements we denote by $\|\cdot\|$ the spectral norm, _i.e._ , the
largest singular value. Let also $\mathbb{R}(\lambda)^{p\times m}$ be the set
of $p\times m$ TFMs, which are matrices having real-rational functions as
entries. Henceforth, we denote such TFMs through the use of boldface symbols.

Additionally, we point out that our results apply to both continuous- and
discrete-time LTI systems. Thus, we assimilate the indeterminate $\lambda$
with the complex variables $s$ or $z$ appearing in the Laplace or
$\mathcal{Z}$-transform, respectively, depending upon the type of system which
is under investigation.

###  2.2 Realization Theory

The main objects of study in this paper are LTI systems which are described in
the time domain by the state equations

(1a) $\displaystyle\sigma{x}$ $\displaystyle=Ax+Bu,\ {x}(t_{o})={x}_{o},$ (1b)
$\displaystyle y$ $\displaystyle=Cx+Du,$

with $A\in\mathbb{R}^{n\times n}$ being the _state matrix_ ,
$B\in\mathbb{R}^{n\times m}$ the _input matrix_ , $C\in\mathbb{R}^{p\times n}$
the _output matrix_ and $D\in\mathbb{R}^{p\times m}$ the _feedthrough matrix_
, while $n$ is called the order of the realization. The operator $\sigma$ in
(1a)-(1b) stands for either the _time derivative_ , in the continuous-time
context, or the _forward unit-shift_ , in the discrete-time case. Given any
$n$-dimensional state-space realization (1a)-(1b) of an LTI system, its input-
output representation is expressed via the TFM denoted by

$\mathbf{G}(\lambda)=\left[\begin{array}[]{c|c}A-\lambda I_{n}&B\\\ \hline\cr
C&D\end{array}\right]:=D+C(\lambda I_{n}-A)^{-1}B.$

A pair $(A,B)$ or a realization (1a)-(1b) for which $\begin{bmatrix}A-\lambda
I_{n}&B\end{bmatrix}$ has full row rank at $\lambda\in\mathbb{C}$ is called
_controllable at $\lambda$_. Similarly, a pair $(C,A)$ or a realization
(1a)-(1b) that is _observable at $\lambda$_ is one for which the pair
$(A^{\top},C^{\top})$ is controllable at $\lambda$. A _controllable_
/_observable_ matrix pair or realization (1a)-(1b) is one that is
controllable/observable at any $\lambda\in\mathbb{C}$. A realization whose
order is the smallest out of all others for the same $\mathbf{G}(\lambda)$ is
called _minimal_ (see Section 6.5.1 of [12]). Any two minimal realizations
$(A,B,C,D)$ and $(\widehat{A},\widehat{B},\widehat{C},\widehat{D})$ of
$\mathbf{G}(\lambda)$ having order $\nu$ are related via a state equivalence
transformation $x=T{\widehat{x}}$, with $T$ invertible in
$\mathbb{R}^{\nu\times\nu}$, given by

(2)
$\begin{array}[]{llll}A=T\widehat{A}T^{-1},&B=T\widehat{B},&C=\widehat{C}T^{-1},&D=\widehat{D}.\end{array}$

######  Definition 2.1.

Any two realizations which are related by a state equivalence transformation,
as in (2), are called _equivalent_.

Let $\mathbb{S}$ denote the _domain of stability_ , which is either the open
left-half plane, _i.e._ , $\left\\{\lambda\in\mathbb{C}\ :\
\text{Re}(\lambda)<0\right\\}$ in the case of continuous-time systems or the
open unit disk, _i.e._ , $\left\\{\lambda\in\mathbb{C}\ :\
|\lambda|<1\right\\}$ for discrete-time ones. A _stabilizable_ realization
(1a)-(1b) or pair $(A,B)$ is one that is controllable at any
$\lambda\in\mathbb{C}\backslash\mathbb{S}$. By duality, a _detectable_
realization (1a)-(1b) or pair $(C,A)$ is one that is observable at any
$\lambda\in\mathbb{C}\backslash\mathbb{S}$.

###  2.3 Rational Matrix Theory

The matrix polynomial $A-\lambda E$ is called a _pencil_. If it is square and
$\det(A-\lambda E)\not\equiv 0$, we refer to it as _regular_ (see chapter 12
of [8] for more details). Note also that a pencil is merely a polynomial TFM.
The finite poles and zeros of any TFM $\mathbf{G}(\lambda)$ are rigorously
defined in terms of the _Smith-McMillan Form_ (see Section 6.5.2 of [12]),
whereas the infinite poles and zeros of $\mathbf{G}(\lambda)$ are those of
$\mathbf{G}\left(\frac{1}{\lambda}\right)$ at $\lambda=0$. Any realization
(1a)-(1b) that is both controllable and observable is also minimal (see [36])
and the poles of its TFM coincide with the eigenvalues of the regular pencil
$A-\lambda I_{n}$, known as the realization’s _pole pencil_.

The _McMillan degree_ of $\mathbf{G}(\lambda)$ is the sum of the degrees of
its poles (see Section 6.5.3 of [12]), both finite and infinite. A TFM’s
_normal rank_ , which is its rank for almost all points in
$\mathbb{C}_{\infty}$, is also rigorously defined via the TFM’s Smith-McMillan
Form. If a TFM has no poles at $\\{\infty\\}$, it is called _proper_.
Otherwise, it is called _improper_. A TFM
$\mathbf{G}\in\mathbb{R}(\lambda)^{p\times m}$ for which
$\lim_{\lambda\rightarrow\infty}\mathbf{G}(\lambda)=O$ is called _strictly
proper_. Finally, a TFM is deemed _stable_ if it has no poles located in
$\mathbb{C}_{\infty}\backslash\mathbb{S}$.

Two TFMs $\mathbf{N}(\lambda)\in\mathbb{R}(\lambda)^{p\times m}$ and
$\mathbf{M}(\lambda)\in\mathbb{R}(\lambda)^{p\times p}$, with
$\mathbf{M}(\lambda)$ having full normal rank, that satisfy
$\mathbf{G}(\lambda)=\mathbf{M}^{-1}(\lambda)\mathbf{N}(\lambda)$ are said to
be a _left factorization_ of $\mathbf{G}(\lambda)$. When, additionally,
$\begin{bmatrix}\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix}$ has full
normal rank (this is always ensured by $\mathbf{M}(\lambda)$ being invertible)
along with no zeros in $\mathbb{C}_{\infty}$ (see also [20]), we say that
$\mathbf{M}(\lambda)$ and $\mathbf{N}(\lambda)$ constitute a _fully left
coprime factorization_ (FLCF) of $\mathbf{G}(\lambda)$ (see [31, 20, 21] for
equivalent characterizations). When
$\begin{bmatrix}\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix}$ has full
row normal rank along with no poles or zeros located outside of $\mathbb{S}$,
the pair is called an _LCF over $\mathbb{S}$_.

##  3 Representations for LTI Networks

This section contains a discussion based upon various representations for
networks of LTI systems, with an emphasis on those tentatively prototyped in
[28] and now discussed in a broader context.

###  3.1 State-Space Realizations

We start with the given system $\mathbf{G}(\lambda)$, described by the state
equations of order $n$ given explicitly by

(3a) $\displaystyle\sigma{\widetilde{x}}$
$\displaystyle=\widetilde{A}\widetilde{x}+\widetilde{B}u,\
\widetilde{x}(t_{o})=\widetilde{x}_{o},$ (3b) $\displaystyle y$
$\displaystyle=\widetilde{C}\widetilde{x}.$

The following hypothesis is in effect throughout the paper.

The matrix $\widetilde{C}$ from (3b) has $p<n$ and full row rank.

######  Remark 3.1.

Note that Assumption 3.1 is in no way restrictive from a practical standpoint,
where oftentimes $n\gg p$. The lack of full row rank for the matrix
$\widetilde{C}$ implies that some of the sensor measurements are redundant and
can be amalgamated via linear combinations to produce a new, full row rank
$C$-matrix, all while preserving the property of observability (see Section
3.4 of [36] for the underlying connection with reduced-order observers).
Notice also that when $n=p$, the matrix $\widetilde{C}$ is invertible and the
problem can be reduced to the trivial, full state measurement case, in which
the main representations discussed in the sequel take on the classical form
given in (3a).

We now choose any matrix $\overline{C}$ such that
$T:=\begin{bmatrix}\widetilde{C}^{\top}&\overline{C}^{\top}\end{bmatrix}^{\top}$
is nonsingular, noticing that such a $\overline{C}$ always exists due to
$\widetilde{C}$ having full row rank, and we apply the state equivalence
transformation given by $T$ on (3a)-(3b), in order to get

(4a) $\displaystyle\begin{bmatrix}\sigma{y}\\\ \sigma{q}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{bmatrix}\begin{bmatrix}y\\\
q\end{bmatrix}+\begin{bmatrix}B_{1}\\\ B_{2}\end{bmatrix}u,\
\begin{bmatrix}y(t_{o})\\\ q(t_{o})\end{bmatrix}=\begin{bmatrix}y_{o}\\\
q_{o}\end{bmatrix},\ \begin{bmatrix}y\\\ q\end{bmatrix}=T\widetilde{x},$ (4b)
$\displaystyle y$ $\displaystyle=\begin{bmatrix}\hskip 5.69054ptI_{p}&\hskip
6.40186ptO\hskip 4.97922pt\end{bmatrix}\begin{bmatrix}y\\\ q\end{bmatrix}.$

Henceforth, we will assume that the following statement holds.

The pair $(\widetilde{C},\widetilde{A})$ in (3a)-(3b) is observable.

######  Remark 3.2.

Notice that, due to the particular structure of the output matrix from (4b),
namely that it is given by $\begin{bmatrix}I_{p}&O\end{bmatrix}$, Assumption
3.1 is equivalent to the pair $(A_{12},A_{22})$ from (4a) being observable.

Taking now the Laplace or $\mathcal{Z}$-transform of (4a), depending on the
time domain, and considering all initial conditions to be 0, we get that

(5) $\small\begin{bmatrix}\lambda I_{p}-A_{11}&-A_{12}\\\ -A_{21}&\lambda
I_{n-p}-A_{22}\end{bmatrix}\small\begin{bmatrix}\mathbf{Y}(\lambda)\\\
\mathbf{Q}(\lambda)\end{bmatrix}=\small\begin{bmatrix}B_{1}\\\
B_{2}\end{bmatrix}\mathbf{U}(\lambda).$

By left-multiplying (5) with the following TFM

(6) $\mathbf{\Omega}(\lambda):=\small\begin{bmatrix}I_{p}&A_{12}(\lambda
I_{n-p}-A_{22})^{-1}\\\ O&I_{n-p}\\\ \end{bmatrix},$

we obtain the identity

(7) $\footnotesize\begin{bmatrix}(\lambda I_{p}-A_{11})-A_{12}(\lambda
I_{n-p}-A_{22})^{-1}A_{21}&O\\\
*&*\end{bmatrix}\begin{bmatrix}\mathbf{Y}(\lambda)\\\
\mathbf{Q}(\lambda)\end{bmatrix}=\begin{bmatrix}B_{1}+A_{12}(\lambda
I_{n-p}-A_{22})^{-1}B_{2}\\\ *\end{bmatrix}\mathbf{U}(\lambda),\normalsize$

where $*$ denotes entries which are not of interest. Taking the inverse
Laplace or $\mathcal{Z}$-transform in the first block-row of (7), we are able
to express (recalling that we consider all initial conditions to be 0) the
following input-output relationship

$\lambda\mathbf{Y}(\lambda)=\big{(}A_{11}+A_{12}(\lambda
I_{n-p}-A_{22})^{-1}A_{21}\big{)}\mathbf{Y}(\lambda)+\big{(}B_{1}+A_{12}(\lambda
I_{n-p}-A_{22})^{-1}B_{2}\big{)}\mathbf{U}(\lambda),$

from which we introduce the notation

$\begin{array}[]{ll}\mathbf{W}(\lambda):=A_{11}+A_{12}(\lambda
I_{n-p}-A_{22})^{-1}A_{21},&\mathbf{V}(\lambda):=B_{1}+A_{12}(\lambda
I_{n-p}-A_{22})^{-1}B_{2}.\end{array}$

In doing so, we finally get the following equation, which describes the
relationship between the network’s manifest variables

(8)
$\lambda\mathbf{Y}(\lambda)=\mathbf{W}(\lambda)\mathbf{Y}(\lambda)+\mathbf{V}(\lambda)\mathbf{U}(\lambda).$

###  3.2 Network Realization Functions

Promisingly, the identity from (8) generates numerous avenues of
investigation. We proceed to investigate one of them, by first pointing out
that a routine computation produces the fact that
$\displaystyle\mathbf{G}(\lambda)=\big{(}\lambda
I_{p}-\mathbf{W}(\lambda)\big{)}^{-1}\mathbf{V}(\lambda)$. Notice that, since
$\mathbf{W}(\lambda)$ is always proper, it follows that $\big{(}\lambda
I_{p}-\mathbf{W}(\lambda)\big{)}$ is always invertible as a TFM. Let
$\mathbf{D}(\lambda)$ denote the TFM obtained by taking the diagonal entries
of $\mathbf{W}(\lambda)$, that is
$\displaystyle\mathbf{D}(\lambda):=\mathrm{diag}\\{\mathbf{W}_{11}(\lambda),\mathbf{W}_{22}(\lambda),\dots,\mathbf{W}_{pp}(\lambda)\\}$.
Then,

$\mathbf{G}(\lambda)=\Big{[}\big{(}\lambda
I_{p}-\mathbf{D}(\lambda)\big{)}-\big{(}\mathbf{W}(\lambda)-\mathbf{D}(\lambda)\big{)}\Big{]}^{-1}\mathbf{V}(\lambda)=\\\
=\Big{[}I_{p}-\big{(}\lambda
I_{p}-\mathbf{D}(\lambda)\big{)}^{-1}\big{(}\mathbf{W}(\lambda)-\mathbf{D}(\lambda)\big{)}\Big{]}^{-1}\big{(}\lambda
I_{p}-\mathbf{D}(\lambda)\big{)}^{-1}\mathbf{V}(\lambda),$

where $(\mathbf{W}-\mathbf{D})$ has only zeros on its diagonal and, by
denoting

(9a) $\displaystyle\mathbf{\Phi}(\lambda)$ $\displaystyle:=\big{(}\lambda
I_{p}-\mathbf{D}(\lambda)\big{)}^{-1}\big{(}\mathbf{W}(\lambda)-\mathbf{D}(\lambda)\big{)},$
(9b) $\displaystyle\mathbf{\Gamma}(\lambda)$ $\displaystyle:=\big{(}\lambda
I_{p}-\mathbf{D}(\lambda)\big{)}^{-1}\mathbf{V}(\lambda),$

we get that
$\displaystyle\mathbf{G}(\lambda)=\big{(}I_{p}-\mathbf{\Phi}(\lambda)\big{)}^{-1}\mathbf{\Gamma}(\lambda)$
or, equivalently, that

(10)
$\mathbf{Y}(\lambda)=\mathbf{\Phi}(\lambda)\mathbf{Y}(\lambda)+\mathbf{\Gamma}(\lambda)\mathbf{U}(\lambda).$

######  Remark 3.3.

The splitting and the “extraction” of the diagonal in (9a) are intended to
ensure that $\mathbf{\Phi}(\lambda)$ has the sparsity (and the meaning) of the
adjacency matrix of the graph, _i.e._ , the square matrix which indicates the
adjacency between any two of the graph’s vertices, thus describing the causal
relationships between the system’s various outputs. Consequently,
$\mathbf{\Phi}(\lambda)$ will always have zero entries on its diagonal, which
is a key factor in facilitating any implementations based upon (9a)-(9b).

We now formally introduce the notion of NRF pair, discussed at length in [24].

######  Definition 3.4.

Given a $\mathbf{G}(\lambda)\in\mathbb{R}^{p\times m}(\lambda)$ and any two
TFMs $\mathbf{\Phi}(\lambda)\in\mathbb{R}^{p\times p}(\lambda)$ and
$\mathbf{\Gamma}(\lambda)\in\mathbb{R}^{p\times m}(\lambda)$, satisfying
$\displaystyle\mathbf{G}(\lambda)=\big{(}I_{p}-\mathbf{\Phi}(\lambda)\big{)}^{-1}\mathbf{\Gamma}(\lambda)$
and with $\mathbf{\Phi}(\lambda)$ having zero entries on its diagonal, the
pair $\small\big{(}\mathbf{\Phi}(\lambda),\mathbf{\Gamma}(\lambda)\big{)}$ is
called an _NRF pair_ of $\mathbf{G}(\lambda)$.

######  Remark 3.5.

Note that the outputs of the network’s nodes need not be scalar and that
constraining only the diagonal entries of $\mathbf{\Phi}(\lambda)$ represents
the most generic case. Indeed, when implementing controllers via their NRF
pairs, it may be convenient for certain nodes to compute more than one command
signal, with the latter representing our network’s outputs. Therefore, it is
natural to impose identically zero elements on the block-diagonal of
$\mathbf{\Phi}(\lambda)$, in order to avoid the redundant use of local
information. This merely amounts to an additional sparsity constraint on the
entries of $\mathbf{\Phi}(\lambda)$ and we point out that the results from
[24] are not dependent upon the structure of the aforementioned TFM, except
for its identically zero diagonal. Moreover, as explained in Remark III.8 of
[24], the block-row implementations which arise naturally from imposing
identically zero blocks on the block-diagonal of $\mathbf{\Phi}(\lambda)$
retain all the desirable properties of the row implementations provided in the
results of Section III from [24].

######  Remark 3.6.

Another benefit of the these NRF representations consists in the preservation
of the network’s adjacency graph, both with respect to the input-output
relationships that characterize it, as modeled by $\mathbf{\Gamma}(\lambda)$,
and also of the intrinsic interconnections of its manifest variables, as
encoded through the structure of $\mathbf{\Phi}(\lambda)$. These graph-
oriented representations have received significant attention in the context of
Kron reduction for electrical networks (see [2, 4, 6] and the references
therein).

It is for the reasons discussed in this subsection that NRF pairs present
themselves as tractable representations for the purpose of network modeling
and control. Their versatility in all areas of control theory, such as system
identification, controller synthesis and even model reduction, is one of the
chief motivators for investigating their relationships with the other types of
representations from this paper.

###  3.3 An Example of NRF Modelling

$z_{1}$${\bf\Gamma_{1}(\lambda)}$$u_{1}$${\bf\Phi_{2}(\lambda)}$${\bf\Gamma_{2}(\lambda)}$$z_{2}$$u_{2}$${\bf\Phi_{3}(\lambda)}$${\bf\Phi_{1}(\lambda)}$${\bf\Gamma_{3}(\lambda)}$$z_{3}$$u_{3}$
Figure 1: Three-hop network architecture

We now showcase a practical example, in order to illustrate the inherent
potential of NRF representations and to provide motivation for the means
(which will be discussed in the sequel) of obtaining them from the various
coprime factorizations of the network’s TFM.

We will focus on the 3-hop ring network in Fig. 1, where all the
$\mathbf{\Phi}_{i}(\lambda)$ and $\mathbf{\Gamma}_{i}(\lambda)$ blocks
represent scalar TFMs of LTI systems, with all $\mathbf{\Gamma}_{i}(\lambda)$
blocks being strictly proper. We will denote by $\mathbf{G}(\lambda)$ the TFM
from all the input signals $u_{i}$ to all the outputs $y_{i}$. By directly
inspecting the signal flow graph in Fig. 1, we can write

(11) $\small\begin{bmatrix}\mathbf{U}_{1}(\lambda)\\\
\mathbf{U}_{2}(\lambda)\\\
\mathbf{U}_{3}(\lambda)\end{bmatrix}=\small\begin{bmatrix}O&O&\mathbf{\Phi}_{1}(\lambda)\\\
\mathbf{\Phi}_{2}(\lambda)&O&O\\\
O&\mathbf{\Phi}_{3}(\lambda)&O\end{bmatrix}\normalsize\small\begin{bmatrix}\mathbf{U}_{1}(\lambda)\\\
\mathbf{U}_{2}(\lambda)\\\
\mathbf{U}_{3}(\lambda)\end{bmatrix}+\small\begin{bmatrix}\mathbf{\Gamma}_{1}(\lambda)&O&O\\\
O&\mathbf{\Gamma}_{2}(\lambda)&O\\\
O&O&\mathbf{\Gamma}_{3}(\lambda)\end{bmatrix}\normalsize\small\begin{bmatrix}\mathbf{Z}_{1}(\lambda)\\\
\mathbf{Z}_{2}(\lambda)\\\ \mathbf{Z}_{3}(\lambda)\end{bmatrix},\normalsize$

where $\mathbf{U}_{i}(\lambda)$ and $\mathbf{Y}_{i}(\lambda)$ are the Laplace
or $\mathcal{Z}$-transforms of $u_{i}$ and, respectively, of $y_{i}$, while
also introducing the notation

(12)
$\mathbf{\Phi}(\lambda):=\small\begin{bmatrix}O&O&\mathbf{\Phi}_{1}(\lambda)\\\
\mathbf{\Phi}_{2}(\lambda)&O&O\\\
O&\mathbf{\Phi}_{3}(\lambda)&O\end{bmatrix},\quad\mathbf{\Gamma}(\lambda):=\small\begin{bmatrix}\mathbf{\Gamma}_{1}(\lambda)&O&O\\\
O&\mathbf{\Gamma}_{2}(\lambda)&O\\\
O&O&\mathbf{\Gamma}_{3}(\lambda)\end{bmatrix},\normalsize$

in order to state that
$\big{(}\mathbf{\Phi}(\lambda),\mathbf{\Gamma}(\lambda)\big{)}$ is the NRF
pair associated with the TFM $\mathbf{G}(\lambda)$.

######  Remark 3.7.

A noteworthy observation consists in the fact that the structure of the
subsystem interconnections in Fig. 1 is no longer recognizable from the input-
output relation described by the TFM of the system,
$\displaystyle\mathbf{G}(\lambda)=\big{(}I-\mathbf{\Phi}(\lambda)\big{)}^{-1}\mathbf{\Gamma}(\lambda)$,
since $\mathbf{G}(\lambda)$ does not have any sparsity pattern, nor (in
general) does it have any other particularities. Notice, however, that the
aforementioned structure is successfully captured in the sparsity patterns of
$\mathbf{\Phi}(\lambda)$ and $\mathbf{\Gamma}(\lambda)$. This remarkable
property makes the NRF a well-suited theoretical instrument for modeling any
LTI network.

Additionally, we wish to further illustrate how the NRF determines, via (10),
the topology of the LTI network describing the given TFM
$\mathbf{G}(\lambda)$. By considering $\mathbf{\Phi}_{13}(\lambda)$
identically zero, which would imply the “breaking” of the ring network from
Fig. 1, the latter becomes a cascade connection and $\mathbf{G}(\lambda)$ can
be seen as a “line” network. This is relevant in the motion control of
vehicles moving in “platoon” formation [25].

###  3.4 System Response-Type Realizations

With the utility of NRF representation now established, we return to the pair
of TFMs introduced just before (8), and which were employed in obtaining
(9a)-(9b). Given the need to impose a certain sparsity structure on the
$\mathbf{\Phi}(\lambda)$ term of an NRF pair, in order to facilitate
distributed implementations (as discussed in Remark 3.5), and the opaqueness
of the NRF formalism with respect to state dynamics, it becomes desirable to
investigate a new type of representation that alleviates these limitations. It
is in this context that we introduce one of the central objects of this paper,
via the following definition.

######  Definition 3.8.

Let the system described by the realization (4a)-(4b), of order $n$, have a
TFM be denoted $\mathbf{G}(\lambda)$. Then, a pair
$\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ of proper TFMs, where
$\mathbf{W}(\lambda)\in\mathbb{R}^{p\times p}(\lambda)$ and
$\mathbf{V}(\lambda)\in\mathbb{R}^{p\times m}(\lambda)$, with
$\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ having a
McMillan degree of at most $n-p$ and which satisfy

(13) $\mathbf{G}(\lambda)=\big{(}\lambda
I_{p}-\mathbf{W}(\lambda)\big{)}^{-1}\mathbf{V}(\lambda),$

is called a System Response-Type Realization of the system given in (4a)-(4b).

######  Remark 3.9.

It is worthwhile to point out that the two TFMs which make up the SRTR pair
denoted $\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ fulfill the
same role as the TFMs $\widetilde{\mathbf{R}}^{+}(z)$ and
$\widetilde{\mathbf{N}}(z)$ from the state iteration equation of (18) from
[35]. Denoting by $\mathbf{B}(z)$ the $\mathcal{Z}$-transform of the state
vector belonging to the controller, the aforementioned state iteration is
expressed in the same manner as (8) via the identity

(14)
$z\mathbf{B}(z)=\widetilde{\mathbf{R}}^{+}(z)\mathbf{B}(z)+\widetilde{\mathbf{N}}(z)\mathbf{Y}(z).$

It is this very fact which brought about the designation of _System Response-
Type Realization_ and which can be exploited to obtain sparse and scalable
representations for systems in the context of distributed control. Note that,
in contrast to the implementations from [35], the one from (8) is also
available for continuous-time systems.

######  Remark 3.10.

In the sequel, SRTR pairs of controllers will be employed to implement control
laws in a distributed fashion. However, in contrast to the approach from [35]
and its implementations of type (14), we point out that closed-loop maps are
not intrinsically linked to the definition of an SRTR pair. Therefore, the
newly introduced representation bears all the hallmarks of the so-called
_implementation matrices_ from [13], in which a mutation of the classical SLS
paradigm allows for the decoupling of controller synthesis from closed-loop
design. Although the framework proposed in the sequel shares this benefit, we
continue to refer to SRTR pairs as such due to (14) being the inspiration for
the proposed representation and, more importantly, due to not wishing to
overshadow their significant potential for system analysis, which would
inevitably result from naming them “implementation matrices”.

######  Remark 3.11.

Another feature of the SRTR representation is that any pair
$\big{(}\mathbf{W}(\lambda),$ $\mathbf{V}(\lambda)\big{)}$ possesses the same
sparsity pattern with its subsequent NRF pair
$\big{(}\mathbf{\Phi}(\lambda),\mathbf{\Gamma}(\lambda)\big{)}$, obtained via
(9a)-(9b). Therefore, $\mathbf{W}(\lambda)$ is lower triangular if and only if
$\mathbf{\Phi}(\lambda)$ is lower triangular and, similarly,
$\mathbf{V}(\lambda)$ is tridiagonal if and only if $\mathbf{\Gamma}(\lambda)$
is tridiagonal.

In light of Remark 3.11 and the TFMs from (9a)-(9b), it becomes clear that, in
order to obtain a rich parametrization of factored representations, for a
given $\mathbf{G}(\lambda)$ and having a particular sparsity pattern, it
suffices to study the set of all SRTR pairs
$\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ associated with the
investigated network. This can be achieved via the following result, that will
prove critical to many of the results presented in the sequel.

######  Theorem 3.12.

For the system described by the realization given in (4a)-(4b), we have that:

  1. $\mathbf{a)}$

An SRTR pair $\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$
associated with the aforementioned system is expressed via

(15)
$\small\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}=\left[\begin{array}[]{c|cc}A_{22}+KA_{12}-\lambda
I_{n-p}&KA_{11}-KA_{12}K+A_{21}-A_{22}K&KB_{1}+B_{2}\\\ \hline\cr
A_{12}&A_{11}-A_{12}K&B_{1}\\\ \end{array}\right],$

  1. $\mathbf{a)}$

where $K$ is any matrix in $\mathbb{R}^{(n-p)\times p}$;

  2. $\mathbf{b)}$

All SRTR pairs associated with the system described by (4a)-(4b) are expressed
via (15) for some $K\in\mathbb{R}^{(n-p)\times p}$ if and only if the
aforementioned $n$-th order realization is minimal.

######  Proof 3.13.

See Appendix D.

######  Remark 3.14.

The poles of $\mathbf{W}(\lambda)$ and $\mathbf{V}(\lambda)$ can be placed at
will in the complex plane, by a suitable choice of the matrix $K$ and the
observability of the pair $(A_{12},A_{22})$ (recalling Assumption 3.1). Thus,
when implementing a controller with a realization of type (4a)-(4b) via its
SRTR pair, it may be possible to obtain a distributed control law of type (8)
using TFMs whose entries are all stable. If this is the case, then this
implementation can be employed even for a plant which is not strongly
stabilizable, _i.e._ , there exist no (centralized) controllers having stable
TFMs which render all closed-loop maps stable. This possibility will be
investigated in the sequel.

$\begin{array}[]{ccl}\begin{bmatrix}\sigma{y}\\\
\sigma{q}\end{bmatrix}&=&\begin{bmatrix}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{bmatrix}\begin{bmatrix}y\\\
q\end{bmatrix}+\begin{bmatrix}B_{1}\\\ B_{2}\end{bmatrix}u\\\
y&=&\begin{bmatrix}\hskip 5.69054ptI_{p}&\hskip 4.2679ptO\hskip
5.69054pt\end{bmatrix}\hskip 1.42262pt\begin{bmatrix}y\\\
q\end{bmatrix}\end{array}$
$\begin{array}[]{l}\lambda\mathbf{Y}(\lambda)=\mathbf{W}(\lambda)\mathbf{Y}(\lambda)+\mathbf{V}(\lambda)\mathbf{U}(\lambda)\\\
\phantom{}\vspace{-2mm}\\\ \big{(}\lambda
I_{p}-\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}\text{ FLCF}\end{array}$
$\begin{array}[]{lcl}\mathbf{Y}(\lambda)=\mathbf{\Phi}(\lambda)\mathbf{Y}(\lambda)+\mathbf{\Gamma}(\lambda)\mathbf{U}(\lambda)\\\
\phantom{}\vspace{-2mm}\\\ \mathbf{\Phi}_{ii}(\lambda)=0,\ \forall\,1\leq
i\leq p\end{array}$
$\begin{array}[]{l}\mathbf{Y}(\lambda)=\mathbf{M}^{-1}(\lambda)\mathbf{N}(\lambda)\mathbf{U}(\lambda)\\\
\phantom{}\vspace{-2mm}\\\
\big{(}\mathbf{M}(\lambda),\mathbf{N}(\lambda)\big{)}\text{ LCF over
$\mathbb{S}$}\end{array}$ Theorem 3.12(78)-(79)Theorem C.3
$\begin{array}[]{c}\eqref{Kontroller}\\\ \text{via}\\\ \eqref{dc2}\end{array}$
Theorem 4.6Theorem 4.10 (9a)-(9b)(9) in [24] Figure 2: Diagram showing the
means of switching between various LTI network representations

##  4 Main Results

The ultimate goal of this line of research is the computation of controllers
whose factorized representation possesses a desired sparsity pattern or which
inherits the structure of a closely related factorization, from which we may
shift by employing numerically reliable procedures. This would allow us, for
instance, to compute controllers that can be implemented as a “ring” network
(recall Fig. 1) or as a “line” network, which is important for motion control
of vehicles moving in “platoon” formation (see [25]). A key insight in this
regard is the fact that the most prominent classical results in control
theory, such as the Youla Parameterization, state the expression of a
stabilizing controller via a coprime factorization over $\mathbb{S}$ of its
TFM.

As a further step towards employing Youla-like methods for the synthesis of
controllers featuring either structured NRF or SRTR pairs, we need to
understand the connections between the (stable) left coprime factorizations
(of a given stabilizing controller) and its NRF- or SRTR-based factorizations.
To help in keeping track of these various representations, the diagram
displayed in Fig. 2 at the top of this page illustrates how the various
results presented in this paper interact with one another.

######  Remark 4.1.

A poignant insight into the relationship between the SRTR and NRF
representations is hinted at in Fig. 2. As can be seen from the indicated
diagram, an SRTR pair bears much stronger connections with a network’s state
dynamics than its NRF counterpart. Indeed, it is this very affinity which
serves to motivate the study of SRTR representations, in contrast to the more
input-output oriented perspective offered by an NRF pair. This also makes SRTR
pairs a far more appealing tool for system analysis than their correspondents
in the NRF formalism, as the latter was primarily developed with the aim of
controller synthesis (see [24] and [27]).

###  4.1 Coprimeness of SRTR Pairs

In this subsection, we will prove that (by chance, rather than by design), for
any SRTR $\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ associated
with a system described by a minimal realization (4a)-(4b) and having
$\mathbf{G}(\lambda)$ as its TFM, it follows that $\displaystyle\big{(}\lambda
I_{p}-\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ is an FLCF of
$\mathbf{G}(\lambda)$. Before doing so, we recall the pivotal role of
controllability, as established by point $\mathbf{b)}$ in Theorem 3.12 and,
from now onward, we assume that the following statement holds.

The realization (4a)-(4b) is controllable.

######  Remark 4.2.

A direct consequence of point $\mathbf{b)}$ in Theorem 3.12 is that the set of
all SRTR pairs that can be obtained from a system described by a minimal
realization (4a)-(4b), and having a TFM denoted $\mathbf{G}(\lambda)$,
coincides with the set of all SRTR pairs expressed via (15) for any other
system described by a minimal realization of the same $\mathbf{G}(\lambda)$.
Given the fact that, in the sequel, we consider only realizations of type
(4a)-(4b) which are minimal, the subsequently presented SRTR-based results are
free to employ any minimal realization of type (4a)-(4b), for a given TFM.

As previously discussed, coprimeness is an important property for output
feedback stabilization, since classical results require stable coprime factors
of the plant, in addition to the stable coprime factors of the controllers.
Therefore, we now characterize a class of SRTR-based coprime factorizations
via the following result.

######  Theorem 4.3.

Consider a system given by a minimal realization (4a)-(4b), having
$\mathbf{G}(\lambda)$ as its TFM. Then, for any SRTR pair
$\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ of the latter system
expressed as in (15), we have that the pair $\displaystyle\big{(}\lambda
I_{p}-\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ is an FLCF of
$\mathbf{G}(\lambda)$.

######  Proof 4.4.

See Appendix D.

######  Remark 4.5.

While any SRTR pair $\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$
produced by point $\mathbf{b)}$ of Theorem 3.12 and associated with the system
having $\mathbf{G}(\lambda)$ as its TFM offers an FLCF
$\mathbf{G}(\lambda)=\big{(}\lambda
I_{p}-\mathbf{W}(\lambda)\big{)}^{-1}\mathbf{V}(\lambda)$, the NRF
representation
$\displaystyle\mathbf{G}(\lambda)=\big{(}I_{p}-\mathbf{\Phi}(\lambda)\big{)}^{-1}\mathbf{\Gamma}(\lambda)$
is _not guaranteed_ to be fully left coprime. This is owed to the fact that
some of the finite poles belonging to $\big{(}\lambda
I_{p}-\mathbf{D}(\lambda)\big{)}$ may not cancel out when multiplying its
inverse with $\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ to obtain, via
(9a)-(9b), the fact that

$\begin{bmatrix}I_{p}-\mathbf{\Phi}&\mathbf{\Gamma}\end{bmatrix}=\big{(}\lambda
I_{p}-\mathbf{D}(\lambda)\big{)}^{-1}\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}.$

We point out that, due to the diagonal structure of $\big{(}\lambda
I_{p}-\mathbf{D}(\lambda)\big{)}^{-1}$, the statements from Remark 3.11
regarding the sparsity pattern of the resulting NRF are reinforced.

The inherent coprimeness of SRTR pairs turns out to be an important property,
allowing for the commutation between these pairs and the classical coprime
factorizations of the network’s model. Building upon this property, we proceed
to investigate in the sequel reliable means of shifting between these two sets
of coprime representations.

###  4.2 Getting from SRTR pairs to Left Coprime Factorizations

In this subsection, we show that any SRTR pair
$\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ with both
$\mathbf{W}(\lambda)$ and $\mathbf{V}(\lambda)$ having no poles located
outside of $\mathbb{S}$ can produce an entire class of factorizations which
are left coprime over $\mathbb{S}$. Notice that, for any SRTR pair
$\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$, the TFM
$\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ is improper and
has exactly $p$ poles at infinity of multiplicity one, and recall that the
poles of $\mathbf{W}(\lambda)$ and $\mathbf{V}(\lambda)$ can be placed at
will, as per Remark 3.14.

We aim to showcase the means of shifting from a stable SRTR pair
$\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ of the system having
$\mathbf{G}(\lambda)$ as its TFM to an LCF over $\mathbb{S}$ of
$\mathbf{G}(\lambda)=\mathbf{M}^{-1}(\lambda)\mathbf{N}(\lambda)$. Crucial to
this endeavor, we provide in Appendix C the characterization for a class of
LCFs over $\mathbb{S}$ for our network’s TFM. It is this very result which we
now employ, in order to obtain a procedure which extracts an LCF over
$\mathbb{S}$ from a stable SRTR pair.

######  Theorem 4.6.

Consider a system given by a minimal realization (4a)-(4b), having
$\mathbf{G}(\lambda)$ as its TFM, and let
$\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ be a stable SRTR pair
associated with this system. Define

(16) $\mathbf{\Theta}(\lambda):=\left[\begin{array}[]{c|c}A_{x}-\lambda
I_{p}&B_{x}\\\ \hline\cr C_{x}&O\end{array}\right],$

for any $A_{x}\in\mathbb{R}^{p\times p}$ having all its eigenvalues located in
$\mathbb{S}$, along with any $B_{x}\in\mathbb{R}^{p\times p}$ and any
$C_{x}\in\mathbb{R}^{p\times p}$ which are both invertible. Then, we have
that:

  1. $\mathbf{a)}$

The following TFM

(17) $\begin{bmatrix}\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix}{=}\
\mathbf{\Theta}(\lambda)\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}\normalsize$

  1. $\mathbf{a)}$

designates an LCF over $\mathbb{S}$ of
$\mathbf{G}(\lambda)=\mathbf{M}^{-1}(\lambda)\mathbf{N}(\lambda)$;

  2. $\mathbf{b)}$

The multiplication with $\mathbf{\Theta}(\lambda)$ from (17) preserves the
poles of the SRTR pair in the $\mathbf{M}(\lambda)$ and $\mathbf{N}(\lambda)$
factors.

######  Proof 4.7.

See Appendix D.

######  Remark 4.8.

Notice that, for any _diagonal_ $A_{x}$ having all its eigenvalues located in
$\mathbb{S}$, the factor $\mathbf{\Theta}(\lambda)=(\lambda I_{p}-A_{x})^{-1}$
can be used to produce an LCF over $\mathbb{S}$ of $\mathbf{G}(\lambda)$ with
precisely the same sparsity structure as the SRTR pair
$\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$.

###  4.3 Getting from Left Coprime Factorizations to SRTR pairs

In this subsection, we show that, given an LCF over $\mathbb{S}$ of
$\mathbf{G}(\lambda)=\mathbf{M}^{-1}(\lambda)\mathbf{N}(\lambda)$, which is
described by a realization of type (4a)-(4b) and for which
$\begin{bmatrix}\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix}$ has the
same McMillan degree as $\mathbf{G}(\lambda)$, it is possible to use a certain
type of realization for the aforementioned factorization, in order to retrieve
a stable SRTR pair $\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$.

To begin with, recall from the previous subsection that Appendix C includes a
characterization for a particular class of LCFs over $\mathbb{S}$ for our
network’s TFM. Thus, we start with a given LCF over $\mathbb{S}$ of
$\mathbf{G}(\lambda)$ from this class, where the latter is described by an
$n$-th order minimal realization. In this case, let the given LCF over
$\mathbb{S}$ have a state-space representation as in (39) from the Appendix C,
to which we apply a state equivalence transformation described by
$T\in\mathbb{R}^{n\times n}$, so that $\displaystyle
CT^{-1}=\begin{bmatrix}I_{p}&O\end{bmatrix}$. Such a $T$ matrix always exists,
due to Assumption 3.1, and it follows that (39) turns into

(18)
$\begin{bmatrix}\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix}=\footnotesize\left[\begin{array}[]{cc|cc}A_{11}+F_{1}-\lambda
I_{p}&A_{12}&F_{1}&B_{1}\\\ A_{21}+F_{2}&A_{22}-\lambda I_{n-p}&F_{2}&B_{2}\\\
\hline\cr U&O&U&O\end{array}\right].\normalsize$

The key to retrieving stable SRTR pairs rests with a particular class of
solutions to the continuous-time non-symmetric algebraic Riccati equation
(CTNARE) given by

(19) $K(A_{11}+F_{1})-KA_{12}K+(A_{21}+F_{2})-A_{22}K=O.$

As will be shown in the sequel, it is these solutions which underpin the
correspondence between the set of systems from (18) and their related SRTR
pairs. Thus, given their paramount importance, we now formally introduce these
particular solutions.

######  Definition 4.9.

A matrix $K$ that is a solution of (19) and which ensures that
$(A_{11}+F_{1}-A_{12}K)$ has all its eigenvalues located in $\mathbb{S}$ is
called a _right stabilizing solution_ of the CTNARE.

The existence of these solutions is shown in Appendix B to be a generic
property (in light of Assumption 3.1), with the aforementioned appendix
offering means of reliably computing them. This being said, we proceed to
employ these right stabilizing solutions, in order to state the central result
of this subsection.

######  Theorem 4.10.

Consider a system given by a minimal realization (4a)-(4b), having
$\mathbf{G}(\lambda)$ as its TFM, and such that
$\mathbf{G}(\lambda)=\mathbf{M}^{-1}(\lambda)\mathbf{N}(\lambda)$ is an LCF
over $\mathbb{S}$, having a realization of type (18). Let also $K$ be a right
stabilizing solution of the CTNARE from (19) and denote
$A_{x}:=A_{11}+F_{1}-A_{12}K$. Then, we have that:

  1. $\mathbf{a)}$

The realization

(20)
$\begin{bmatrix}\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix}=\left[\scriptsize\begin{array}[]{cc|cc}A_{x}-\lambda
I_{p}&A_{12}&A_{x}-A_{11}+A_{12}K&B_{1}\\\ O&A_{22}+KA_{12}-\lambda
I_{n-p}&KA_{12}K+A_{22}K-KA_{11}-A_{21}&KB_{1}+B_{2}\\\ \hline\cr U&O&U&O\\\
\end{array}\right]\normalsize$

  1. $\mathbf{a)}$

is equivalent to the one from (18);

  1. $\mathbf{b)}$

A stable SRTR pair $\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ of
the system can be obtained, from the available LCF over $\mathbb{S}$, via

(21)
$\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}=\begin{bmatrix}\lambda
I_{p}&O\end{bmatrix}+(\lambda
I_{p}-A_{x})U^{-1}\begin{bmatrix}-\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix};$

  1. $\mathbf{c)}$

The stable SRTR pair from (15), with $K$ taken as a right stabilizing solution
of (19), is identical to the stable SRTR pair from (21) for the same $K$.

######  Proof 4.11.

See Appendix D.

######  Remark 4.12.

The realization of the stable SRTR pair, mentioned in point $\mathbf{c)}$ of
Theorem 4.10, can also be obtained from (20). Notice that the state-space
representation from (15) can be formed by simply extracting the appropriate
submatrices which make up the realization given in (20).

$r$$z$${\bf K(\lambda)}$$u$$v$$w$${\bf G(\lambda)}$$\zeta$$y$ Figure 3:
Feedback loop of the plant $\bf G$ with the controller $\bf K$

###  4.4 Stable and Stabilizing SRTR Implementations

In this subsection, we elaborate upon Remark 3.14 and present the means of
implementing distributed controllers via their SRTR representations. The
results given here are, in many ways, the analogues of Theorems III.4 and
III.6 from [24], thus highlighting the connections being showcased in Fig. 2.
Before doing so, we introduce the following definitions.

######  Definition 4.13.

Let $\mathbf{K}\in\mathbb{R}(\lambda)^{p\times m}$ and
$\mathbf{G}\in\mathbb{R}(\lambda)^{m\times p}$. A closed-loop interconnection
(such as the one in Fig. 3) between $\mathbf{K}(\lambda)$ and
$\mathbf{G}(\lambda)$ for which the closed-loop maps from all exogenous
signals ($r$, $w$ and $\zeta$) to all internal ones ($y$, $u$, $z$ and $v$)
are stable TFMs is called _internally stable_.

######  Definition 4.14.

A system, described by its TFM $\mathbf{K}\in\mathbb{R}(\lambda)^{p\times m}$,
which ensures that the closed-loop interconnection with
$\mathbf{G}\in\mathbb{R}(\lambda)^{m\times p}$ from Fig. 3 is internally
stable, is termed a _stabilizing controller_ for $\mathbf{G}(\lambda)$.

We start by proving that the specialized implementations of type (8),
corresponding to the TFMs of stabilizing controllers, are themselves
stabilizing.

$r$$z$$\mathbf{K}_{d}(\lambda)$$u$$v$$\bullet$$\delta_{u}$$w$${\bf
G(\lambda)}$$\zeta$$y$ Figure 4: Feedback loop of the plant ${\bf G}$ with the
controller ${\bf K}$ in an SRTR-based implementation
$\lambda\mathbf{U}(\lambda)={\bf
W}(\lambda)(\mathbf{U}(\lambda)+\mathbf{\Delta_{U}}(\lambda))+{\bf
V}(\lambda)\mathbf{Z}(\lambda)$

######  Theorem 4.15.

Let $\mathbf{K}\in\mathbb{R}(\lambda)^{p\times m}$ be a stabilizing controller
for $\mathbf{G}\in\mathbb{R}(\lambda)^{m\times p}$, given by a minimal
realization (4a)-(4b) and by a stable SRTR pair
$\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ associated with the
latter via (15), as in Definition 3.8. Then, the implementation

(22a) $\displaystyle\mathbf{K}_{d}(\lambda):=$
$\displaystyle\begin{bmatrix}\frac{1}{\lambda}\mathbf{W}(\lambda)&\frac{1}{\lambda}\mathbf{V}(\lambda)\end{bmatrix},$
(22b) $\displaystyle\mathbf{U}(\lambda)=$ $\displaystyle\
\mathbf{K}_{d}(\lambda)\begin{bmatrix}\mathbf{U}(\lambda)^{\top}+\mathbf{\Delta}^{\top}_{\mathbf{U}}(\lambda)&\mathbf{Z}^{\top}(\lambda)\end{bmatrix}^{\top},$

internally stabilizes the closed-loop interconnection in Fig. 4, where
$\delta_{u}$ designates a bounded communication disturbance and
$\mathbf{\Delta_{U}}(\lambda)$ denotes its Laplace or $\mathcal{Z}$-transform.

######  Proof 4.16.

See Appendix D.

######  Remark 4.17.

Notice that the stabilizing and distributed controller from (22a) always has a
stable TFM in the discrete-time case, as previously anticipated in Remark
3.14. Recalling the parallels discussed in Remark 3.9, this conclusion comes
naturally and serves to further emphasize the close connections between the
SRTR representations and the celebrated SLS framework [35], for discrete-time
systems.

######  Remark 4.18.

In the continuous-time case, the sparsity structure of the distributed
controller from (22a) is much more versatile than the ones obtained in [10].
We do not restrict $\mathbf{K}_{d}(\lambda)$ to possessing only banded nonzero
entries and the generic sparsity structure it inherits from
$\mathbf{W}(\lambda)$ and $\mathbf{V}(\lambda)$ offers more flexibility with
respect to the network’s communication infrastructure. Note that, in contrast
to the results from [10], we make no assumptions on the spatial invariance of
the investigated networks.

Although the TFM from (22a) is guaranteed to be stable in the discrete-time
case, we now show that this feature is unavailable for continuous-time
controllers, when considering implementations obtained via SRTR pairs produced
by Theorem 3.12.

######  Proposition 4.19.

In the continuous-time case, $\mathbf{K}_{d}(s)$ from (22a) is guaranteed to
be an unstable TFM when it is formed from any stable SRTR pair, obtained from
a minimal realization (4a)-(4b) of $\mathbf{K}(s)$ via (15), as considered in
Theorem 4.15.

######  Proof 4.20.

See Appendix D.

We now show that a particularly useful state-space implementation of (22b) can
be used to drive the closed-loop system’s state vector to the origin for any
finite initial conditions. This critical result is formalized via the
following theorem.

######  Theorem 4.21.

Let $\mathbf{G}\in\mathbb{R}(\lambda)^{m\times p}$ be given by a stabilizable
and detectable realization and let $\mathbf{K}\in\mathbb{R}(\lambda)^{p\times
m}$ be a stabilizing controller of $\mathbf{G}(\lambda)$. Let
$\mathbf{K}(\lambda)$ be given by a minimal realization (4a)-(4b) and by a
stable SRTR pair $(\mathbf{W}(\lambda),\mathbf{V}(\lambda))$ associated with
the latter via (15). Let $e_{i}$ be the $i^{\text{th}}$ vector in the
canonical basis of $\mathbb{R}^{p}$. By implementing stabilizable and
detectable realizations for $e_{i}^{\top}\mathbf{K}_{d}(\lambda)$, with $i\in
1:p$, and by computing the command signals as in (22b), the state dynamics of
the closed-loop system from Fig. 4 are made asymptotically stable (see Section
5.3 in [40]).

######  Proof 4.22.

See Appendix D.

######  Remark 4.23.

For the means to obtain realizations of the type described in Theorem 4.21,
consult Remark III.7 in [24]. Fortuitously, Remark III.8 in [24] also holds
for block-row implementations (recall Remark 3.5) of the control laws from
(22a)-(22b). Additionally, the synthesis techniques described in [24] and [27]
can be employed in order to obtain the stabilizing controllers that are
required by the results given in this subsection. Such a design scenario, in
which the techniques from [24] and [27] are used to obtain a stabilizing
controller which generates sparse SRTR-based control laws, will be showcased
in Section 5 through a comprehensive numerical example.

######  Remark 4.24.

Among the chief benefits of the distributed control schemes proposed in
Theorems 4.15 and 4.21 is that they provide stability guarantees in both the
continuous- and discrete-time contexts. In contrast to this statement, we now
refer to the proof of sufficiency for Theorem 2 from [35] and to the proof of
Lemma 4 from [35]. The fact that $\frac{1}{z}I_{p}$ is a discrete-time stable
TFM has been employed in these proofs so as to provide guarantees for both
internal stability (recall Definition 4.13) and for the fact that the free
evolution of the plant’s and the controller’s states tends to zero from any
finite initial conditions (see the proof of Theorem 4.21 in Appendix D), in
closed-loop configuration. Thus, the arguments made in [35] cannot be employed
for the continuous-time setting, since $\frac{1}{s}I_{p}$ is not a stable TFM
in the latter time domain.

Given that we now posses theoretical guarantees for closed-loop distributed
stabilization via SRTR-based control laws, it becomes natural to investigate
the means by which these representations can be constrained to possess various
sparsity structures, meant to promote decentralized implementation. In the
following subsection, we turn our attention to this problem, with the aim of
finding reliable numerical procedures that enforce a desired sparsity pattern
upon an SRTR pair.

###  4.5 Forming structured and stabilizing STRT-based implementations

In light of Theorem 4.10 and of the results presented in Section III of [27],
it is tempting to envision a procedure that produces stable SRTR pairs by way
of structured LCFs over $\mathbb{S}$. However, a closer inspection of Theorem
4.21 suggests a different approach.

Recall that, as per Remark 4.23, we aim to implement state-space realizations
of (block-)rows belonging to the TFM from (22a). Moreover, notice that the
latter has the same sparsity structure as the TFM given in (15) and that, for
the purpose of computational efficiency when calculating the command signals,
we wish to obtain realizations for these (block-)rows which have orders that
are as small as possible.

It turns out that a convenient solution to these objectives can be obtained by
employing the class of realizations parametrized by the constant matrices
$K\in\mathbb{R}^{(n-p)\times p}$ and given explicitly in (15). Given two
binary matrices, which encode the desired sparsity structure of the stable
SRTR pair, and a set of integers, which impose the orders of the sought-after
realizations, the following result offers tractable solutions.

######  Theorem 4.25.

Consider a system given by a minimal realization (4a)-(4b) and let $e_{q,i}$
denote the $i^{th}$ vector from the canonical basis of $\mathbb{R}^{q}$, for
any nonzero $q\in\mathbb{N}$. For each $i\in 1:p$, denote by
$Q_{i}\in\mathbb{R}^{(n-p)\times(n-p)}$ any orthogonal matrix which satisfies

(23)
$e_{p,i}^{\top}A_{12}Q_{i}^{\top}=\left\|e_{p,i}^{\top}A_{12}\right\|e_{(n-p),(n-p)}^{\top}\
,$

along with an integer $n_{i}\in 1:(n-p)$. Consider also
$\mathcal{B}_{\mathbf{W}}\in\mathbb{R}^{p\times p}$ and
$\mathcal{B}_{\mathbf{V}}\in\mathbb{R}^{p\times m}$ with
$e_{p,i}^{\top}\mathcal{B}_{\mathbf{W}}e_{p,j}\in\\{0,1\\}$ and
$e_{p,i}^{\top}\mathcal{B}_{\mathbf{V}}e_{m,k}\in\\{0,1\\}$, for any $i,j\in
1:p$ and any $k\in 1:m$. Assume that, for some $K\in\mathbb{R}^{(n-p)\times
p}$ and for each $i\in 1:p$, we have:

  1. $i)$

$e_{p,i}^{\top}(A_{11}-A_{12}K)e_{p,j}\left(1-e_{p,i}^{\top}\mathcal{B}_{\mathbf{W}}e_{p,j}\right)=0,\
\forall j\in 1:p$;

  2. $ii)$

$e_{p,i}^{\top}B_{1}e_{m,k}\left(1-e_{p,i}^{\top}\mathcal{B}_{\mathbf{V}}e_{m,k}\right)=0,\
\forall k\in 1:m$;

and, whenever $\left\|e_{p,i}^{\top}A_{12}\right\|\neq 0$, we also have:

  1. $iii)$

$\begin{bmatrix}O&I_{n_{i}}\end{bmatrix}Q_{i}(KA_{11}-KA_{12}K+A_{21}-A_{22}K)e_{p,j}\left(1-e_{p,i}^{\top}\mathcal{B}_{\mathbf{W}}e_{p,j}\right)=O,\forall
j\in 1:p$;

  2. $iv)$

$\begin{bmatrix}O&I_{n_{i}}\end{bmatrix}Q_{i}(KB_{1}+B_{2})e_{m,k}\left(1-e_{p,i}^{\top}\mathcal{B}_{\mathbf{V}}e_{m,k}\right)=O,\forall
k\in 1:m$;

  3. $v)$

$\begin{bmatrix}O&I_{n_{i}}\end{bmatrix}Q_{i}(A_{22}+KA_{12})Q_{i}^{\top}\begin{bmatrix}I_{n-p-
n_{i}}&O\end{bmatrix}^{\top}=O$;

  4. $vi)$

$\left\\{\lambda\in\mathbb{C}\ :\ \det\left(\lambda
I_{n_{i}}-\begin{bmatrix}O&I_{n_{i}}\end{bmatrix}Q_{i}(A_{22}+KA_{12})Q_{i}^{\top}\begin{bmatrix}O&I_{n_{i}}\end{bmatrix}^{\top}\right)=0\right\\}\subseteq\mathbb{S}$.

Then, it follows that:

  1. $\mathbf{a)}$

The TFM
$e_{p,i}^{\top}\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$
can be expressed, when $\left\|e_{p,i}^{\top}A_{12}\right\|=0$, via the
constant matrix

(4.9a)
$e_{p,i}^{\top}\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}=e_{p,i}^{\top}\begin{bmatrix}A_{11}-A_{12}K&B_{1}\end{bmatrix};$

  1. $\mathbf{a)}$

or, when $\left\|e_{p,i}^{\top}A_{12}\right\|\neq 0$, via the realization of
order $n_{i}$ given by

(4.9b)
$e_{p,i}^{\top}\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}=\left[\begin{array}[]{c|c}\begin{bmatrix}O&I_{n_{i}}\end{bmatrix}Q_{i}(A_{22}+KA_{12})Q_{i}^{\top}\begin{bmatrix}O&I_{n_{i}}\end{bmatrix}^{\top}-\lambda
I_{n_{i}}&\widetilde{B}_{i2}\\\ \hline\cr
e_{p,i}^{\top}A_{12}Q_{i}^{\top}\begin{bmatrix}O&I_{n_{i}}\end{bmatrix}\phantom{}^{\top}&\widetilde{D}_{i}\\\
\end{array}\right],$

  1. $\mathbf{a)}$

having defined

(25a) $\displaystyle\widetilde{B}_{i2}:=$
$\displaystyle\begin{bmatrix}O&I_{n_{i}}\end{bmatrix}Q_{i}\begin{bmatrix}KA_{11}-KA_{12}K+A_{21}-A_{22}K&KB_{1}+B_{2}\end{bmatrix},$
(25b) $\displaystyle\widetilde{D}_{i}:=$ $\displaystyle\
e_{p,i}^{\top}\begin{bmatrix}A_{11}-A_{12}K&B_{1}\end{bmatrix};$

  1. $\mathbf{b)}$

The SRTR pair $(\mathbf{W}(\lambda),\mathbf{V}(\lambda))$ obtained from $K$
via the realization from (15) is formed of stable TFMs that, for any $i,j\in
1:p$ and any $k\in 1:m$, satisfy

(26a) $\displaystyle e_{p,i}^{\top}\mathcal{B}_{\mathbf{W}}e_{p,j}=0\
\Longrightarrow$ $\displaystyle\ \
e^{\top}_{p,i}\mathbf{W}(\lambda)e_{p,j}\equiv 0,$ (26b) $\displaystyle
e_{p,i}^{\top}\mathcal{B}_{\mathbf{V}}e_{m,k}=0\ \Longrightarrow$
$\displaystyle\ \ e^{\top}_{p,i}\mathbf{V}(\lambda)e_{m,k}\equiv 0.$

######  Proof 4.26.

See Appendix D.

######  Remark 4.27.

Notice the fact that the proposed procedure also caters to the situations in
which we desire to implement block-rows of the obtained SRTR pair. Indeed, the
sparsity pattern and stability constraints are satisfied if and only if all
rows of the considered block-rows abide by the aforementioned constraints.
Additionally, the dimensionality aspect of the problem can also be tackled by
minimizing the individual orders of the realizations belonging to a block-
row’s component rows. Should the block-row problem be tackled directly, the
algebraic conditions from points $i)$ through $vi)$ of Theorem 4.25 can easily
be altered to accommodate this change of perspective, with the resulting
procedure amounting to a mere mutation of the provided technique.

######  Remark 4.28.

In contrast to the approach from [17], our technique places little emphasis on
the sparsity pattern belonging to the realization of type (4a)-(4b), employed
in the enforcement of sparsity patterns on the resulting SRTR pairs.
Furthermore, as will be shown in the numerical example from Section 5, the
proposed technique can produce SRTR pairs with distinctive network topologies
and high degrees of sparsity, even when the realization employed by the
procedure possesses a greatly reduced degree of sparsity. In spite of this,
the arguments made in [17] remain relevant even in the considered case, which
is to say that the discussed problem’s feasibility is highly dependent upon
the network’s intrinsic flow of information. Thus, the aforementioned example
will employ the procedure from [27] to obtain a controller with just such an
information flow, imparted by a structured NRF representation.

Before moving on to the numerical example, we first discuss the computational
aspects of the algorithms used to satisfy conditions $i)$ through $vi)$ from
Theorem 4.25. Firstly, the orthogonal matrices $Q_{i}$ with $i\in 1:p$, which
compress the individual rows of $A_{12}$, are straightforward to compute via
the use of permutations and Householder transformations. Additionally, notice
that conditions $i)$, $iv)$ and $v)$ only amount to a set of equality
constraints in terms of linear combinations of the entries belonging to $K$,
whereas condition $ii)$ must only be checked against the desired sparsity
pattern.

######  Remark 4.29.

It has been shown via the identity from (80) in Appendix D that $B_{1}$ is
invariant with respect to the class of all minimal realizations of a system
showcasing the structure from (4a)-(4b). Therefore, the choice of minimal
realization to be employed in Theorem 4.25 does not restrict the result’s
applicability and means of inducing sparsity in $B_{1}$ will be presented in
the numerical example from Section 5, by employing the synthesis procedures
from [27], as previously indicated in Remark 4.28.

We conclude that the main difficulty in applying Theorem 4.25 rests with
satisfying conditions $iii)$ and $vi)$. Notably, the former involves a series
of equality constraints containing bilinear terms of the entries which belong
to the matrix $K$, whereas the latter requires that we restrict the
eigenvalues of an affine expression of $K$ to $\mathbb{S}$. Fortunately, these
types of problems have been jointly tackled in Section IV of [27] and
Algorithm 1 therein can easily be adapted to accommodate the conditions laid
out in Theorem 4.25. While the procedures from [27] deal with continuous-time
versions of condition $vi)$ from the aforementioned theorem, we point out that
the discrete-time case can be handled just as elegantly, by employing Lemma 6
from [17] alongside the convexification techniques given in [27].
Alternatively, one may employ Lemma 8 from [17] to restrict the eigenvalues of
$A_{22}+KA_{12}$ to $\mathbb{S}$ which, combined with condition $v)$ of
Theorem 4.25, is sufficient to satisfy condition $vi)$ of the aforementioned
theorem.

######  Remark 4.30.

The high computational performance of the indicated algorithm, as highlighted
in Table 1 of [27] and discussed comparatively in Section V.B of [27], serves
to justify the chosen approach and its distinctive formulation, given in
Theorem 4.25. Furthermore, a code implementation for this type of numerical
procedure is publicly available, and can be accessed via the link in the
footnote from the first page of [27].

##  5 Numerical example

We now present an example of SRTR-based design for distributed control. To
this end, we return to the network illustrated in Fig. 1 and modeled in
Section 3.3. Recall that, although this network possesses a highly particular
topology, the latter’s structure is completely obscured when inspecting the
network via its TFM. The aim of this section is to illustrate how the
framework developed in this paper can be used to obtain a controller, for the
aforementioned network, which has the same ring topology and which can be
implemented in a distributed manner, via a set of sparse and stabilizing
control laws. The chosen example, inspired by the one showcased in Section V
of [27], will be presented in the continuous-time setting. We point out that,
in doing so, our aim is to provide further contrast with respect to the
framework discussed in [35], which caters exclusively to discrete-time
systems.

Consider a network of the type described in Section 3.3 which, instead of the
$3$ nodes shown in Fig. 1, is more generically formed from the ring-like
interconnection of $p\in\mathbb{N}$ nodes, with $p\geq 2$. Thus, its NRF-like
dynamics are captured via the identities

(27) $\mathbf{Y}_{(i\text{ mod
}p)+1}(s)=\mathbf{\Phi}_{\mathbf{G}}(s)\mathbf{Y}_{((i-1)\text{ mod
}p)+1}(s)+\mathbf{\Gamma}_{\mathbf{G}}(s)\mathbf{U}_{(i\text{ mod }p)+1}(s),\
\forall i\in 1:p,$

where $\mathbf{Y}_{i}(s)$ and $\mathbf{U}_{i}(s)$ denote the Laplace
transforms of the network’s scalar outputs and inputs, respectively, while the
interconnection and input dynamics are given by the scalar TFMs
$\mathbf{\Phi}_{\mathbf{G}}(s)$ and $\mathbf{\Gamma}_{\mathbf{G}}(s)$,
respectively. Notice that these two scalar TFMs do not depend upon $i\in 1:p$,
which indicates homogeneous dynamics among the network’s nodes (see [25] for a
practical scenario where such a situation can occur).

The aim is to design a set of control laws which exhibit the same type of
ring-like structure, _i.e._ , each command signal may be computed using only
information that is local or which originates from the node directly behind it
in the ring. Moreover, given the homogeneous dynamics occurring in each of the
controlled network’s nodes, we wish to preserve this trait with respect to the
controller’s SRTR implementation. Therefore, we desire control laws of type
(22b) which, for all $i\in 1:p$, are given by

$\displaystyle\mathbf{U}_{(i\text{ mod
}p)+1}(s)=\tfrac{1}{s}\begin{bmatrix}\mathbf{W}_{local}(s)&\mathbf{V}_{local}(s)\end{bmatrix}\begin{bmatrix}\mathbf{U}_{(i\text{
mod }p)+1}^{\top}(s)&\mathbf{Y}_{(i\text{ mod
}p)+1}^{\top}(s)\end{bmatrix}^{\top}+$ (28)
$\displaystyle+\tfrac{1}{s}\begin{bmatrix}\mathbf{W}_{prev}(s)&\mathbf{V}_{prev}(s)\end{bmatrix}\begin{bmatrix}\mathbf{U}_{((i-1)\text{
mod }p)+1}^{\top}(s)&\mathbf{Y}_{((i-1)\text{ mod
}p)+1}^{\top}(s)\end{bmatrix}^{\top}.$

Once more, note that the four stable and scalar TFMs, $\mathbf{W}_{local}(s)$,
$\mathbf{V}_{local}(s)$, $\mathbf{W}_{prev}(s)$ and $\mathbf{V}_{prev}(s)$,
which form the sought-after control law, do not depend upon $i\in 1:p$.

In this example, we consider that the network is characterized by $p=6$ and by

(29) $\mathbf{\Phi}_{\mathbf{G}}(s)=\left[\begin{array}[]{r|r}-0.5-s&-0.1\\\
\hline\cr 1&0\end{array}\right]=\frac{-1}{10s+5}\
,\quad\mathbf{\Gamma}_{\mathbf{G}}(s)=\left[\begin{array}[]{r|r}1-s&1\\\
\hline\cr 1&0\end{array}\right]=\frac{1}{s-1}\ .$

By employing the identities from (27) along with the realizations from (29),
and by defining the matrix which models the interconnection structure of the
$\mathbf{\Phi}_{\mathbf{G}}(s)$ terms

(30) $F_{b}:={\scriptsize\begin{bmatrix}0&0&0&0&0&1\\\ 1&0&0&0&0&0\\\
0&1&0&0&0&0\\\ 0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\end{bmatrix}},$

we will express a state-space realization for our network as follows

(31)
$\mathbf{G}(s)=\left[\begin{array}[]{cc|c}(-0.5-s)I_{6}-0.1F_{b}&-0.1F_{b}&O\\\
O&(1-s)I_{6}&I_{6}\\\ \hline\cr I_{6}&I_{6}&O\end{array}\right].$

The state vector of the realization from (31) is, as per the identities in
(29), the concatenation of the state vectors belonging to all
$\mathbf{\Phi}_{\mathbf{G}}(s)$ subsystems and of those belonging to all
$\mathbf{\Gamma}_{\mathbf{G}}(s)$ subsystems making up the interconnection.
Moreover, routine computations show that the network’s realization from (31)
is minimal, which also makes it both stabilizable and detectable. Thus,
Theorem 4.21 can be freely applied.

######  Remark 5.1.

The observation pertaining to the state vector of the realization from (31) is
crucial. By devising distributed control laws which stabilize the state
dynamics of the system given in (31) in closed-loop configuration, this
controller’s distributed implementation will guarantee that the states of all
the $\mathbf{\Phi}_{\mathbf{G}}(s)$ and $\mathbf{\Gamma}_{\mathbf{G}}(s)$
subsystems forming the network will remain bounded when they are excited by
bounded exogenous signals (as a consequence of the closed-loop state dynamics
being made asymptotically stable, see Section 5.3 in [40] and Corollary III.9
from [24], along with its proof), and that these states (along with the state
variables of the controller’s implementation) will tend to zero when evolving
freely from any finite initial conditions.

In order to produce the control laws described by the identities from (28), we
first employ the numerical procedures from [27] to obtain a controller whose
NRF representation has the desired type of sparsity structure. This controller
is given by a minimal state-space realization of order $n=12$ and type
(4a)-(4b) denoted by

(32) $\mathbf{K}(s)=\left[\begin{array}[]{cc|c}A_{11}-\lambda
I_{6}&A_{12}&B_{1}\\\ A_{21}&A_{22}-\lambda I_{6}&B_{2}\\\ \hline\cr
I_{6}&O&O\end{array}\right],$

where we have that

$\displaystyle A_{11}=$
$\displaystyle\left[\scriptsize\begin{array}[]{rrrrrr}-11.8035&-2.5928&0.6483&-0.1498&-0.0059&1.5415\\\
1.5415&-11.8035&-2.5928&0.6483&-0.1498&-0.0059\\\
-0.0059&1.5415&-11.8035&-2.5928&0.6483&-0.1498\\\
-0.1498&-0.0059&1.5415&-11.8035&-2.5928&0.6483\\\
0.6483&-0.1498&-0.0059&1.5415&-11.8035&-2.5928\\\
-2.5928&0.6483&-0.1498&-0.0059&1.5415&-11.8035\end{array}\right],$
$\displaystyle A_{12}=$
$\displaystyle\left[\scriptsize\begin{array}[]{rrrrrr}-7.3460&-12.0158&29.1222&-12.1621&9.7523&\hskip
4.2679pt-5.1182\\\ -16.0509&10.1202&25.3239&-8.1033&-15.2385&4.0548\\\
-23.7679&21.4320&8.6752&13.6752&1.5280&-5.1783\\\
-27.0198&9.8895&-11.8960&-7.4120&16.8991&2.4006\\\
-26.0184&-17.3463&-14.7520&-7.3268&-4.8881&-6.5282\\\
-15.8018&-29.0018&9.5454&10.1566&5.3141&1.5347\end{array}\right],$
$\displaystyle A_{21}=$
$\displaystyle\left[\scriptsize\begin{array}[]{rrrrrr}0.1795&0.0818&0.4039&0.4133&0.4137&\phantom{-}0.3992\\\
0.4333&-0.0452&-0.2691&-0.3175&0.0051&0.4693\\\
-0.4047&-0.5266&-0.2458&-0.0123&0.2775&0.1615\\\
-0.4200&0.5344&0.0288&-0.4924&0.4689&0.0625\\\
-0.2869&-0.2283&0.5717&-0.3033&-0.4110&0.4398\\\
-0.2019&0.2441&-0.2441&0.2778&-0.2152&0.2834\end{array}\right],$
$\displaystyle A_{22}=$
$\displaystyle\left[\scriptsize\begin{array}[]{rrrrrr}-2.2413&1.0141&0.3885&-0.5740&-0.4535&0.2336\\\
-1.0913&-2.1134&-2.7942&-0.1511&0.1878&0.0216\\\
-0.3266&2.7957&-2.1347&-0.4690&0.0492&0.1115\\\
0.6269&0.2681&0.3424&-2.8351&4.0507&-0.0716\\\
0.2603&-0.3236&-0.2731&-4.0526&-2.8072&0.0350\\\
0.1026&0.1898&-0.0850&0.0140&-0.1408&-4.6168\end{array}\right],$
$\displaystyle B_{1}=$
$\displaystyle\left[\scriptsize\begin{array}[]{rrrrrr}-1.0779&0.0000&0.0000&0.0000&0.0000&15.8441\\\
15.8441&-1.0779&0.0000&0.0000&0.0000&0.0000\\\
0.0000&15.8441&-1.0779&0.0000&0.0000&0.0000\\\
0.0000&0.0000&15.8441&-1.0779&0.0000&0.0000\\\
0.0000&0.0000&0.0000&15.8441&-1.0779&0.0000\\\
0.0000&0.0000&0.0000&0.0000&15.8441&-1.0779\end{array}\right],$ $\displaystyle
B_{2}=$
$\displaystyle\left[\scriptsize\begin{array}[]{rrrrrr}0.6417&1.1275&1.4058&1.4121&1.0183&0.5263\\\
-0.1623&-1.0111&-0.7933&0.4321&1.4028&1.0261\\\
-1.5048&-0.7775&0.3057&0.7985&-0.0171&-1.2371\\\
0.5132&-0.2538&-0.0823&0.3877&-0.1357&0.1614\\\
0.2803&0.2855&-0.5768&-0.2496&-0.0568&-0.3892\\\
0.0177&0.0315&0.0806&0.1186&0.1352&0.0834\end{array}\right].$

######  Remark 5.2.

Notably, the obtained controller’s TFM displays no distinctive sparsity
pattern, having only nonzero entries. This is straightforward to check by
simply computing the value of
$\mathbf{K}(0)=\begin{bmatrix}-I_{6}&O\end{bmatrix}\begin{bmatrix}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{bmatrix}^{-1}\begin{bmatrix}B_{1}^{\top}&B_{2}^{\top}\end{bmatrix}^{\top}$.
As discussed throughout Section 3, the classical TFM-based representation of a
system is oftentimes opaque to its intrinsic structure, as otherwise captured
by NRF or SRTR pairs.

For the obtained realization of the controller, given in (32), we will employ
a set of orthogonal matrices in order to compress each individual row of
$A_{12}$. In the notation employed in Theorem 4.25, these matrices are given
explicitly by

$\displaystyle Q_{1}=$
$\displaystyle\left[\scriptsize\begin{array}[]{rrrrrr}-0.1411&0.2689&-0.3353&0.8029&-0.3313&-0.2025\\\
-0.0389&0.0741&-0.0924&0.2212&0.9087&-0.3313\\\
0.0942&-0.1795&0.2239&0.4639&0.2212&0.8029\\\
-0.0393&0.0750&0.9065&0.2239&-0.0924&-0.3353\\\
0.0315&0.9399&0.0750&-0.1795&0.0741&0.2689\\\
0.9834&0.0315&-0.0393&0.0942&-0.0389&-0.1411\end{array}\right],$
$\displaystyle Q_{2}=$
$\displaystyle\left[\scriptsize\begin{array}[]{rrrrrr}0.1118&-0.4201&-0.2234&0.6982&0.2790&-0.4425\\\
-0.0216&0.0813&0.0432&-0.1350&0.9460&0.2790\\\
-0.0541&0.2033&0.1081&0.6621&-0.1350&0.6982\\\
0.0173&-0.0651&0.9654&0.1081&0.0432&-0.2234\\\
0.0326&0.8776&-0.0651&0.2033&0.0813&-0.4201\\\
0.9913&0.0326&0.0173&-0.0541&-0.0216&0.1118\end{array}\right],$ $\displaystyle
Q_{3}=$
$\displaystyle\left[\scriptsize\begin{array}[]{rrrrrr}-0.1428&0.0421&0.3770&0.2392&0.5909&-0.6553\\\
0.0510&-0.0150&-0.1346&-0.0854&0.7891&0.5909\\\
0.0206&-0.0061&-0.0545&0.9654&-0.0854&0.2392\\\
0.0325&-0.0096&0.9141&-0.0545&-0.1346&0.3770\\\
0.0036&0.9989&-0.0096&-0.0061&-0.0150&0.0421\\\
0.9877&0.0036&0.0325&0.0206&0.0510&-0.1428\end{array}\right],$ $\displaystyle
Q_{4}=$
$\displaystyle\left[\scriptsize\begin{array}[]{rrrrrr}0.0662&0.4659&-0.2043&-0.3280&0.2726&-0.7449\\\
-0.0103&-0.0728&0.0319&0.0512&0.9574&0.2726\\\
0.0124&0.0876&-0.0384&0.9384&0.0512&-0.3280\\\
0.0078&0.0546&0.9761&-0.0384&0.0319&-0.2043\\\
-0.0177&0.8756&0.0546&0.0876&-0.0728&0.4659\\\
0.9975&-0.0177&0.0078&0.0124&-0.0103&0.0662\end{array}\right],$ $\displaystyle
Q_{5}=$
$\displaystyle\left[\scriptsize\begin{array}[]{rrrrrr}-0.1800&-0.1348&-0.2020&-0.4067&-0.4782&-0.7173\\\
-0.0501&-0.0375&-0.0563&-0.1133&0.8668&-0.4782\\\
-0.0426&-0.0319&-0.0478&0.9037&-0.1133&-0.4067\\\
-0.0212&-0.0159&0.9762&-0.0478&-0.0563&-0.2020\\\
-0.0141&0.9894&-0.0159&-0.0319&-0.0375&-0.1348\\\
0.9811&-0.0141&-0.0212&-0.0426&-0.0501&-0.1800\end{array}\right],$
$\displaystyle Q_{6}=$
$\displaystyle\left[\scriptsize\begin{array}[]{rrrrrr}0.0423&0.1465&0.2800&0.2632&-0.7996&-0.4356\\\
0.0236&0.0816&0.1559&0.1466&0.5547&-0.7996\\\
-0.0078&-0.0269&-0.0513&0.9518&0.1466&0.2632\\\
-0.0083&-0.0286&0.9454&-0.0513&0.1559&0.2800\\\
-0.0043&0.9850&-0.0286&-0.0269&0.0816&0.1465\\\
0.9988&-0.0043&-0.0083&-0.0078&0.0236&0.0423\end{array}\right].$

With all of these elements now computed, we proceed to employ the procedure
given in Section 4.5, with the aim of obtaining a structured and stabilizing
SRTR representation for our controller. To this end, we will employ Theorem
4.25 and we point out the fact that $\left\|e_{p,i}^{\top}A_{12}\right\|\neq
0,\ \forall i\in 1:6$. We then inspect the structure of the control law given
in (28) and we select the two binary matrices from the statement of Theorem
4.25 as follows
$\mathcal{B}_{\mathbf{W}}=\mathcal{B}_{\mathbf{V}}=I_{6}+F_{b}$, recalling the
matrix defined in (30).

Additionally, it is desirable to find low-order state-space realizations for
each row of the TFM given in (22a) and, therefore, we choose $n_{i}=1,\
\forall i\in 1:6$. Finally, recall that we wish to obtain an SRTR-based
control law in which each command signal is computed via the expression given
in (28). For the controller produced by the procedures from [27], we obtain
precisely these types of control laws when, in addition to the constraints
stated in Theorem 4.25, we also impose that $A_{22}+KA_{12}$ be diagonal and
that it have the same element on each position of its diagonal.

######  Remark 5.3.

Notice that if the latter two constraints from the previous paragraph which
pertain to $A_{22}+KA_{12}$ are satisfied, then condition $v)$ from Theorem
4.25 is implicitly fulfilled. Therefore, the constraints arising from
condition $v)$ can be omitted from the optimization procedure’s statement, for
the sake of avoiding redundancy.

A solution to the modified problem described in Remark 5.3 is given by the
matrix

$K=\left[\scriptsize\begin{array}[]{rrrrrr}-0.0259&0.0404&0.0610&0.0693&0.0776&0.0821\\\
0.0563&-0.0062&-0.0746&-0.0792&0.0163&0.1318\\\
-0.1053&-0.0838&-0.0574&0.0127&0.1007&0.0124\\\
0.0257&0.1610&-0.1578&-0.0006&0.1571&-0.1560\\\
-0.1736&0.1068&0.0956&-0.1956&0.0737&0.0580\\\
0.1935&-0.1927&0.1912&-0.1835&0.1914&-0.1767\end{array}\right],$

which produces, via (15), the SRTR-based control laws of type (28) given by

(33) $\begin{array}[]{ll}\mathbf{W}_{local}(s)=\dfrac{-5.255s-55.9}{s+9.34}\
,&\mathbf{V}_{local}(s)=\dfrac{-1.078s-94.28}{s+9.34}\ ,\\\ \phantom{}\\\
\mathbf{W}_{prev}(s)\hskip 0.7113pt=\dfrac{-15.84}{s+9.34}\
,&\mathbf{V}_{prev}(s)\hskip 0.7113pt=\dfrac{15.84s-15.84}{s+9.34}\ .\\\
\end{array}$

The obtained SRTR pair of the controller from (32) is (continuous-time)
stable, as can be seen from (33). Additionally, since this SRTR pair belongs
to a stabilizing controller of the network, whose realization from (31) is
both stabilizable and detectable, the obtained control laws of type (22b) can
be implemented as in Theorem 4.21 with the aim of stabilizing the network’s
state dynamics, as given in (31).

##  6 Conclusions

In this paper, we have introduced the SRTR representation for networked
systems and we have presented a comprehensive discussion on the intrinsic
connections between SRTR pairs, their NRF counterparts introduced in [24], the
response TFMs from [35] with their associated implementation matrices in [13],
and the consecrated coprime factorizations treated extensively in [31]. Akin
to our previous results from [24], we have provided a unitary mathematical
framework that is not restricted to either continuous- (see [10]) or discrete-
time (see [35]) network models, while also highlighting the proposed
paradigm’s rich potential for system analysis. Finally, we have extended the
closed-loop internal stability guarantees from [24] to the current setting,
thus paving the way for distributed SRTR-based implementations of control laws
and validating our proposed framework’s capacity for controller design.

##  Appendix A Rosenbrock-type realizations

We state here several results related to a type of realization which has been
fully investigated in [21]. These remarkable representations are the
generalizations of state-space [36], descriptor [30] and pencil [19, 20]
realizations, being instrumental in constructing many of the proofs given in
Appendix D.

######  Definition A.1.

A collection of $4$ _polynomial_ matrices
$(T(\lambda),U(\lambda),V(\lambda),W(\lambda))$ with coefficients in
$\mathbb{R}$ and with $\det\left(T(\lambda)\right)\not\equiv 0$ that satisfy

(34)
$\mathbf{G}(\lambda)=V(\lambda)T^{-1}(\lambda)U(\lambda)+W(\lambda)=:\left[\footnotesize\begin{array}[]{r|r}-T(\lambda)&U(\lambda)\\\
\hline\cr V(\lambda)&W(\lambda)\end{array}\right]$

is called a _Rosenbrock-type realization_ of
$\mathbf{G}(\lambda)\in\mathbb{R}(\lambda)^{p\times m}$.

######  Definition A.2.

For a Rosenbrock-type realization of $\mathbf{G}(\lambda)$ as in (34), the
polynomial matrix denoted as

(35)
$\mathcal{S}_{\mathbf{G}}(\lambda):=\left[\small\begin{array}[]{rr}-T(\lambda)&U(\lambda)\\\
V(\lambda)&W(\lambda)\end{array}\right]$

is called a _system matrix_ of $\mathbf{G}(\lambda)$.

######  Proposition A.3.

$\mathbf{G}(\lambda)$ has full row/column normal rank if and only if
$\mathcal{S}_{\mathbf{G}}(\lambda)$ has full row/column normal rank.

The use of Rosenbrock-type realizations in Appendix D becomes justified due to
the fact that improper TFMs, such as the $\lambda I_{p}-\mathbf{W}(\lambda)$
factor generated by an SRTR pair, cannot be modeled by using state-space
realizations. Furthermore, we are particularly interested in a certain class
of Rosenbrock-type realizations.

######  Definition A.4.

A representation of type (34) for which
$\begin{bmatrix}-T(\lambda)&U(\lambda)\end{bmatrix}$ and
$\begin{bmatrix}-T^{\top}(\lambda)&V^{\top}(\lambda)\end{bmatrix}^{\top}$ have
full row and, respectively, column rank $\forall\lambda\in\mathbb{C}$ is
called an _irreducible_ Rosenbrock-type realization.

The next result plays a key role in proving Theorem 4.3.

######  Theorem A.5.

[21, pp. 111, Theorem 4.1] Let $\mathbf{G}(\lambda)$ be given by an
irreducible realization of type (34). Then:

  1. $\mathbf{a)}$

The finite poles of $\mathbf{G}(\lambda)$ are the finite zeros of
$T(\lambda)$;

  2. $\mathbf{b)}$

The finite zeros of $\mathbf{G}(\lambda)$ are the finite zeros of
$\mathcal{S}_{\mathbf{G}}(\lambda)$.

##  Appendix B Non-symmetrical Algebraic Riccati Equations

We show here that the right stabilizing solutions of (19), as introduced in
Definition 4.9, can be readily expressed and reliably computed by employing a
particular invariant subspace (see Section 1.3 of [9]) of the matrix

(36) $A^{+}:=\begin{bmatrix}\phantom{-(}A_{11}+F_{1}\phantom{)}&-A_{12}\\\
-(A_{21}+F_{2})&\phantom{-}A_{22}\end{bmatrix}.$

Let $\begin{bmatrix}V_{1}^{\top}&V_{2}^{\top}\end{bmatrix}^{\top}$ have $w$
columns and full column rank, for some integer $w\in 1:n$, with $V_{1}$ having
$p$ rows and $V_{2}$ having $n-p$ rows. By point 2 of Theorem 1.3.2 in [9], we
have that if

(37) $\begin{bmatrix}\phantom{-(}A_{11}+F_{1}\phantom{)}&-A_{12}\\\
-(A_{21}+F_{2})&\phantom{-}A_{22}\end{bmatrix}\begin{bmatrix}V_{1}\\\
V_{2}\end{bmatrix}=\begin{bmatrix}V_{1}\\\ V_{2}\end{bmatrix}S$

for some $w\times w$ matrix $S$, then
$\begin{bmatrix}V_{1}^{\top}&V_{2}^{\top}\end{bmatrix}^{\top}$ is a basis that
spans an invariant subspace of dimension $w$ belonging to $A^{+}$. By the same
result, we also have that the collection (which allows for repeated elements)
of $w$ eigenvalues belonging to $S$ is included in the collection (also
allowing for repeated elements) of $n$ eigenvalues belonging to $A^{+}$.

######  Remark B.1.

An invariant subspace of $A^{+}$ will be termed _disconjugate_ if it admits a
basis $\begin{bmatrix}V_{1}^{\top}&V_{2}^{\top}\end{bmatrix}^{\top}$ with
$V_{1}$ having full column rank. Moreover, the disconjugacy of an invariant
subspace is _independent_ with respect to the choice of basis for it.

######  Definition B.2.

When $S$ has all its eigenvalues located in $\mathbb{S}$, the subspace being
spanned by the basis from $\eqref{eq:invar}$ is called _$\mathbb{S}$
-invariant_.

Using these notions, we now express the following result, which characterizes
the right stabilizing solutions of CTNAREs.

######  Proposition B.3 (Corollary 4.2 in [11]).

The CTNARE from (19) has a right stabilizing solution $K$ if and only if
$A^{+}$ has a disconjugate $\mathbb{S}$-invariant subspace of dimension $p$.
In such cases, if we denote any basis for such a subspace as
$\begin{bmatrix}V_{1}^{\top}&V_{2}^{\top}\end{bmatrix}^{\top}$, then $V_{1}$
is invertible and $K=V_{2}V_{1}^{-1}$ is a right stabilizing solution of (19).

######  Remark B.4.

Since we consider only LCFs over $\mathbb{S}$ with representations of type
(18), and since $A^{+}$ and the state matrix of the state-space realization
from (18) are related by a similarity transformation given by
$\footnotesize\begin{bmatrix}I_{p}&O\\\ O&-I_{n-p}\end{bmatrix}$, it follows
that $A^{+}$ has all its eigenvalues located in $\mathbb{S}$. Therefore, all
its invariant subspaces, such as those spanned by the eigenvectors of $A^{+}$,
are actually $\mathbb{S}$-invariant. By freely placing the eigenvalues of
$A^{+}$ in distinct points of $\mathbb{S}\cap\mathbb{R}$ (due to Assumption
3.1), the $n\times n$ matrix $T_{eig}$ whose columns are chosen from the
resulting real eigenvectors of $A^{+}$ (selecting one eigenvector for each
distinct eigenvalue) will be invertible. Thus,
$\begin{bmatrix}I_{p}&O\end{bmatrix}T_{eig}$ has full row rank or,
equivalently, at least one $p\times p$ nonzero minor. Due to this minor
existing, we may always choose those $p$ columns of $T_{eig}$ that correspond
to (one of) the nonzero $p\times p$ minor(s) of
$\begin{bmatrix}I_{p}&O\end{bmatrix}T_{eig}$ to obtain a basis for a
disconjugate $\mathbb{S}$-invariant subspace of dimension $p$ belonging to
$A^{+}$. Therefore, Assumption 3.1 ensures that the CTNARE from (19) can
always be made to have a (real) right stabilizing solution.

##  Appendix C Auxiliary Results

We discuss here a number of technical results, which will be employed in
proving the main theorems presented in this paper. The first of these results
gives necessary and sufficient conditions, under the assumptions made in
Sections 3 and 4, for both of the TFMs that make up an SRTR pair to be stable.

######  Lemma C.1.

The TFM $\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$,
as given in (15), is stable if and only if $A_{22}+KA_{12}$ has all its
eigenvalues located in $\mathbb{S}$.

######  Proof C.2.

The sufficiency of $A_{22}+KA_{12}$ having all its eigenvalues located in
$\mathbb{S}$ is clear, since this implies that the TFM from (15) cannot have
poles located outside of $\mathbb{S}$. For the necessity of this condition, we
begin by showing that (15) is a minimal realization. Notice first that the
pair $(A_{12},A_{22}+KA_{12})$ is observable $\forall\
K\in\mathbb{R}^{(n-p)\times p}$ if and only if the pair $(A_{12},A_{22})$ is
observable. The latter is ensured via Assumption 3.1 and via Remark 3.2.
Moving on, define now the following pencil

(38) $\left[\begin{array}[]{c}S_{1}(\lambda)\\\ \hdashline
S_{2}(\lambda)\end{array}\right]:=\left[\footnotesize\begin{array}[]{ccc}A_{11}-A_{12}K-\lambda
I_{p}&A_{12}&B_{1}\\\ \hdashline
KA_{11}-KA_{12}K+A_{21}-A_{22}K&A_{22}+KA_{12}-\lambda
I_{n-p}&KB_{1}+B_{2}\end{array}\right]=\\\ =\begin{bmatrix}I_{p}&O\\\
K&I_{n-p}\end{bmatrix}\begin{bmatrix}A-\lambda
I_{n}&B\end{bmatrix}\begin{bmatrix}I_{p}&O&O\\\ -K&I_{n-p}&O\\\
O&O&I_{m}\end{bmatrix},\normalsize$

and notice that, due to Assumption 4.1,
$\begin{bmatrix}S_{1}^{\top}(\lambda)&S_{2}^{\top}(\lambda)\end{bmatrix}^{\top}$
has full row rank $\forall\lambda\in\mathbb{C}$ and $\forall
K\in\mathbb{R}^{(n-p)\times p}$. Then, $S_{2}(\lambda)$ must share this
property, which makes the realization from (15) controllable. Since this
realization is also observable, we get that it is minimal and that all the
eigenvalues of $A_{22}+KA_{12}$ are the poles of its TFM. To be stable, these
eigenvalues must be located in $\mathbb{S}$, thus proving the statement’s
necessity.

The following result is a classical state-space derivation, originating from
[18] and generalized in [15] (see also Proposition 1 in [15]). It proves
crucial in establishing the connections between stable SRTR pairs and LCFs
over $\mathbb{S}$, in Sections 4.2 and 4.3.

######  Theorem C.3.

Let $\mathbf{G}(\lambda)\in\mathbb{R}^{p\times m}(\lambda)$ be a strictly
proper TFM. All LCFs over $\mathbb{S}$ of
$\mathbf{G}(\lambda)=\mathbf{M}^{-1}(\lambda)\mathbf{N}(\lambda)$, with
$\begin{bmatrix}\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix}$ having
the same McMillan degree as $\mathbf{G}(\lambda)$, are given by

(39)
$\begin{bmatrix}\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix}\normalsize=\left[\begin{array}[]{c|cc}A+FC-\lambda
I_{n}&F&B\\\ \hline\cr UC&U&O\end{array}\right]\normalsize,$

where $A,B,C,F$ and $U$ are real matrices such that:

  1. $\mathbf{a)}$

$U$ is an invertible matrix in $\mathbb{R}^{p\times p}$;

  2. $\mathbf{b)}$

$F$ is a matrix which ensures that $A+FC$ has all its eigenvalues located in
$\mathbb{S}$;

  3. $\mathbf{c)}$

$\mathbf{G}(\lambda)=\left[\begin{array}[]{c|c}A-\lambda I_{n}&B\\\ \hline\cr
C&O\end{array}\right]$ is both controllable and observable.

A direct consequence of the previous result is the following lemma, which will
be employed in proving the main result of Section 4.2.

######  Lemma C.4.

Let $\mathbf{G}(\lambda)$ be given by a minimal realization of type (4a)-(4b)
and let $A_{x}\in\mathbb{R}^{p\times p}$ have all its eigenvalues located in
$\mathbb{S}$. Let both $B_{x}\in\mathbb{R}^{p\times p}$ and
$C_{x}\in\mathbb{R}^{p\times p}$ be invertible and let
$K\in\mathbb{R}^{(n-p)\times p}$ ensure that $A_{22}+KA_{12}$ has all its
eigenvalues located in $\mathbb{S}$. Then, the system

(40)
$\begin{bmatrix}\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix}=\left[\scriptsize\begin{array}[]{cc|cc}A_{x}-\lambda
I_{n-p}&B_{x}A_{12}&A_{x}B_{x}-B_{x}A_{11}+B_{x}A_{12}K&B_{x}B_{1}\\\
O&A_{22}+KA_{12}-\lambda I_{p}&KA_{12}K+A_{22}K-KA_{11}-A_{21}&KB_{1}+B_{2}\\\
\hline\cr C_{x}&O&C_{x}B_{x}&O\\\ \end{array}\right]$

designates an LCF over $\mathbb{S}$ of
$\mathbf{G}(\lambda)=\mathbf{M}^{-1}(\lambda)\mathbf{N}(\lambda)$.

######  Proof C.5.

Let $\mathbf{G}(\lambda)$ be given by a minimal realization of type (4a)-(4b),
namely

$\mathbf{G}(\lambda)=\left[\begin{array}[]{cc|c}A_{11}-\lambda
I_{p}&A_{12}&B_{1}\\\ A_{21}&A_{22}-\lambda I_{n-p}&B_{2}\\\ \hline\cr
I_{p}&O&O\\\ \end{array}\right].$

We wish to retrieve (40) by employing Theorem C.3. Thus, we first apply a
state equivalence transformation given by $\displaystyle
T:=\footnotesize\begin{bmatrix}B_{x}&O\\\ K&I_{n-p}\end{bmatrix}$ in order to
obtain the minimal realization expressed as

(41)
$\mathbf{G}(\lambda)=\left[\small\begin{array}[]{cc|c}B_{x}(A_{11}-A_{12}K)B_{x}^{-1}-\lambda
I_{p}&B_{x}A_{12}&B_{x}B_{1}\\\
\big{(}KA_{11}-KA_{12}K+A_{21}-A_{22}K\big{)}B_{x}^{-1}&KA_{12}+A_{22}-\lambda
I_{n-p}&KB_{1}+B_{2}\\\ \hline\cr B_{x}^{-1}&O&O\\\
\end{array}\right].\normalsize$

Define now
$F:=\scriptsize\begin{bmatrix}A_{x}B_{x}-B_{x}A_{11}+B_{x}A_{12}K\\\
KA_{12}K+A_{22}K-KA_{11}-A_{21}\end{bmatrix}$ and $U:=C_{x}B_{x}$. Since
$B_{x}$ and $C_{x}$ are invertible, then so is $U$. Moreover, denote by
$A_{\mathbf{G}}$ and $C_{\mathbf{G}}$ the state and, respectively, the output
matrix of the realization from (41). Since $A_{x}$ and $A_{22}+KA_{12}$ have
all their eigenvalues located in $\mathbb{S}$, we have that
$A_{\mathbf{G}}+FC_{\mathbf{G}}=\scriptsize\begin{bmatrix}A_{x}&B_{x}A_{12}\\\
O&A_{22}+KA_{12}\end{bmatrix}$ has all its eigenvalues located in
$\mathbb{S}$. Since (41) is minimal, points $\mathbf{a)}$, $\mathbf{b)}$ and
$\mathbf{c)}$ of Theorem C.3 are satisfied and, thus, (40) designates a left
coprime factorization over $\mathbb{S}$ of $\mathbf{G}(\lambda)$.

The final auxiliary result stated here will be crucial in proving Theorem
3.12.

######  Lemma C.6.

Let $\mathbf{G}_{1}\in\mathbb{R}(\lambda)^{p\times p}$ and
$\mathbf{G}_{2}\in\mathbb{R}(\lambda)^{p\times m}$ such that we can express
the following realization of order $n-p>0$, given explicitly by

(42)
$\begin{bmatrix}\mathbf{G}_{1}(\lambda)&\mathbf{G}_{2}(\lambda)\end{bmatrix}=\left[\scriptsize\begin{array}[]{c|cc}{A}-\lambda
I_{n-p}&{B}_{1}&{B}_{2}\\\ \hline\cr{C}&{D}_{1}&{D}_{2}\end{array}\right].$

Then, we can obtain the following realization of order $n$ for the TFM defined
as

(43)
$\overline{\mathbf{G}}(\lambda):=\big{(}I_{p}-\lambda^{-1}\mathbf{G}_{1}(\lambda)\big{)}^{-1}\big{(}\lambda^{-1}\mathbf{G}_{2}(\lambda)\big{)}=\left[\scriptsize\begin{array}[]{cc|c}{D}_{1}-\lambda
I_{p}&{C}&{D}_{2}\\\ {B}_{1}&{A}-\lambda I_{n-p}&{B}_{2}\\\ \hline\cr
I_{p}&O&O\end{array}\right].$

######  Proof C.7.

By first employing the realization
$\frac{1}{\lambda}I_{p}=\left[\scriptsize\begin{array}[]{c|c}-\lambda
I_{p}&I_{p}\\\ \hline\cr I_{p}&O\end{array}\right]$ alongside the one given in
(42), routine state-space-based computations yield the following identity

(44) $\displaystyle\overline{\mathbf{G}}(\lambda)=$ $\displaystyle\
\big{(}I_{p}-\lambda^{-1}\mathbf{W}(\lambda)\big{)}^{-1}\big{(}\lambda^{-1}\mathbf{V}(\lambda)\big{)}$
(53) $\displaystyle=$
$\displaystyle\left(I_{p}-\left[\footnotesize\begin{array}[]{c|c}-\lambda
I_{p}&I_{p}\\\ \hline\cr
I_{p}&O\end{array}\right]\left[\footnotesize\begin{array}[]{c|c}{A}-\lambda
I_{n-p}&{B}_{1}\\\
\hline\cr{C}&{D}_{1}\end{array}\right]\right)^{-1}\left[\footnotesize\begin{array}[]{c|c}-\lambda
I_{p}&I_{p}\\\ \hline\cr
I_{p}&O\end{array}\right]\left[\footnotesize\begin{array}[]{c|c}{A}-\lambda
I_{n-p}&{B}_{2}\\\ \hline\cr{C}&{D}_{2}\end{array}\right]$ (60)
$\displaystyle=$
$\displaystyle\footnotesize\left[\begin{array}[]{cc|c}-\lambda
I_{p}&{C}&{D}_{1}\\\ O&{A}-\lambda I_{n-p}&{B}_{1}\\\ \hline\cr-
I_{p}&O&I_{p}\end{array}\right]^{-1}\footnotesize\left[\begin{array}[]{cc|c}-\lambda
I_{p}&{C}&{D}_{1}\\\ O&{A}-\lambda I_{n-p}&{B}_{1}\\\ \hline\cr
I_{p}&O&O\end{array}\right]$ (66) $\displaystyle=$
$\displaystyle\scriptsize\left[\begin{array}[]{cccc|c}{D}_{1}-\lambda
I_{p}&{C}&{D}_{1}&O&O\\\ {B}_{1}&{A}-\lambda I_{n-p}&{B}_{1}&O&O\\\
O&O&-\lambda I_{p}&{C}&{D}_{2}\\\ O&O&O&{A}-\lambda I_{n-p}&{B}_{2}\\\
\hline\cr I_{p}&O&I_{p}&O&O\end{array}\right].$

Applying now a state equivalence transformation
$T_{o}:=\tiny\begin{bmatrix}I_{p}&O&I_{p}&O\\\ O&I_{n-p}&O&I_{n-p}\\\
O&O&I_{p}&O\\\ O&O&O&I_{n-p}\end{bmatrix}$ to the state-space realization of
order $2n$ from (44), we obtain the fact that

(72)
$\displaystyle\overline{\mathbf{G}}(\lambda)=\left[\scriptsize\begin{array}[]{cccc|c}{D}_{1}-\lambda
I_{p}&{C}&O&O&{D}_{2}\\\ {B}_{1}&{A}-\lambda I_{n-p}&O&O&{B}_{2}\\\
O&O&-\lambda I_{p}&{C}&{D}_{2}\\\ O&O&O&{A}-\lambda I_{n-p}&{B}_{2}\\\
\hline\cr I_{p}&O&O&O&O\end{array}\right].$

Finally, after eliminating the last $n$ unobservable components from the state
vector of the realization from (72), we retrieve the realization of
$\overline{\mathbf{G}}(\lambda)$ given in (43).

##  Appendix D Proofs of the main results

Proof of Theorem 3.12 We begin by stating that, throughout this entire proof,
the TFM of the system given by (4a)-(4b) will be denoted by
$\mathbf{G}(\lambda)$, just as in Definition 3.8. In order to prove point
$\mathbf{a)}$, we begin by first defining the following matrix
$A_{K}:=KA_{11}-KA_{12}K+A_{21}-A_{22}K$ and by employing Lemma C.6 for the
$\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ pair along with its
associated realization given in (15), to get that

(73)
$\overline{\mathbf{G}}(\lambda):=\big{(}I_{p}-\lambda^{-1}\mathbf{W}(\lambda)\big{)}^{-1}\big{(}\lambda^{-1}\mathbf{V}(\lambda)\big{)}=\left[\tiny\begin{array}[]{cc|c}A_{11}-A_{12}K-\lambda
I_{p}&A_{12}&B_{1}\\\ A_{K}&A_{22}+KA_{12}-\lambda I_{n-p}&KB_{1}+B_{2}\\\
\hline\cr I_{p}&O&O\end{array}\right],$

and by applying a state equivalence transformation given by
$\widetilde{T}:=\scriptsize\begin{bmatrix}I_{p}&O\\\ -K&I_{n-p}\end{bmatrix}$
to the realization in (73), we obtain precisely the realization from
(4a)-(4b). Therefore

(74)
$\mathbf{G}(\lambda)=\overline{\mathbf{G}}(\lambda)=\big{(}I_{p}-\lambda^{-1}\mathbf{W}(\lambda)\big{)}^{-1}\big{(}\lambda^{-1}\mathbf{V}(\lambda)\big{)}=\big{(}\lambda
I_{p}-\mathbf{W}(\lambda)\big{)}^{-1}\mathbf{V}(\lambda).$

In addition to the equality from (74), notice that $\mathbf{W}(\lambda)$ and
$\mathbf{V}(\lambda)$ are expressed via state-space realizations, which means
that they are guaranteed to be proper TFMs. Moreover, from the expression of
$\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ given in
(15), we get that the latter TFM’s McMillan degree is at most $n-p$. thus, by
Definition 3.8, any
$\displaystyle\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ obtained
via (15) is indeed an SRTR pair of the system from the result’s statement.

To prove point $\mathbf{b)}$, we first show that minimality is necessary for
(15) to produce, given the plant’s realization, every SRTR pair associated
with the latter realization and having a McMillan degree of at most $n-p$. Let
us assume that this is not the case and consider the system described by the
realization of type (4a)-(4b)

$\mathbf{G}(\lambda)=\left[\scriptsize\begin{array}[]{cc|c}-1-\lambda&1&1\\\
0&-1-\lambda&0\\\ \hline\cr 1&0&0\end{array}\right],$

which is observable (as per Assumption 3.2), but not controllable, and for
which (15) produces the class of SRTR pairs

(77)
$\displaystyle\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}=$
$\displaystyle\left[\footnotesize\begin{array}[]{c|cc}k-1&-k^{2}&k\\\
\hline\cr
1&-1-k&1\end{array}\right]=\begin{bmatrix}-\frac{(k+1)\lambda+1}{\lambda+1-k}&\
\frac{\lambda+1}{\lambda+1-k}\end{bmatrix},\ \forall\ k\in\mathbb{R}.$

Note also that
$\begin{bmatrix}\overline{\mathbf{W}}(\lambda)&\overline{\mathbf{V}}(\lambda)\end{bmatrix}=\begin{bmatrix}-2&\frac{\lambda+2}{\lambda+1}\end{bmatrix}$
forms an SRTR pair of the same system having a McMillan degree equal to $1$
and that this representation cannot be obtained for any real $k$ in (77). This
contradiction shows, therefore, that minimality of the realization from
(4a)-(4b) is necessary for (15) to characterize the desired class.

To show the sufficiency of starting from a system given by minimal realization
(4a)-(4b) of order $n$, having $\mathbf{G}(\lambda)$ as its TFM, we will first
consider an arbitrary SRTR pair
$\displaystyle\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$
associated with the aforementioned system. Thus, by Definition 3.8, we can
always obtain a realization of order $n-p$ for the following TFM

(78)
$\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}=\left[\scriptsize\begin{array}[]{c|cc}\overline{A}-\lambda
I_{n-p}&\overline{B}_{1}&\overline{B}_{2}\\\ \hline\cr\vspace{-2mm}\hfil\\\
\overline{C}&\overline{D}_{1}&\overline{D}_{2}\end{array}\right],$

which we will employ, in order to obtain a minimal realization of
$\mathbf{G}(\lambda)$. Since (13) holds, then
$\mathbf{G}(\lambda)=\big{(}I_{p}-\lambda^{-1}\mathbf{W}(\lambda)\big{)}^{-1}\big{(}\lambda^{-1}\mathbf{V}(\lambda)\big{)}$
and, by employing Lemma C.6 for the
$\big{(}\mathbf{W}(\lambda),\mathbf{V}(\lambda)\big{)}$ pair along with its
associated realization from (78), we get that

(79)
$\mathbf{G}(\lambda)=\left[\scriptsize\begin{array}[]{cc|c}\overline{D}_{1}-\lambda
I_{p}&\overline{C}&\overline{D}_{2}\\\ \overline{B}_{1}&\overline{A}-\lambda
I_{n-p}&\overline{B}_{2}\\\ \hline\cr I_{p}&O&O\end{array}\right].$

Notice that the obtained realization from (79) has the same order as the
McMillan degree of $\mathbf{G}(\lambda)$, making it minimal. We proceed to
exploit the minimality of this realization and its particular structure, in
order to show that (78) is related to (15) by a state equivalence
transformation, for some $K\in\mathbb{R}^{(n-p)\times p}$.

Consider all matrices $\overline{T}:=\scriptsize\begin{bmatrix}I_{p}&O\\\
LK&L\end{bmatrix}$, where $K$ is arbitrary in $\mathbb{R}^{(n-p)\times p}$ and
$L$ is any invertible matrix in $\mathbb{R}^{(n-p)\times(n-p)}$. Then,
$\overline{T}^{-1}=\scriptsize\begin{bmatrix}I_{p}&O\\\
-K&L^{-1}\end{bmatrix}$ represents the class of all invertible matrices in
$\mathbb{R}^{n\times n}$ having the first block-row equal to
$\begin{bmatrix}I_{p}&O\end{bmatrix}$. Thus, we get that
$\begin{bmatrix}I_{p}&O\end{bmatrix}=\begin{bmatrix}I_{p}&O\end{bmatrix}\overline{T}^{-1}$
if and only if $\overline{T}^{-1}$ has the indicated structure. Starting the
minimal realization of type (4a)-(4b) from the result’s statement and by
applying the state-equivalence transformation given by $\overline{T}$, we
obtain the class of all minimal realizations of $\mathbf{G}(\lambda)$ having
an output matrix equal to $\begin{bmatrix}I_{p}&O\end{bmatrix}$, given by

(80)
$\mathbf{G}(\lambda)=\left[\scriptsize\begin{array}[]{cc|c}A_{11}-A_{12}K-\lambda
I_{p}&A_{12}L^{-1}&B_{1}\\\
L(KA_{11}-KA_{12}K+A_{21}-A_{22}K)&L(A_{22}+KA_{12})L^{-1}-\lambda
I_{n-p}&L(KB_{1}+B_{2})\\\ \hline\cr I_{p}&O&O\end{array}\right].$

Thus, there must exist a $K$ and an invertible $L$ such that the rightmost
term from (79) coincides with (80), which makes (78) coincide with the
following realization

(81)
$\scriptsize\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}=\left[\scriptsize\begin{array}[]{c|cc}L(A_{22}+KA_{12})L^{-1}-\lambda
I_{n-p}&L(KA_{11}-KA_{12}K+A_{21}-A_{22}K)&L(KB_{1}+B_{2})\\\ \hline\cr
A_{12}L^{-1}&A_{11}-A_{12}K&B_{1}\\\ \end{array}\right]\normalsize.$

Apply now a state equivalence transformation $T_{L}:=L^{-1}$ to (81) in order
to conclude that, when starting from a minimal realization of type (4a)-(4b)
of $\mathbf{G}(\lambda)$, there must exist a $K\in\mathbb{R}^{(n-p)\times p}$
which allows
$\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ to be
expressed via (15).

Proof of Theorem 4.3 In order to prove our statement, we first require a
Rosenbrock-type realization of $\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$. Obtain first a
state-space realization of
$\begin{bmatrix}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ by
using (15) and multiplying its input and feedthrough matrices to the right
with $\footnotesize\begin{bmatrix}-I_{p}&O\\\ O&I_{m}\end{bmatrix}$. Write now
that $\begin{bmatrix}\lambda
I_{p}&O\end{bmatrix}=\left[\footnotesize\begin{array}[]{c|cc}I_{p}&-\lambda
I_{p}&O\\\ \hline\cr I_{p}&O&O\end{array}\right]$ and, by expressing the
parallel connection formula between this Rosenbrock-type representation and
the computed realization of
$\begin{bmatrix}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$, use it
to form the sought-after Rosenbrock-type realization of
$\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$, given explicitly
by

(82) $\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}=\left[\footnotesize\begin{array}[]{c|c}\widetilde{A}-\lambda\widetilde{E}&\widetilde{B}-\lambda\widetilde{F}\\\
\hline\cr\widetilde{C}&\widetilde{D}\end{array}\right]:=\\\
:=\left[\footnotesize\begin{array}[]{cc|cc}A_{22}+KA_{12}-\lambda
I_{n-p}&O&KA_{12}K+A_{22}K-KA_{11}-A_{21}&KB_{1}+B_{2}\\\ O&I_{p}&-\lambda
I_{p}&O\\\ \hline\cr A_{12}&I_{p}&-A_{11}+A_{12}K&B_{1}\\\
\end{array}\right].\normalsize$

If the realization from (82) is irreducible then, by Theorem A.5, the finite
zeros of $\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ are the finite
zeros of the realization’s system matrix, namely

(83) $\mathcal{S}(\lambda):=\scriptsize\begin{bmatrix}A_{22}+KA_{12}-\lambda
I_{n-p}&O&KA_{12}K+A_{22}K-KA_{11}-A_{21}&KB_{1}+B_{2}\\\ O&I_{p}&-\lambda
I_{p}&O\\\ A_{12}&I_{p}&-A_{11}+A_{12}K&B_{1}\\\
\end{bmatrix}.\normalsize\vspace{-1mm}$

Additionally, the infinite zeros of $\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ can be deduced by
using an irreducible Rosenbrock-type realization of
$\begin{bmatrix}\frac{1}{\lambda}I_{p}-\mathbf{W}\left(\frac{1}{\lambda}\right)&\mathbf{V}\left(\frac{1}{\lambda}\right)\end{bmatrix}$
and by computing the latter TFM’s zeros at $\lambda=0$.

Thus, we divide the proof into two distinct parts. In part I), we show that
the realization from (82) is indeed irreducible and that
$\mathcal{S}(\lambda)$ from (83) has full row rank
$\forall\lambda\in\mathbb{C}$. In part II), we compute an irreducible
Rosenbrock-type realization for
$\begin{bmatrix}\frac{1}{\lambda}I_{p}-\mathbf{W}\left(\frac{1}{\lambda}\right)&\mathbf{V}\left(\frac{1}{\lambda}\right)\end{bmatrix}$
and show that its own system matrix has full row rank at $\lambda=0$.
Therefore, $\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ must have no
finite or infinite zeros, and it must also have full row normal rank, due to
Proposition A.3.

I) Due to Assumption 3.1 and Remark 3.2, notice that

$\footnotesize\begin{bmatrix}\widetilde{A}-\lambda\widetilde{E}\\\
\widetilde{C}\end{bmatrix}=\footnotesize\begin{bmatrix}A_{22}+KA_{12}-\lambda
I_{n-p}&O\\\ O&I_{p}\\\
A_{12}&I_{p}\end{bmatrix}\normalsize=\footnotesize\begin{bmatrix}I_{n-p}&-K&K\\\
O&I_{p}&O\\\ O&O&I_{p}\end{bmatrix}\footnotesize\begin{bmatrix}A_{22}-\lambda
I_{n-p}&O\\\ O&I_{p}\\\ A_{12}&I_{p}\end{bmatrix}\vspace{-1mm}$

has full column rank $\forall\lambda\in\mathbb{C}$. Moreover, due to
Assumption 4.1, note also that the pencil from the right-hand term of the
following equality

$\begin{array}[]{c}\footnotesize\begin{bmatrix}A_{22}+KA_{12}-\lambda
I_{n-p}&O&KA_{12}K+A_{22}K-KA_{11}-A_{21}&KB_{1}+B_{2}\\\ O&I_{p}&-\lambda
I_{p}&O\end{bmatrix}\normalsize=\\\ \footnotesize\begin{bmatrix}K&I_{n-p}\\\
-I_{p}&O\end{bmatrix}\small\begin{bmatrix}\lambda
I_{p}-A_{11}&-A_{12}&B_{1}&-I_{p}\\\ -A_{21}&\lambda I_{n-p}-A_{22}&B_{2}&K\\\
\end{bmatrix}\scriptsize\begin{bmatrix}O&O&I_{p}&O\\\ -I_{n-p}&O&-K&O\\\
O&O&O&I_{m}\\\ A_{12}&I_{p}&A_{12}K-A_{11}&B_{1}\\\ \end{bmatrix}\end{array}$

has full row rank $\forall\lambda\in\mathbb{C}$, which means that so does
$\small\begin{bmatrix}\widetilde{A}-\lambda\widetilde{E}&\widetilde{B}-\lambda\widetilde{F}\end{bmatrix}$.

By Definition A.4, the realization from (82) is irreducible and we now inspect
its finite zeros. It follows from the following identity

(84) $\begin{array}[]{c}\footnotesize\begin{bmatrix}A_{22}+KA_{12}-\lambda
I_{n-p}&O&KA_{12}K+A_{22}K-KA_{11}-A_{21}&KB_{1}+B_{2}\\\ O&I_{p}&-\lambda
I_{p}&O\\\ A_{12}&I_{p}&-A_{11}+A_{12}K&B_{1}\\\ \end{bmatrix}\normalsize=\\\
\footnotesize\begin{bmatrix}K&I_{n-p}&O\\\ O&O&I_{p}\\\
I_{p}&O&I_{p}\end{bmatrix}\footnotesize\begin{bmatrix}\lambda
I_{p}-A_{11}&-A_{12}&B_{1}&O\\\ -A_{21}&\lambda I_{n-p}-A_{22}&B_{2}&O\\\
-\lambda
I_{p}&O&O&I_{p}\end{bmatrix}\footnotesize\begin{bmatrix}O&O&I_{p}&O\\\
-I_{n-p}&O&-K&O\\\ O&O&O&I_{m}\\\
O&I_{p}&O&O\end{bmatrix}\normalsize\end{array}$

that $\mathcal{S}(\lambda)$ has full row rank $\forall\lambda\in\mathbb{C}$
since so does the pencil from the right-hand term in (84), once again due to
Assumption 4.1. We conclude, by Proposition A.3, that $\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ has full row
normal rank and, by Theorem A.5, no finite zeros.

II) Recall the Rosenbrock-type realization from (82) to get

(85)
${\begin{bmatrix}\frac{1}{\lambda}I_{p}-\mathbf{W}\left(\frac{1}{\lambda}\right)&\mathbf{V}\left(\frac{1}{\lambda}\right)\end{bmatrix}}=\widetilde{C}(\lambda\widetilde{A}-\widetilde{E})^{-1}(\widetilde{F}-\lambda\widetilde{B})+\widetilde{D}.$

Remark that, since the realization given by (82) is irreducible, then we have
that
$\begin{bmatrix}\widetilde{E}-\lambda\widetilde{A}&\widetilde{F}-\lambda\widetilde{B}\end{bmatrix}$
and
$\begin{bmatrix}\widetilde{E}^{\top}-\lambda\widetilde{A}^{\top}&\widetilde{C}^{\top}\end{bmatrix}^{\top}$
have full row and, respectively, column rank
$\forall\lambda\in\mathbb{C}\backslash\\{0\\}$. It is straightforward to check
that
$\begin{bmatrix}\widetilde{E}&\widetilde{F}\end{bmatrix}=\footnotesize\begin{bmatrix}I_{n-p}&O&O&O\\\
O&O&I_{p}&O\end{bmatrix}$ has full row rank, while
$\begin{bmatrix}\widetilde{E}^{\top}&\widetilde{C}^{\top}\end{bmatrix}^{\top}=\footnotesize\begin{bmatrix}I_{n-p}&O&A_{12}^{\top}\\\
O&O&I_{p}\end{bmatrix}^{\top}$ has full column rank. Therefore, we conclude
that (85) is an irreducible Rosebrock-type realization and we proceed to
define the associated system matrix
$\widetilde{\mathcal{S}}(\lambda):=\small\begin{bmatrix}\widetilde{E}-\lambda\widetilde{A}&\widetilde{F}-\lambda\widetilde{B}\\\
\widetilde{C}&\widetilde{D}\end{bmatrix}$ in order to investigate the latter’s
zeros at $\lambda=0$. Notice that
$\normalsize\widetilde{\mathcal{S}}(0)=\scriptsize\begin{bmatrix}I_{n-p}&O&O&O\\\
O&O&I_{p}&O\\\ A_{12}&I_{p}&-A_{11}+A_{12}K&B_{1}\end{bmatrix}$ has full row
rank and, thus, it follows that (85) has no zeros at $\lambda=0$, which
implies that $\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ does not have any
zeros at infinity.

By the conclusions drawn in parts I) and II) of the proof, we get the fact
that $\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ has full row
normal rank along with no zeros, be they finite or infinite. Then, it follows
that this compound TFM designates an FLCF of $\mathbf{G}(\lambda)$.

Proof of Theorem 4.6 Point $\mathbf{a})$ boils down to the displacement of
poles at infinity, via the procedure from Theorem 3.1 of [20]. We will apply
this result using only state-space-based arguments, to avoid using the
_pencil_ realizations from [20].

Begin by recalling Theorem 3.12 to conclude that the given SRTR pair must be
expressed through a realization of type (15) for some $K$ and a minimal
realization of type (4a)-(4b), mentioned in the result’s statement. Thus, we
can write that

(86) $\normalsize\mathbf{\Theta}(\lambda)\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}=\begin{bmatrix}\lambda\mathbf{\Theta}(\lambda)&O\end{bmatrix}+\mathbf{\Theta}(\lambda)\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}\footnotesize\begin{bmatrix}-I_{p}&O\\\
O&I_{m}\end{bmatrix}\normalsize,$

where we have that

(87c)
$\displaystyle\begin{bmatrix}\lambda\mathbf{\Theta}(\lambda)&O\end{bmatrix}=$
$\displaystyle\begin{bmatrix}C_{x}(\lambda I_{p}-A_{x})^{-1}(\lambda
I_{p}-A_{x}+A_{x})B_{x}&O\end{bmatrix}=\left[\footnotesize\begin{array}[]{c|cc}A_{x}-\lambda
I_{p}&A_{x}B_{x}&O\\\ \hline\cr C_{x}&C_{x}B_{x}&O\end{array}\right]$ (87g)
$\displaystyle=$
$\displaystyle\left[\footnotesize\begin{array}[]{cc|cc}A_{x}-\lambda
I_{p}&B_{x}A_{12}&A_{x}B_{x}&O\\\ O&A_{22}+KA_{12}-\lambda I_{n-p}&O&O\\\
\hline\cr C_{x}&O&C_{x}B_{x}&O\end{array}\right].$

The identity given in (87g) is obtained by introducing $n-p$ uncontrollable
modes in the realization from (87c). We denote
$A_{K}:=KA_{11}-KA_{12}K+A_{21}-A_{22}K$ and, using (15) and (16), we write

(88)
$\mathbf{\Theta}(\lambda)\footnotesize\begin{bmatrix}\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}\begin{bmatrix}-I_{p}&O\\\
O&I_{m}\end{bmatrix}=\left[\footnotesize\begin{array}[]{cc|cc}A_{x}-\lambda
I_{p}&B_{x}A_{12}&B_{x}(A_{12}K-A_{11})&B_{x}B_{1}\\\ O&A_{22}+KA_{12}-\lambda
I_{n-p}&-A_{K}&KB_{1}+B_{2}\\\ \hline\cr
C_{x}&O&O&O\end{array}\right].\normalsize$

Notice now that the state-space realizations from (87g) and (88) have the same
state and output matrices and, therefore, the parallel connection from (86)
can be computed by adding together their input and feedthrough matrices, from
which we directly obtain (40). Since the submatrices from the employed
realization of the SRTR pair come from a minimal realization of type (4a)-(4b)
for $\mathbf{G}(\lambda)$, we apply Lemma C.4 to get that the system from (86)
describes an LCF over $\mathbb{S}$ of $\mathbf{G}(\lambda)$.

To prove point $\mathbf{b})$, begin by recalling the proof of Lemma C.1, in
order to state that the realization from (15) is minimal, with all the
eigenvalues of $A_{22}+KA_{12}$ being poles of the TFM. Notice also that, due
to the Assumption 3.1, Remark 3.2 and the invertibility of $B_{x}$ and
$C_{x}$, the realization given in (40) is observable. Additionally, recall the
matrix pencil from (38), which has full row rank
$\forall\,\lambda\in\mathbb{C}$. It follows that

$\begin{bmatrix}B_{x}&O\\\
O&I_{p}\end{bmatrix}\begin{bmatrix}B_{x}^{-1}(A_{x}-\lambda
I_{p})&S_{1}(\lambda)\\\
O&S_{2}(\lambda)\end{bmatrix}\begin{bmatrix}I_{p}&O&B_{x}&O\\\ O&O&-I_{p}&O\\\
O&I_{n-p}&O&O\\\ O&O&O&I_{m}\end{bmatrix}$

has full row rank $\forall\,\lambda\in\mathbb{C}$, since $B_{x}$ is
invertible, which makes (40) controllable. Since it is also observable, we get
that the realization from (40) is minimal, with all the eigenvalues of its
pole pencil being poles of its associated TFM. Therefore, due to the
minimality of both realizations from (15) and (40) and due to the shared
eigenvalues of $A_{22}+KA_{12}$, the poles of (15) are preserved among those
of (40).

Proof of Theorem 4.10 To prove point $\mathbf{a)}$, we first obtain from the
definition of $A_{x}$ in the Theorem’s statement and from the fact that $K$ is
a solution of (19) the following identity $\begin{bmatrix}F_{1}\\\
F_{2}\end{bmatrix}{=}\begin{bmatrix}A_{x}-A_{11}+A_{12}K\\\
A_{22}K-KA_{x}-A_{21}\end{bmatrix}$. We then plug this equality into the
expression of (18) and we apply a state equivalence transformation which is
given by $T:=\begin{bmatrix}I_{p}&O\\\ K&I_{n-p}\end{bmatrix}$, in order to
obtain the realization from (20).

To prove point $\mathbf{b)}$, we will simply reverse the procedure used to
prove point $\mathbf{a)}$ in Theorem 4.6. This is done by choosing
$B_{x}=I_{p}$, $C_{x}=U$ and $A_{x}$ as in the Theorem’s statement, and then
left-multiplying (20) with the inverse of (16), as in (21). By doing so, we
get that

(90) $\displaystyle\footnotesize\begin{array}[]{r}(\lambda
I_{p}-A_{x})U^{-1}\begin{bmatrix}-\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix}\end{array}=$
$\displaystyle\ (\lambda
I_{p}-A_{x})\begin{bmatrix}I_{p}&O\end{bmatrix}\times$
$\displaystyle\times\footnotesize\begin{bmatrix}(\lambda
I_{p}-A_{x})^{-1}&(\lambda I_{p}-A_{x})^{-1}A_{12}(\lambda
I_{n-p}-A_{22}-KA_{12})^{-1}\\\ O&(\lambda
I_{n-p}-A_{22}-KA_{12})^{-1}\end{bmatrix}\times$ (91)
$\displaystyle\times\footnotesize\begin{bmatrix}A_{11}-A_{12}K+A_{x}&B_{1}\\\
KA_{11}-KA_{12}K+A_{21}-A_{22}K&KB_{1}+B_{2}\end{bmatrix}+\begin{bmatrix}A_{x}-\lambda
I_{p}&O\end{bmatrix}.$

Denoting $A_{K}:=KA_{11}-KA_{12}K+A_{21}-A_{22}K$, the TFM in (91) is equal to

(93) $\displaystyle\footnotesize\begin{array}[]{r}(\lambda
I_{p}-A_{x})U^{-1}\end{array}\footnotesize\begin{bmatrix}-\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix}=$
$\displaystyle\footnotesize\begin{bmatrix}A_{x}-\lambda
I_{p}&O\end{bmatrix}+\footnotesize\begin{bmatrix}I_{p}&A_{12}(\lambda
I_{n-p}-A_{22}-KA_{12})^{-1}\end{bmatrix}\times$
$\displaystyle\qquad\qquad\quad\,\times\footnotesize\begin{bmatrix}A_{11}-A_{12}K-A_{x}&B_{1}\\\
A_{K}&KB_{1}+B_{2}\end{bmatrix}$ (96) $\displaystyle=$
$\displaystyle\footnotesize\begin{bmatrix}-\lambda
I_{p}&O\end{bmatrix}+\left[\footnotesize\begin{array}[]{c|cc}A_{22}+KA_{12}-\lambda
I_{n-p}&A_{K}&KB_{1}+B_{2}\\\ \hline\cr
A_{12}&A_{11}-A_{12}K&B_{1}\end{array}\right].$

Recall Definition 3.8 and Theorem 3.12 to notice that the system from the
right-hand term in the final equality of (96), which is in parallel connection
to $\begin{bmatrix}-\lambda I_{p}&O\end{bmatrix}$, has a realization of type
(15). Therefore, its TFM designates an SRTR pair of the system and we proceed
to rewrite (96) as follows

$(\lambda
I_{p}-A_{x})U^{-1}\begin{bmatrix}-\mathbf{M}(\lambda)&\mathbf{N}(\lambda)\end{bmatrix}=\begin{bmatrix}\mathbf{W}(\lambda)-\lambda
I_{p}&\mathbf{V}(\lambda)\end{bmatrix},$

from which we retrieve (21), along with the stable SRTR pair.

Point $\mathbf{c)}$ follows directly by inspecting the realization of the TFM
which designates the SRTR pair obtained in (96) and by recalling that $K$ is
the solution of (19).

$\begin{bmatrix}\frac{1}{\lambda}\mathbf{W}^{\top}(\lambda)\\\
\frac{1}{\lambda}\mathbf{V}^{\top}(\lambda)\end{bmatrix}^{\top}$$\zeta$$w$$\begin{bmatrix}I_{m}\\\
\mathbf{G}(\lambda)\end{bmatrix}$–$v$$y-\zeta$$u$$z-r$$\delta_{u}$$r$ Figure
5: Equivalent feedback interconnection

Proof of Theorem 4.15 The proof boils down to showing that
$\mathbf{K}_{d}(\lambda)$ is a stabilizing controller for
$\begin{bmatrix}I_{p}&\mathbf{G}^{\top}(\lambda)\end{bmatrix}^{\top}$,
considering a standard feedback loop as in Fig. 3. This represents a
sufficient condition for the closed-loop interconnection shown in Fig. 5,
located at the top of this page, to be internally stable. The desired result
follows by noticing that Fig. 5 is equivalent to Fig. 4. In order to show that
$\mathbf{K}_{d}(\lambda)$ is a stabilizing controller for
$\begin{bmatrix}I_{p}&\mathbf{G}^{\top}(\lambda)\end{bmatrix}^{\top}$, we
point out (see Section 5.3 of [40]) that an equivalent condition for this is
the fact that all the entries of $\mathbf{H}(\lambda)$ are stable, where

(97) $\mathbf{H}(\lambda):=\small\begin{bmatrix}I_{p}\\\ I_{p}\\\
\mathbf{G}(\lambda)\end{bmatrix}\left(I_{p}-\begin{bmatrix}\frac{1}{\lambda}\mathbf{W}(\lambda)&\frac{1}{\lambda}\mathbf{V}(\lambda)\end{bmatrix}\begin{bmatrix}I_{p}\\\
\mathbf{G}(\lambda)\end{bmatrix}\right)^{-1}\begin{bmatrix}I_{p}&\frac{1}{\lambda}\mathbf{W}(\lambda)&\frac{1}{\lambda}\mathbf{V}(\lambda)\end{bmatrix}.\normalsize$

Express now any LCF over $\mathbb{S}$ of
$\mathbf{G}^{\top}(\lambda)=\mathbf{M}_{\mathbf{G}}^{-1}(\lambda)\mathbf{N}_{\mathbf{G}}(\lambda)$
to get that
$\mathbf{G}(\lambda)=\mathbf{N}^{\top}_{\mathbf{G}}(\lambda)\mathbf{M}_{\mathbf{G}}^{-\top}(\lambda)$,
which designates a _right coprime factorization (RCF) over $\mathbb{S}$_ of
$\mathbf{G}(\lambda)$. Then, select any stable $\mathbf{\Theta}(\lambda)$ as
in Theorem 4.6 and apply this result, to get that

(98)
$\begin{bmatrix}\mathbf{M}_{\mathbf{K}}(\lambda)&\mathbf{N}_{\mathbf{K}}(\lambda)\end{bmatrix}:=\mathbf{\Theta}(\lambda)\begin{bmatrix}\lambda
I_{p}-\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$

designates an LCF over $\mathbb{S}$ of the stabilizing controller
$\mathbf{K}(\lambda)=\mathbf{M}_{\mathbf{K}}^{-1}(\lambda)\mathbf{N}_{\mathbf{K}}(\lambda)$.

Since $\mathbf{K}(\lambda)$ is a stabilizing controller for
$\mathbf{G}(\lambda)$, we have by point $5$ of Lemma 5.10 in [40] that
$(\mathbf{M}_{\mathbf{K}}(\lambda)\mathbf{M}^{\top}_{\mathbf{G}}(\lambda)-\mathbf{N}_{\mathbf{K}}(\lambda)\mathbf{N}^{\top}_{\mathbf{G}}(\lambda))^{-1}$
must be stable. Finally, by (98), we have the fact that
$\mathbf{\Theta}(\lambda)=\mathbf{M}_{\mathbf{K}}(\lambda)(\lambda
I_{p}-\mathbf{W}(\lambda))^{-1}$ and, by noticing that
$\mathbf{M}_{\mathbf{K}}(\lambda)$, $\mathbf{W}(\lambda)$ along with
$\mathbf{\Theta}(\lambda)$ are all stable TFMs, we get that
$\lambda\mathbf{\Theta}(\lambda)=\mathbf{M}_{\mathbf{K}}(\lambda)+\mathbf{\Theta}(\lambda)\mathbf{W}(\lambda)$
is also a stable TFM. With this fact established, return to (97) and rewrite
it as

(99) $\displaystyle\mathbf{H}(\lambda)=$
$\displaystyle\footnotesize\begin{bmatrix}I_{p}\\\ I_{p}\\\
\mathbf{G}(\lambda)\end{bmatrix}\left(I_{p}-\left(\lambda
I_{p}-\mathbf{W}(\lambda)\right)^{-1}\mathbf{V}(\lambda)\mathbf{G}(\lambda)\right)^{-1}\times$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\left(\lambda
I_{p}-\mathbf{W}(\lambda)\right)^{-1}\begin{bmatrix}\lambda
I_{p}&\mathbf{W}(\lambda)&\mathbf{V}(\lambda)\end{bmatrix}$ $\displaystyle=$
$\displaystyle\footnotesize\begin{bmatrix}\mathbf{M}_{\mathbf{G}}^{\top}(\lambda)\\\
\mathbf{M}_{\mathbf{G}}^{\top}(\lambda)\\\
\mathbf{N}_{\mathbf{G}}^{\top}(\lambda)\end{bmatrix}\left(\mathbf{M}_{\mathbf{K}}(\lambda)\mathbf{M}_{\mathbf{G}}^{\top}(\lambda)-\mathbf{N}_{\mathbf{K}}(\lambda)\mathbf{N}^{\top}_{\mathbf{G}}(\lambda)\right)^{-1}\begin{bmatrix}\lambda\mathbf{\Theta}(\lambda)&\mathbf{\Theta}(\lambda)\mathbf{W}(\lambda)&\mathbf{N}_{\mathbf{K}}(\lambda)\end{bmatrix}.$

Notice that all the (block-)entries of the TFMs from the final equality in
(99) are products of stable TFMs, which makes all of them stable. Thus, the
closed-loop interconnections showcased in Figs. 4 and 5 are internally stable.
This confirms that the special implementation given in (22a)-(22b) is indeed
stabilizing, in both the continuous- and the discrete-time case.

Proof of Proposition 4.19 We will prove our statement by showing that the TFM
given in (22a) is guaranteed to have $p$ poles in $s=0$ for any stable SRTR
pair obtained via a minimal realization (4a)-(4b), making it continuous-time
unstable. Begin by considering the minimal state-space realization
$\frac{1}{s}I_{p}=\left[\footnotesize\begin{array}[]{c|c}-sI_{p}&I_{p}\\\
\hline\cr I_{p}&O\end{array}\right]$ and let the stable SRTR pair
$(\mathbf{W}(s),\mathbf{V}(s))$ be as in (15). Then, a realization of
$\mathbf{K}_{d}(s)$ is

(100)
$\mathbf{K}_{d}(s)=\left[\footnotesize\begin{array}[]{c|c}A_{d}-sI_{n}&B_{d}\\\
\hline\cr C_{d}&D_{d}\end{array}\right]:=\\\
:=\left[\scriptsize\begin{array}[]{cc|cc}-sI_{p}&A_{12}&A_{11}-A_{12}K&B_{1}\\\
O&A_{22}+KA_{12}-sI_{n-p}&KA_{11}-KA_{12}K+A_{21}-A_{22}K&KB_{1}+B_{2}\\\
\hline\cr I_{p}&O&O&O\end{array}\right].\normalsize$

Notice now that, due to Assumption 3.1 and due to Remark 3.2, the matrix
pencil

$\begin{bmatrix}A_{d}-sI_{n}\\\
C_{d}\end{bmatrix}=\footnotesize\begin{bmatrix}I_{p}&O&O\\\ K&I_{n-p}&O\\\
O&O&I_{p}\end{bmatrix}\footnotesize\begin{bmatrix}-sI_{p}&A_{12}\\\
sK&A_{22}-sI_{n-p}\\\ I_{p}&O\end{bmatrix}\normalsize$

has full column rank $\forall s\in\mathbb{C}$, making (100) observable.
Finally, consider the identity

(101)
$\small\begin{bmatrix}A_{d}-sI_{n}&B_{d}\end{bmatrix}=\footnotesize\begin{bmatrix}I_{p}&O\\\
K&I_{n-p}\end{bmatrix}\footnotesize\begin{bmatrix}-sI_{p}&A_{11}-sI_{p}&A_{12}&B_{1}\\\
sK&A_{21}&A_{22}-sI_{n-p}&B_{2}\end{bmatrix}\tiny\begin{bmatrix}I_{p}&O&-I_{p}&O\\\
O&O&\phantom{-}I_{p}&O\\\ O&I_{n-p}&-K&O\\\
O&O&\phantom{-}O&I_{m}\end{bmatrix}.\normalsize$

Due to Assumption 4.1, the left-hand pencil in (101) has full row rank
$\forall s\in\mathbb{C}$, making the realization from the final equality in
(100) controllable. Thus, we conclude that the realization of
$\mathbf{K}_{d}(s)$ from (100) is minimal, which implies that all $p$
eigenvalues of $A_{d}$ at $s=0$ are poles of the system’s TFM, making it
continuous-time unstable.

Proof of Theorem 4.21 The proof largely follows the same reasoning as the one
belonging to Theorem III.6 in [24] and can be broken down into its two
constituent parts: the discrete-time case $\mathbf{I)}$ and the continuous-
time one $\mathbf{II)}$, respectively.

$\mathbf{I)}$ Begin with the discrete-time case and recall from Remark 4.17
that $\mathbf{K}_{d}(z)$ is stable, when starting from a stable SRTR pair, and
that each of its rows is also a stable TFM. Thus, the stabilizable and
detectable realizations for each such row, represented by
$e_{i}^{\top}\mathbf{K}_{d}(z)=\left[\footnotesize\begin{array}[]{c|c}\widetilde{A}_{di}-zI_{n_{i}}&\widetilde{B}_{di}\\\
\hline\cr\vspace{-1.5mm}\hfil\\\
\widetilde{C}_{di}&\widetilde{D}_{di}\end{array}\right]$, possess pole pencils
$\widetilde{A}_{di}-zI_{n_{i}}$ which have all their eigenvalues located in
$\mathbb{S}$. Using these $p$ representations, form the realization

(102)
$\mathbf{K}_{d}(z)=\left[\begin{array}[]{c:c:c}\mathbf{K}_{d}^{\top}(z)e_{1}&\dots&\mathbf{K}_{d}^{\top}(z)e_{p}\end{array}\right]^{\top}=\left[\scriptsize\begin{array}[]{ccc|c}\widetilde{A}_{d1}-zI_{n_{1}}&&&\widetilde{B}_{d1}\\\
&\ddots&&\vdots\\\ &&\widetilde{A}_{dp}-zI_{n_{p}}&\widetilde{B}_{dp}\\\
\hline\cr&&&\vspace{-1mm}\hfil\\\ \widetilde{C}_{d1}&&&\widetilde{D}_{d1}\\\
&\ddots&&\vdots\\\ &&\widetilde{C}_{dp}&\widetilde{D}_{dp}\end{array}\right].$

Due to its block-diagonal structure, the pole pencil of the realization from
(102) will have all its eigenvalues located in $\mathbb{S}$, which ensures
that the obtained realization for $\mathbf{K}_{d}(z)$ will be stabilizable and
detectable, while Theorem 4.15 shows that $\mathbf{K}_{d}(z)$ is a stabilizing
controller for $\begin{bmatrix}I&\mathbf{G}^{\top}(z)\end{bmatrix}^{\top}$.
Recall that
$\mathbf{G}(z)=\left[\footnotesize\begin{array}[]{c|c}A-zI_{n}&B\\\ \hline\cr
C&D\end{array}\right]$ has a stabilizable and detectable realization and let
$\begin{bmatrix}I&\mathbf{G}^{\top}(z)\end{bmatrix}^{\top}=\left[\footnotesize\begin{array}[]{c|cc}A^{\top}-zI_{n}&O&C^{\top}\\\
\hline\cr\vspace{-2mm}\hfil\\\
B^{\top}&I_{m}&D^{\top}\end{array}\right]^{\top}$, with the latter having a
realization possessing the same state variables as the realization of
$\mathbf{G}(z)$ and which is both stabilizable and detectable (since so is the
realization of $\mathbf{G}(z)$). For these realizations of $\mathbf{G}(z)$ and
$\begin{bmatrix}I&\mathbf{G}^{\top}(z)\end{bmatrix}^{\top}$, it is
straightforward to check that the closed-loop state dynamics in Figs. 4 and 5
are described by the same set of equations (with respect to the same sets of
inputs and outputs). Thus, by the same arguments as those in part
$\mathbf{(IV)}$ of the proof of Theorem III.6 in [24], we employ the discrete-
time versions of Definition 5.2 and of Lemmas 5.2 and 5.3 from [40] for the
closed-loop interconnection in Fig. 5 to get the desired result.

$\mathbf{II)}$ For the continuous-time case, start by recalling the
realization from (100), which was shown in the proof of Proposition 4.19 to be
minimal. Since the chosen SRTR pair is stable, Lemma C.1 ensures that
$A_{22}+KA_{12}$ has all its eigenvalues located in $\mathbb{S}$. Thus,
$\mathbf{K}_{d}(s)$ has $p$ unstable poles in $s=0$ along with $n-p$ poles
located in $\mathbb{S}$. Obtain now a realization for each
$e_{i}^{\top}\mathbf{K}_{d}(s)$ as follows

(103)
$e_{i}^{\top}\mathbf{K}_{d}(s)=\left[\small\begin{array}[]{c|c}A_{di}-sI_{n-p+1}&B_{di}\\\
\hline\cr C_{di}&D_{di}\end{array}\right]:=\\\
:=\left[\scriptsize\begin{array}[]{cc|cc}-s&e_{i}^{\top}A_{12}&e_{i}^{\top}(A_{11}-A_{12}K)&e_{i}^{\top}B_{1}\\\
O&A_{22}+KA_{12}-sI_{n-p}&KA_{11}-KA_{12}K+A_{21}-A_{22}K&KB_{1}+B_{2}\\\
\hline\cr 1&O&O&O\end{array}\right],\normalsize$

and notice that (103) is implicitly controllable and observable at $\forall
s\in\mathbb{C}\backslash\\{\mathbb{S}\cup\\{0\\}\\}$.

We now prove controllability and observability at $s=0$ for the realizations
of type (103). The latter property follows from the fact that $A_{22}+KA_{12}$
has all its eigenvalues located in $\mathbb{S}$. Since the pencil from (38)
has full row rank for $s=0$, then

$\begin{bmatrix}A_{di}&B_{di}\end{bmatrix}=\begin{bmatrix}e_{i}^{\top}&O\\\
O&I_{n-p}\end{bmatrix}\begin{bmatrix}O&S_{1}(0)\\\
O&S_{2}(0)\end{bmatrix}\small\begin{bmatrix}1&O&O&O\\\ O&O&I_{p}&O\\\
O&I_{n-p}&O&O\\\ O&O&O&I_{m}\end{bmatrix}\normalsize$

has full row rank, which makes (103) controllable at $s=0$.

We conclude that the realization from (103) is both stabilizable and
detectable and, since $A_{22}+KA_{12}$ has all its eigenvalues located in
$\mathbb{S}$, each $e_{i}^{\top}\mathbf{K}_{d}(s)$ has one pole in $s=0$.
Therefore, the pole pencil of any stabilizable and detectable realization
belonging to $e_{i}^{\top}\mathbf{K}_{d}(s)$ will have only one unstable
eigenvalue, at $s=0$. Compute now just such realizations for each
$e_{i}^{\top}\mathbf{K}_{d}(s)=\left[\footnotesize\begin{array}[]{c|c}\widetilde{A}_{di}-sI_{n_{i}}&\widetilde{B}_{di}\\\