#  On minimal LPV state-space representations in innovation form: an algebraic
characterization

Elie Rouphael, Mihaly Petreczky, Lotfi Belkoura E. Rouphael, M. Petreczky and
L. Belkoura are with the Centre de Recherche en Informatique, Signal et
Automatique de Lille, UMR CNRS 9189, Villeneuve dAscq 59651, France :
{elie.rouphael,mihaly.petreczky<EMAIL_ADDRESS>
###### Abstract

In this paper a definition of the concept of minimal state-space
representations in innovation form for LPV is proposed. We also present
algebraic conditions for a stochastic LPV state-space representation to be
minimal in forward innovation form and discuss an algorithm for transforming
any stochastic LPV state-space representation to a minimal one in innovation
form.

##  I Introduction

Identification of _Linear Parameter-Varying_ (LPV) models has gained
significant attention, see [2, 12, 15, 22, 9, 24, 26, 28, 25] and the
references therein. In particular, there is a rich literature on subspace
identification of LPV state-space representations, see for instance [26, 9,
27, 28, 6, 7] and the references therein.

Despite these advances, the theoretical analysis of system identification
algorithms, especially subspace methods, for stochastic LPV state-space
representations remains challenging. As the history of LTI system
identification indicates [13, 10, 4] that such a theoretical analysis requires
a good understanding of the notion of minimal state-space representations in
innovation form [13, 10]. The latter notion is not yet fully understood for
LPV state-space representations.

#### Contribution

In this paper LPV state-space representations with affine dependence on
parameters (abbreviated by _LPV-SSA_) are considered. We restrict attention to
autonomous (without control input) stationary stochastic LPV-SSAs (_asLPV-SSA_
for short). _The main technical contributions are new algebraic conditions for
an asLPV-SSA to be minimal in innovation form_. These conditions depend only
the the matrices of the system representation. In order to present these
results, the paper also provides a _systematic overview of the results on
existence and minimality of asLPV-SSA in innovation form_. These results can
be derived from realization theory of stochastic bilinear systems [20], but
they have not been stated explicitly for LPV-SSAs. In particular, we state
that any asLPV-SSA can be converted to a minimal asLPV-SSA in innovation form
while preserving the output. Moreover, any two minimal asLPV-SSA in innovation
form are isomorphic, if they have the same output

#### Motivation for studying asLPV-SSA

Under suitable technical assumptions, any stochastic LPV-SSA can be decomposed
into a noiseless deterministic LPV-SSA which is driven only by the control
input, and asLPV-SSA driven by the noise, [14]. Moreover, the identification
of these two subsystems can be carried out separately [14]. Note that
minimality and uniqueness of deterministic LPV state-space representations is
well understood [19]. Hence, in order to understand the notion of minimality
and innovation representation for stochastic LPV-SSA with control input, the
first step is to understand these notions for asLPV-SSA.

#### Motivation for minimal LPV-SSAs in innovation form

In the formulation of the identification problem for LPV-SSA [26, 9, 27, 28,
6] the stated goal is usually to find an LPV-SSA which is isomorphic111Often,
it is required that the isomorphism does not depend on the scheduling., or at
least which is input-output equivalent to the data generating system. However,
in general there may exist LPV-SSAs which generate the same output for some
scheduling signal, but which are not isomorphic (see Example 4 in Section V or
[11]), or which are not input-output equivalent. In the latter case the
systems generate different outputs when the scheduling signal is changed. See
Example 3 and Example 4 of Section V of the present paper.

This implies that in general the system identification problem is ill-posed.
However, it becomes well-posed, if we add the assumption that the underlying
data generating system is a minimal asLPV-SSA in innovation form.

Indeed, if the observed output has an asLPV-SSA representation, it has a
minimal one in innovation form, and all such representations are related by a
constant isomorphism. The necessity of this assumption is illustrated by the
examples in Section V. From [16] it follows that this assumption is
sufficient, as the identification algorithm in [16] returns a minimal asLPV-
SSA in innovation form. We conjecture that the same will be true for most of
the existing subspace identification algorithms [26, 9, 27, 28, 6, 7].

To deal only with minimal asLPV-SSA representations in innovation form, simple
conditions to check minimality and being in innovation form are needed. The
latter is necessary in order to check if the elements of a parametrization of
asLPV-SSAs are minimal and in innovation form, or to construct such
parametrizations.

#### Related work

As it was mentioned above, there is a rich literature on subspace
identification methods for stochastic LPV-SSA representations [7, 8, 6, 26].
However, the cited papers do not deal with the problem of characterizing
minimal stochastic LPV state-space representations in innovation form. In [5,
6] the existence of an LPV state-space representation in innovation form was
studied, but due to the specific assumptions (deterministic scheduling) and
the definition of the innovation process, the resulting LPV state-space
representation in innovation form had dynamic dependence on the scheduling
parameters. Moreover, [5, 6] do not address the issue of minimality of the
stochastic part of LPV state-space representations.

This paper uses realization theory of stochastic generalized bilinear systems
(_GBS _ for short) of [20]. In particular, asLPV-SSAs correspond to GBSs. The
existence and uniqueness of minimal asLPV-SSAs in innovation form follows from
the results of [20]. The main novelty of the present paper with respect to
[20] is the new algebraic characterization of minimal asLPV-SSAs in innovation
form, and that the results on existence and uniqueness of minimal GBSs are
spelled out explicitly for LPV-SSAs.

The paper [16] used the correspondence between GBSs and asLPV-SSAs to state
existence and uniqueness of minimal asLPV-SSAs in innovation form. However,
[16] did not provide an algebraic characterization of minimality or innovation
form. Moreover, it considered only scheduling signals which were zero mean
white noises. In contrast, in this paper more general scheduling signals are
considered. The present paper is complementary to [16]. This paper explains
when the assumption that the data generating system is minimal asLPV-SSA in
innovation form could be true, while [16] presents an identification algorithm
which is statistically consistent under the latter assumption.

Outline of the paper In Section II we introduce the notations used and we
recall [20], some technical assumptions which are necessary to define of the
stationary LPV-SSA representation. In Section III some principal results on
minimal asLPV-SSAs in innovation form are reviewed. In Section IV we present
the main results of the paper, namely, algebraic conditions for an asLPV-SSA
to be minimal in innovation form. Finally, in Section V numerical examples are
developed to illustrate the contributions.

##  II Preliminaries

In the sequel, we will use the standard terminology of probability theory [3].
In particular, all the random variables and stochastic processes are
understood w.r.t. to a fixed probability space
$\left(\Omega,\mathcal{F},\mathcal{P}\right)$, where $\mathcal{F}$ is a
$\sigma$-algebra over the sample space $\Omega$. The expected value of a
random variable $\mathbf{r}$ is denoted by $E[\mathbf{x}]$ and conditional
expectation w.r.t. $\sigma$\- algebra $\mathcal{F}$ is denoted by
$E\left[\mathbf{r}\mid\mathcal{F}\right]$. All the stochastic processes in
this paper are discrete-time ones defined over the time-axis $\mathbb{Z}$ of
the set of integers: a stochastic process $\mathbf{r}$ is a collection of
random variables $\\{\mathbf{r}(t)\\}_{t\in\mathbb{Z}}$ taking values in some
set $X$.

Next, we define the class of systems studied in this paper. An _autonomous
stochastic linear parameter-varying state-space representation with affine
dependence on scheduling parameter (aLPV-SSA)_ a is system described by

$\mathcal{S}\left\\{\begin{aligned}
&\mathbf{x}(t+1)=\sum_{i=1}^{n_{\mu}}(A_{i}\mathbf{x}(t)+K_{i}\mathbf{v}(t))\bm{\mu}_{i}(t)\\\
&\textbf{y}(t)=C\mathbf{x}(t)+F\mathbf{v}(t)\end{aligned}\right.$ (1)

where $A_{\sigma}\in\mathbb{R}^{n\times n},K_{\sigma}\in\mathbb{R}^{n\times
m}$, $C\in\mathbb{R}^{p\times n}$, $F\in\mathbb{R}^{n_{y}\times m}$, and
$\mathbf{x}$ is the state process,
$\bm{\mu}=\begin{bmatrix}\bm{\mu}_{1},\ldots,\bm{\mu}_{n_{\mu}}\end{bmatrix}^{T}$
is the scheduling process, $\mathbf{v}$ is the noise process and y is the
output process.

Note that all the involved processes are defined for both negative and
positive time. This may create technical problems for the existence of a
solution and the role of initial state. In this paper we will circumvent this
problem by considering LPV-SSA which are mean-square stable in a suitable
sense and state process of which is stationary. In order to define this class
of LPV-SSAs we will have to recall from [20] some notation and terminology.

###  II-A Admissible scheduling and wide-sense stationarity (ZWSII) w.r.t.
scheduling

Below we define the concept of admissible input processes and ZWSII processes
w.r.t. scheduling. These concepts will be used to define the class of aLPV-
SSAs we will work with. To this end, we will need the following notation from
automata theory which will be used for other purposes too

######  Notation 1 (Sequences over $\Sigma$)

Consider the finite set

$\Sigma=\\{1,\ldots,n_{\mu}\\}.$

A _non empty word_ over $\Sigma$ is a finite sequence of letters, i.e.,
$w=\sigma_{1}\sigma_{2}\cdots\sigma_{k}$, where $0<k\in\mathbb{Z}$,
$\sigma_{1},\sigma_{2},\ldots,\sigma_{k}\in\Sigma$. The set of _all_ nonempty
words is denoted by $\Sigma^{+}$. We denote an _empty word_ by $\epsilon$. Let
$\Sigma^{*}=\epsilon\cup\Sigma^{+}$. The concatenation of two nonempty words
$v=a_{1}a_{2}\cdots a_{m}$ and $w=b_{1}b_{2}\cdots b_{n}$ is defined as
$vw=a_{1}\cdots a_{m}b_{1}\cdots b_{n}$ for some $m,n>0$. Note that if
$w=\epsilon$ or $v=\epsilon$, then $v\epsilon=v$ and $\epsilon w=w$, moreover,
$\epsilon\epsilon=\epsilon$. The length of the word $w\in\Sigma^{*}$ is
denoted by $|w|$, and $|\epsilon|=0$.

With the notation above, we can identify the process $\bm{\mu}$ with the
collection $\\{\bm{\mu}_{\sigma}\\}_{\sigma\in\Sigma}$ of its components. We
will say that $\bm{\mu}$ is an _admissible scheduling process_ , if the
collection $\\{\bm{\mu}_{\sigma}\\}_{\sigma\in\Sigma}$ is an admissible
collection of input processes in the sense of [20, Definition 1] for
$S=\Sigma\times\Sigma$. For the convenience of the reader, a version of [20,
Definition 1] is presented in Appendix -A. Before proceeding further we will
present examples of admissible scheduling sequences.

######  Example 1 (White noise scheduling)

The scheduling process
$\bm{\mu}=[\bm{\mu}_{1},\bm{\mu}_{2},\ldots,\bm{\mu}_{n_{\mu}}]^{T}$ is
independent identically distributed (i.i.d.) such that for all
$i,j=2,\ldots,n_{\mu}$, $t\in\mathbb{Z}$, $\bm{\mu}_{i}(t),\bm{\mu}_{j}(t)$
are independent and $\bm{\mu}_{i}(t)$ is zero mean, then $\bm{\mu}$ is
admissible.

######  Example 2 (Discrete valued i.i.d process)

Assume there exists an i.i.d process $\bm{\theta}$ which takes its values from
a finite set $\Sigma=\\{1,\ldots,n_{\mu}\\}$. Let
$\bm{\mu}_{\sigma}(t)=\chi(\bm{\theta}(t)=\sigma)$ for all $\sigma\in\Sigma$,
$t\in\mathbb{Z}$. Then
$\bm{\mu}(t)=\begin{bmatrix}\bm{\mu}_{1}(t)&\ldots&\bm{\mu}_{n_{\mathrm{p}}}\end{bmatrix}^{T}$
is an admissible scheduling processes.

For further examples, see [20].

We assume that the scheduling process $\bm{\mu}$ is admissible throughout the
paper.

Furthermore, we will use the definition of a _Zero Mean Wide Sense Stationary
(abbreviated by ZMWSII) process with respect to $\bm{\mu}$_ from [20,
Definition 2]. For the convenience of the reader this definition is
reformulated in Appendix -A as Definition 5.

In order to explain the intuition behind these definitions, and because we
will use them later on, we define the following products of scheduling
variables along a sequence from $\Sigma^{+}$. For every word $w\in\Sigma^{+}$
where $w=\sigma_{1}\sigma_{2}\cdots\sigma_{k}$, $k\geq 1$,
$\sigma_{1},\ldots,\sigma_{k}\in\Sigma$, we define the process $\bm{\mu}_{w}$
as follows:

$\bm{\mu}_{w}(t)=\bm{\mu}_{\sigma_{1}}(t-k+1)\bm{\mu}_{\sigma_{2}}(t-k+2)\cdots\bm{\mu}_{\sigma_{k}}(t)$
(2)

For an empty word $w=\epsilon$, we set $\bm{\mu}_{\epsilon}(t)=1$.

If $\bm{\mu}$ is admissible, then $\bm{\mu}_{w},\bm{\mu}_{v}$ are jointly
wide-sense stationary. Moreover, $\bm{\mu}_{w}(t)$ and $\bm{\mu}_{v}(t)$ are
uncorrelated, if the last letters of $w$ and $v$ are different, and

$E[\bm{\mu}_{w\sigma}^{2}(t)]=p_{\sigma}E[\bm{\mu}_{w}^{2}(t)]$

for any $w\in\Sigma^{+}$, $\sigma\in\Sigma$. That is, $\bm{\mu}$ uniquely
determines a collection of numbers $\\{p_{\sigma}\\}_{\sigma\in\Sigma}$. This
latter collection will play an important role in the sequel. In particular,
for $\bm{\mu}$ from Example 1, $p_{i}$ is the variance of $\bm{\mu}_{i}$, and
for $\bm{\mu}$ from Example 2, $p_{i}$ is the probability $P(\theta(t)=i)$,
for all $i\in\Sigma$.

In order to explain the significance of these assumptions, let $\mathbf{r}$ be
a stochastic ZMWSII process w.r.t. $\bm{\mu}$, and because we will use them
latter on, we define the following products. Let
$\\{p_{\sigma}\\}_{\sigma\in\Sigma}$ be the constants determined by $\bm{\mu}$
as explained above, and define the products

$\displaystyle p_{w}$ $\displaystyle=p_{\sigma_{1}}p_{\sigma_{2}}\cdots
p_{\sigma_{k}}.$ (3)

For an empty word $w=\epsilon$, we set $p_{\epsilon}=1$. For a stochastic
process $\mathbf{r}\in\mathbb{R}^{n_{u}}$ and for each $w\in\Sigma^{*}$ we
define the stochastic process $\mathbf{z}^{\mathbf{r}}_{w}$ as

$\mathbf{z}^{\mathbf{r}}_{w}(t)=\mathbf{r}(t-|w|)\bm{\mu}_{w}(t-1)\frac{1}{\sqrt{p_{w}}},$
(4)

where $\bm{\mu}_{w}$ and $p_{w}$ are as in (2) and (3). For $w=\epsilon$,
$\mathbf{z}^{\mathbf{r}}_{w}(t)=\mathbf{r}(t)$. The process
$\mathbf{z}^{\mathbf{r}}_{w}$ in (4) is interpreted as the product of the
_past_ of $\mathbf{r}$ and $\bm{\mu}$. The process
$\mathbf{z}^{\mathbf{r}}_{w}$ will be used as predictors for future values of
$\mathbf{r}$ for various choices of $\mathbf{r}$.

Now we will explain the motivation for the concept of ZMWSII. Assume that
$\mathbf{r}$ is ZMWSII w.r.t. $\bm{\mu}$. Then
$\mathbf{z}^{\mathbf{r}}_{w}(t)$ is wide-stationary and square-integrable for
all $w\in\Sigma^{+}$. Moreover, the covariances
$E\left[\mathbf{z}^{\mathbf{r}}_{w}(t)(\mathbf{z}^{\mathbf{r}}_{v}(t))^{T}\right]$
do not depend on $t$. Furthermore $\mathbf{z}^{\mathbf{r}}_{w}(t)$,
$\mathbf{z}^{\mathbf{r}}_{v}(t)$ are orthogonal, if $w$ is not a suffix of $v$
or vice versa. Recall that we say that $w$ is a suffix of $v$, if $v=sw$ for
some $s\in\Sigma^{*}$. Moreover, when $w$ is suffix of $v$, then

$\displaystyle
E\left[\mathbf{z}^{\mathbf{r}}_{w}(t)(\mathbf{z}^{\mathbf{r}}_{v}(t))^{T}\right]=\left\\{\begin{array}[]{ll}E\left[\mathbf{r}(t)(\mathbf{z}^{\mathbf{r}}_{s}(t))^{T}\right]&\text{if
}v=sw\\\
E\left[\mathbf{z}^{\mathbf{r}}_{\sigma}(t)(\mathbf{z}^{\mathbf{r}}_{\sigma}(t))^{T}\right]&\text{if
}v=w=\sigma s\\\ \end{array}\right.$

That is,
$E\left[\mathbf{z}^{\mathbf{r}}_{w}(t)(\mathbf{z}^{\mathbf{r}}_{v}(t))^{T}\right]$
depends only on the difference of $w$ and $v$, i.e., on the prefix of $v$.
This can be viewed as a generalization of wide-sense stationarity, if the
index $w$ and $v$ are viewed as additional multidimensional time instances,
and $\Sigma^{*}$ is viewed as an additional time-axis.

###  II-B Stationary LPV-SSA representation

After identifying the necessary process properties and notations, we are now
ready to present the definition of the _stationary autonoumous stochastic LPV-
SSA_.

###### Definition 1

A _stationary autonomous stochastic LPV-SSA_ , abbreviated as _asLPV-SSA_ , is
a system of the form (1), such that:

(1) $\begin{bmatrix}\mathbf{x}^{T}&\mathbf{v}^{T}\end{bmatrix}^{T}$ is a
ZMWSSI process, and for all $\sigma\in\Sigma$, $w\in\Sigma^{+}$,
$E[\mathbf{z}^{\mathbf{x}}_{\sigma}(t)(\mathbf{z}^{\mathbf{v}}_{\sigma}(t))^{T}]=0,~{}E[\mathbf{v}(t)(\mathbf{z}^{\mathbf{x}}_{w}(t))^{T}]=0.$

(2) $\mathbf{v}$ is ZMWSSI and
$E[\mathbf{v}(t)(\mathbf{z}^{\mathbf{v}}(t))^{T}]=0$ for all $w\in\Sigma^{+}$.

(3) The eigenvalues of the matrix
$\sum_{\sigma\in\Sigma}p_{\sigma}A_{\sigma}\otimes A_{\sigma}$ are inside the
open unit circle.

Note that, condition (2) implies that v is a white noise.

In the terminology of [20], an asLPV-SSA corresponds to a stationary GBS
w.r.t. inputs $\\{\bm{\mu}_{\sigma}\\}_{\sigma\in\Sigma}$. Note that the
processes $\mathbf{x}$ and $\mathbf{y}$ are ZMWSII, in particular, they are
wide-sense stationary, and that $\mathbf{x}$ is orthogonal to the future
values of the noise process $\mathbf{v}$. We should mention that we
concentrate on wide-sense stationary processes, because it is difficult to
estimate the distribution of non-stationary processes. Also, wide-sense
stationary processes solve the problem of the initial state conditions.

The state of an asLPV-SSA is uniquely determined by its matrices and noise
process. In order to present this relationship, we need the following
notation.

######  Notation 2 (Matrix Product)

Consider $n\times n$ square matrices $\\{A_{\sigma}\\}_{\sigma\in\Sigma}$. For
any word $w\in\Sigma^{+}$ of the form
$w=\sigma_{1}\sigma_{2}\cdots\sigma_{k}$, $k\\!>\\!0$ and
$\sigma_{1},\ldots,\sigma_{k}\in\Sigma$, we define

$A_{w}=A_{\sigma_{k}}A_{\sigma_{k-1}}\cdots A_{\sigma_{1}}$

. For an empty word $\epsilon$, let $A_{\epsilon}=I_{n}$.

From [20, Lemma 2] it follows that

$\mathbf{x}(t)=\sum_{v\in\Sigma^{*},\sigma\in\Sigma}\sqrt{p_{\sigma
v}}A_{v}K_{\sigma}\mathbf{z}^{\mathbf{v}}_{\sigma v}(t)\\\ $

where the infinite sum on the right-hand side is absolutely convergent in the
mean square sense. This prompts us to use the following notation.

###### Notation 3

We identify the asLPV-SSA $\mathcal{S}$ of the form (1) with the tuple
$\mathcal{S}=(\\{A_{\sigma},K_{\sigma}\\}_{\sigma=1}^{n_{\mu}},C,F,\mathbf{v})$.

Finally, we need to define what we mean by an asLPV-SSA realization of a
process. An asLPV-SSA $\mathcal{S}$ of the form (1) is a _realization of a
pair $(\tilde{\textbf{y}},\tilde{\bm{\mu}})$_, if
$\tilde{\textbf{y}}=\textbf{y}$, $\tilde{\bm{\mu}}=\bm{\mu}$. If $\mathcal{S}$
is of the form (1), then we call the state-space dimension $n_{x}$ the
_dimension_ of $\mathcal{S}$ and we denote it by $\dim\mathcal{S}$. We say
that the asLPV-SSA $\mathcal{S}$ is a _minimal_ realization of
$(\tilde{\textbf{y}},\tilde{\bm{\mu}})$, if for any asLPV-SSA realization
$\mathcal{S}^{{}^{\prime}}$ of $(\tilde{\textbf{y}},\tilde{\bm{\mu}})$, the
dimension of $\mathcal{S}^{{}^{\prime}}$ is not smaller than
$\dim\mathcal{S}$.

##  III Existence and minilality of asLPV-SSAs in innovation form

In this section we review the principal results on existence and minimality of
asLPV-SSA in innovation form. To this end, in Subsection III-A we recall from
[19] some results on realization theory of deterministic LPV-SSA. In
Subsection III-B we present the definition of asLPV-SSAs in innovation form,
and in Subsection III-C we present results on existence and uniqueness of
minimal asLPV-SSAs. In Subsection III-D we present rank conditions for
minimality of asLPV-SSAs and an algorithm for converting any asLPV-SSA to a
minimal one in innovation form. The results presented in this section follow
from [20], but they have never been formulated explicitly for LPV-SSAs.

###  III-A Deterministic LPV-SSA representation

Recall from [19, 5] that a deterministic LPV state-space representation with
affine dependence (abbreviated as _dLPV-SSA_) is a system of the form:

$\mathscr{S}\left\\{\begin{aligned}
&\mathrm{x}(t+1)=\sum_{i=1}^{n_{\mu}}(\mathcal{A}_{i}\mathrm{x}(t)+\mathcal{B}_{i}\mathrm{u}(t))\mu_{i}(t),~{}\\\
&\mathrm{y}(t)=\mathcal{C}\mathrm{x}(t)+\mathcal{D}\mathrm{u}(t),\end{aligned}\right.$
(5)

where $\mathcal{A}_{i},\mathcal{B}_{i},\mathcal{C},\mathcal{D}$ are matrices
of suitable dimensions,
$\mathrm{x}:\mathbb{Z}\rightarrow\mathbb{R}^{n_{\mathrm{x}}}$ is the state
trajectory $\mathrm{u}:\mathbb{Z}\rightarrow\mathbb{R}^{n_{\mathrm{u}}}$ is
the input trajectory
$\mathrm{y}:\mathbb{Z}\rightarrow\mathbb{R}^{n_{\mathrm{y}}}$ is the output
trajectory with finite support. We identify a dLPV-SSA of the form (5) with
the tuple

$\mathscr{S}=(\\{\mathcal{A}_{i},\mathcal{B}_{i}\\}_{i=0}^{n_{\mu}},\mathcal{C},\mathcal{D})$
(6)

The _sub-Markov function_ of the dLPV-SSA $\mathscr{S}$ is the function
$M_{\mathscr{S}}:\Sigma^{*}\rightarrow\mathbb{R}^{n_{\mathrm{y}}\times
n_{\mathrm{u}}}$, such that for all $w\in\Sigma^{*}$,

$M_{\mathscr{S}}(w)=\left\\{\begin{array}[]{ll}\mathcal{CA}_{s}\mathcal{B}_{\sigma},&w=\sigma
s,\,\sigma\in\Sigma,\,s\in\Sigma^{*}\\\
\mathcal{D}.&w=\epsilon\end{array}\right.$ (7)

The values of $\\{M_{\mathscr{S}}(w)\\}_{w\in\Sigma^{*}}$ are the _sub-Markov
parameters_ of $\mathscr{S}$. From [19] it then follows that two dLPV-SSAs
have the same input-output behavior, if and only if their sub-Markov functions
are equal. For a function
$M:\Sigma^{*}\rightarrow\mathbb{R}^{n_{\mathrm{y}}\times n_{\mathrm{u}}}$, we
will say that the dLPV-SSA $\mathscr{S}$ is a _realization_ of $M$, if $M$
equals the sub-Markov function of $\mathscr{S}$, i.e., $M=M_{\mathscr{S}}$.
For dLPV-SSA (5), we call the integer $n_{x}$ the _dimension_ of
$\mathscr{S}$. We will call a dLPV-SSA _minimal_ , if there exists no other
dLPV-SSA $\mathscr{S}^{{}^{\prime}}$ with a smaller dimension and with the
same sub-Markov function. We call a dLPV-SSA $\mathscr{S}$ a _minimal
realization_ of a function
$M:\Sigma^{*}\rightarrow\mathbb{R}^{n_{\mathrm{y}}\times n_{\mathrm{w}}}$, if
$\mathscr{S}$ is minimal and it is a realization of $M$. From [19], it follows
that a dLPV-SSA is minimal if and only if it is span-reachable and observable,
and the latter properties are equivalent to rank conditions of the extended
$n$-step reachability and observability matrices [19, Definition 1, Theorem
2]. Furthermore, any dLPV-SSA can be transformed to a minimal one with the
same sub-Markov function, using Kalman decomposition [19, Corollary 1]222[19,
Corollary 1] should be applied with zero initial state. For a more detailed
discussion see [19].

###  III-B Definition of asLPV-SSA in innovation form

Next, we define what we mean by asLPV-SSA in innovation form. To this end, we
need to introduce the following notation for orthogonal projection.

######  Notation 4 (Orthogonal projection $E_{l}$)

Recall that the set of square integrable random variables taking values in
$\mathbb{R}$, forms a Hilbert-space with the scalar product defined as
$<\mathbf{z}_{1},\mathbf{z}_{2}>=E[\mathbf{z}_{1}\mathbf{z}_{2}]$. We denote
this Hilbert-space by $\mathcal{H}_{1}$. Let $\mathbf{z}$ be a square
integrable _vector-valued_ random variable taking its values in
$\mathbb{R}^{k}$. Let $M$ be a closed subspace of $\mathcal{H}_{1}$. By the
orthogonal projection of $\mathbf{z}$ onto the subspace $M$, _denoted by
$E_{l}[\mathbf{z}\mid M]$_, we mean the vector-valued square-integrable random
variable
$\mathbf{z}^{*}=\begin{bmatrix}\mathbf{z}_{1}^{*},\ldots,\mathbf{z}_{k}^{*}\end{bmatrix}^{T}$
such that $\mathbf{z}_{i}^{*}\in M$ is the orthogonal projection of the $i$th
coordinate $\mathbf{z}_{i}$ of $\mathbf{z}$ onto $M$, as it is usually defined
for Hilbert spaces. Let $\mathfrak{S}$ be a subset of square integrable random
variables in $\mathbb{R}^{p}$ for some integer $p$, and suppose that $M$ is
generated by the coordinates of the elements of $\mathfrak{S}$, i.e. $M$ is
the smallest (with respect to set inclusion) closed subspace of
$\mathcal{H}_{1}$ which contains the set $\\{\alpha^{T}s\mid
s\in\mathfrak{S},\alpha\in\mathbb{R}^{p}\\}$. Then instead of $E_{l}[z\mid M]$
we use $E_{l}[\mathbf{z}\mid\mathfrak{S}]$.

This said, we define the _innovation process $\mathbf{e}$ of $\mathbf{y}$ with
respect to $\bm{\mu}$_ as follows:

$\mathbf{e}(t)=\mathbf{y}(t)-E_{l}[\mathbf{y}(t)\mid\\{\mathbf{z}^{\mathbf{y}}_{w}(t)\\}_{w\in\Sigma^{+}}]$
(8)

In other words, $\mathbf{e}(t)$ is the difference between the output and its
projection on its past values w.r.t. the scheduling process $\bm{\mu}$, i.e.,
$\mathbf{e}$ is the best predictor of $\mathbf{y}$ using the product of the
output and scheduling past values from (4).

######  Definition 2 (asLPV-SSA in innovation form)

An asLPV-SSA of the form (1) is said to be in _innovation form_ , if it is a
realization of $(\textbf{y},\bm{\mu})$, $F=I_{n_{y}}$, and $\mathbf{v}$ is the
innovation process of $\mathbf{y}$, i.e., $\mathbf{v}=\mathbf{e}$.

###  III-C Existence and uniqueness of minimal asLPV-SSA in innovation form

Let $\mathcal{S}$ be an asLPV-SSA of the form (1) in innovation form with
$F=I_{n_{y}}$. Let
$\widetilde{\mathcal{S}}=(\\{\widetilde{A}_{i},\widetilde{K}_{i},\\}_{i=0}^{n_{\mu}},\widetilde{C},I_{n_{y}},\mathbf{e})$
be another asLPV-SSA of $(\mathbf{y},\bm{\mu})$ in innovation form. We say
that $\mathcal{S}$ and $\widetilde{S}$ are _isomorphic_ , if there exists a
nonsingular matrix $T$ such that

$\widetilde{A}_{i}=TA_{i}T^{-1},\widetilde{K}_{i}=TK_{i},\widetilde{C}=CT^{-1}$

We will say that the process $(\mathbf{y},\bm{\mu})$ is _full rank_ , if for
all $i=1,\ldots,n_{\mu}$,
$Q_{i}=E[\mathbf{e}(t)(\mathbf{e}(t))^{T}\bm{\mu}_{i}^{2}(t)]$ is invertable.
This is a direct extension of the classical notion of a full rank process.

Furthermore, we will say that $\mathbf{y}$ is _Square Integrable_ process
w.r.t. $\bm{\mu}$, abbreviated by _SII_ , if it satisfies [20, Definition
5]333with $\textbf{u}_{\sigma}=\bm{\mu}_{\sigma}$, $\sigma\in\Sigma$ in the
terminology of [20]. For the convenience of the reader, the definition of an
SII process is presented in Appendix -A, Definition 6. From [20, Remark 2] it
follows that if $\mathbf{y},\bm{\mu})$ has a realization by an asLPV-SSA and
$\bm{\mu}$ is bounded, then $\mathbf{y}$ is SII.

######  Theorem 1 (Existence and uniqueness)

Assume that $(\mathbf{y},\bm{\mu})$ has an asLPV-SSA realization and that
$\mathbf{y}$ is full rank and SII w.r.t. $\bm{\mu}$. It follows that:

  1. 1.

$(\textbf{y},\bm{\mu})$ has a minimal asLPV-SSA realization in innovation form

  2. 2.

Any two minimal asLPV-SSA realizations of $(\mathbf{y},\bm{\mu})$ in
innovation form are isomorphic.

The proof of this theorem follows [20, Theorem 2] by using the correspendence
between GBS and asLPV-SSA.

Theorem 1 implies that asLPV-SSAs in innovation form have the useful property
that, when they are minimal, they are unique up to isomorphism, and assuming
that the asLPV-SSA which generates the data is minimal does not restrict the
class of possible output. That is, for system identificaion it is preferable
to consider parameterizing the elements of minimal asLPV-SSAs in innovation
form. This motivates finding conditions for an asLPV-SSA to be minimal in
innovation form and formulating algorithms for transforming an asLPV-SSA to a
minimal one in innovation form. To this end, in Subsection III-D we present a
rank condition for minimality and a minimization algorithm based on the
results of [20]. However, the results of Subsection III-D do not allow
checking that an asLPV-SSA is in innovation form. Moreover, the algebraic
conditions for minimality are difficult to apply. Motivated by this, in
Section IV we present more user-friendly characterizations of minimality and
being in innovation form.

###  III-D Rank conditions and minimization algorithm

In order to present the rank conditions for minimality of asLPV-SSA and the
minimization algorithm, we need to define the _dLPV-SSA associated with asLPV-
SSA_ as

$\mathscr{S}_{\mathcal{S}}=(\\{\sqrt{p_{i}}A_{i},G_{i}\\}_{i=1}^{n_{\mu}},C,I_{n_{y}}),$

where

$G_{i}=\frac{1}{\sqrt{p_{i}}}(A_{i}P_{i}C^{T}+K_{i}Q_{i}F^{T})$ (9)

and $Q_{i}=E[\mathbf{v}(t)\mathbf{v}^{T}(t)\bm{\mu}_{i}^{2}(t)]$ for
$i=1,\ldots,n_{\mu}$, and $\\{P_{i}\\}_{i=1}^{n_{\mu}}$ are the computed as
follows: $P_{i}=\lim_{\mathcal{I}\rightarrow\infty}P_{i}^{\mathcal{I}}$, and
$\\{P_{i}^{N}\\}_{i=1}^{n_{\mu}}$ satisfy the following recursions:

$P_{i}^{\mathcal{I}+1}=p_{i}\sum_{j=0}^{n_{\mu}}(A_{j}P_{j}^{\mathcal{I}}A_{j}^{T}+K_{j}Q_{j}K_{j}^{T})$
(10)

with $P_{i}^{0}=\mathbf{O}_{n_{x},n_{x}}$ where $\mathbf{O}_{n_{x},n_{x}}$
denotes the $n_{x}\times n_{x}$ matrix with all zero entries. Note that the
existence of the limit of $P_{i}^{N}$ when $N$ goes to infinity follows from
[20, Lemma 5]. From [20] it follows that the dLPV-SSA associated with an
asLPV-SSA $\mathscr{S}$ represents a realization of Markov-function
$\Psi_{\mathbf{y}}:\Sigma^{*}\rightarrow\mathbb{R}^{n_{y}\times n_{y}}$,

$\Psi_{\mathbf{y}}(w)=\left\\{\begin{array}[]{ll}E[\mathbf{y}(t)(\mathbf{z}^{\mathbf{y}}_{w}(t))^{T}]&w\in\Sigma^{+}\\\
I_{n_{y}}&w=\epsilon\\\ \end{array}\right.$

computed from covariances of y.

Then from [20] we can derive the following.

######  Theorem 2 (Rank conditions)

An asLPV-SSA is a minimal realization of $(\textbf{y},\bm{\mu})$, if and only
if the associated dLPV-SSA is minimal.

From [19] it follows that minimality of the associated dLPV-SSA can be checked
using rank conditions for the corresponding extended reachability and
observability matrices. Note, however, that minimal asLPV-SSAs, realizing the
same output, may not be isomorphic. In fact, in Section V, Example 3 presents
a counter-example.

Vice versa, with any dLPV-SSA realization of $\Psi_{\mathbf{y}}$ we can
associate an asLPV-SSA in innovation form. This latter relationship is useful
for formulating a minimization algorithm. More precisely, consider a dLPV-SSA

$\mathscr{S}=(\\{\hat{A}_{i},\hat{G}_{i}\\}_{i=1}^{n_{\mu}},\hat{C},I_{n_{y}})$

which is a minimal realization of $\Psi_{\mathbf{y}}$. Define the _asLPV-SSA
$\mathcal{S}_{\mathscr{S}}$ associated with $\mathscr{S}$_ as

$\mathcal{S}_{\mathscr{S}}=(\\{\frac{1}{\sqrt{p_{i}}}\hat{A}_{i},\hat{K}_{i}\\}_{i=1}^{n_{\mu}},\hat{C},I_{n_{y}},\mathbf{e}),$

where
$\hat{K}_{i}=\lim_{\mathcal{I}\rightarrow\infty}\hat{K}_{i}^{\mathcal{I}}$,
and
$\\{\hat{K}_{\sigma}^{\mathcal{I}}\\}_{\sigma\in\Sigma,\mathcal{I}\in\mathbb{N}}$
satisfies the following recursion

$\begin{split}&\hat{P}^{\mathcal{I}+1}_{\sigma}=\sum\limits_{\sigma_{1}\in\Sigma}p_{\sigma}\left(\frac{1}{p_{\sigma_{1}}}\hat{A}_{\sigma_{1}}\hat{P}^{\mathcal{I}}_{\sigma_{1}}(\hat{A}_{\sigma_{1}})^{T}+\hat{K}_{\sigma_{1}}^{\mathcal{I}}\hat{Q}^{\mathcal{I}}_{\sigma_{1}}(\hat{K}^{\mathcal{I}}_{\sigma_{1}})^{T}\right)\\\
&\hat{Q}^{\mathcal{I}}_{\sigma}=p_{\sigma}T_{\sigma,\sigma}^{y}-\hat{C}\hat{P}^{\mathcal{I}}_{\sigma}(\hat{C})^{T}\\\
&\hat{K}^{\mathcal{I}}_{\sigma}=\left(\hat{G}_{\sigma}\sqrt{p_{\sigma}}-\frac{1}{\sqrt{p_{\sigma}}}\hat{A}_{\sigma}\hat{P}^{\mathcal{I}}_{\sigma}(\hat{C})^{T}\right)\left(\hat{Q}^{\mathcal{I}}_{\sigma}\right)^{-1}\end{split}$
(11)

where $\hat{P}^{0}_{\sigma}$ is a $n_{x}\times n_{x}$ zero matrix and
$T_{\sigma,\sigma}^{y}=E\left[\textbf{y}(t)(\mathbf{z}^{\mathbf{y}}_{w}(t))^{T}\right]$.
From [20] it follows that if $\mathscr{S}$ is a minimal dLPV-SSA realization
of $\Psi_{\mathbf{y}}$, then the associated asLPV-SSA
$\mathcal{S}_{\mathscr{S}}$ is an asLPV-SSA of $(\mathbf{y},\bm{\mu})$.

The discussion above suggests the following realization algorithm for
transforming an asLPV-SSA $\mathcal{S}$ to a minimal one, which can be deduced
from [20].

Algorithm 1 Minimization algorithm

Input : The asLPV-SSA
$\mathcal{S}=(\\{A_{\sigma},K_{\sigma}\\}_{\sigma=1}^{n_{\mu}},C,F,\mathbf{v})$
representation matrices.

  1. 1.

Compute the dLPV-SSA $\mathscr{S}_{\mathcal{S}}$ associated with $\mathcal{S}$
and compute
$T_{\sigma,\sigma}^{\textbf{y}}=\frac{1}{p_{\sigma}}(CP_{\sigma}C^{T}+FQ_{\sigma}F^{T})$

  2. 2.

Transform the dLPV-SSA $\mathscr{S}_{\mathcal{S}}$ to a minimal dLPV-SSA
$\mathscr{S}_{m}$ using [19, Corollary 1].

  3. 3.

Construct the asLPV-SSA $\mathcal{S}_{{\mathscr{S}}_{m}}$ associated with
$\mathscr{S}_{m}$.

Output : The asLPV-SSA $\mathcal{S}_{{\mathscr{S}}_{m}}$.

From [20, Theorem 3], it follows that Algorithm 1 returns a minimal
realization of $(\mathbf{y},\bm{\mu})$, if $\mathcal{S}$ is an asLPV-SSA
realization of $(\mathbf{y},\bm{\mu})$ in innovation form. Note that Algorithm
1 requires only the knowledge of the matrices of $\mathcal{S}$ and the noise
covariance matrix
$Q_{\sigma}=E[\mathbf{v}(t)\mathbf{v}^{T}(t)\bm{\mu}_{\sigma}^{2}(t)]$. Also
note that all the steps above are computationally efficient, however, they
require finding the limits of $P_{\sigma}^{\mathcal{I}}$ and
$\hat{K}_{\sigma}^{\mathcal{I}}$ for $\mathcal{I}\rightarrow\infty$
respectively. Also note that, in [16], it exists another minimization
algorithm which uses covariances matrices.

######  Remark 1 (Challenges)

The main disadvantage of verifying the rank condition of Theorem 2 or applying
Algorithm 1 is the necessity of constructing a dLPV-SSA and the necessity to
find the limit of the matrices in (11). The latter represents an extension of
algebraic Riccati equations [20, Remark 7] and even for the linear case
requires attention. Moreover, the rank conditions of Theorem 2 are not easy to
apply to parametrizations: even if the dependence of the matrices
$\\{A_{i},K_{i}\\}_{i=1}^{n_{\mu}}$ and $C$ on a parameter $\theta$ are linear
or polynomial, the dependence of the matrices of the associated dLPV-SSA need
not remain linear or polynomial, due to the definition of $G_{i}$ in (9). For
the same reason, it is difficult to analyze the result of applying Algorithm 1
to elements of a parametrization. Moreover, the conditions of Theorem 2 do not
allow us to check if the elements of a parametrizations are in innovation
form. These shortcomings motivate the contribution of Section IV.

##  IV Main results: algebraic conditions for an asLPV-SSA to be minimal in
innovation form

Motivated by the challenges explained in Remark 1, in this section we present
sufficient conditions for an asLPV-SSA to be minimal and in innovation form.
These conditions depend only on the matrices of the asLPV-SSA in question and
do not require any information on the noise processes.

The first result concerns an algebraic characterization of asLPV-SSA in
innovation form. This characterization does not require any knowledge of the
noise process, only the knowledge of system matrices. In order to streamline
the discussion, we introduce the following definition.

######  Definition 3 (Stably invertable w.r.t. $\bm{\mu}$)

Assume that $\mathcal{S}$ is an asLPV-SSA of the form (1) and $F=I_{n_{y}}$.
We will call $\mathcal{S}$ _stably invertable with respect to $\bm{\mu}$_, or
_stably invertable_ if $\bm{\mu}$ is clear from the context, if the matrix

$\sum_{i=1}^{n_{\mu}}p_{i}(A_{i}-K_{i}C)\otimes(A_{i}-K_{i}C)$ (12)

is stable (all its eigenvalues are inside the complex unit disk).

Note that a system can be stably invertable w.r.t. one scheduling process, and
not to be stably invertable w.r.t. another one. We can now state the result
relating stable invertability to asLPV-SSAs in innovation forms.

######  Theorem 3 (Innovation form condition)

Assume that y is SII and $(\textbf{y},\bm{\mu})$ is full rank. If an asLPV-SSA
realization of $(\textbf{y},\bm{\mu})$ is stably invertable, then it is in
innovation form.

The proof of Theorem 3 can be found in Appendix -B.

Stably invertable asLPV-SSAs can be viewed as optimal predictors. Indeed, let
$\mathcal{S}$ be the asLPV-SSA of the form (1) which is in innovation form,
and let x be the state process of $\mathcal{S}$. It then follows

$\begin{split}&\textbf{x}(t+1)=\sum_{i=1}^{n_{\mu}}(A_{i}-K_{i}C)\textbf{x}(t)+K_{i}\textbf{y}(t))\bm{\mu}_{i}(t),\\\
&\hat{\textbf{y}}(t)=C\textbf{x}(t)\end{split}$ (13)

where
$\hat{\textbf{y}}(t)=E_{l}[\textbf{y}(t)\mid\\{\mathbf{z}_{w}^{\textbf{y}}(t)\\}_{w\in\Sigma^{+}}]$,
i.e., $\hat{\textbf{y}}$ is the best linear prediction of $\textbf{y}(t)$
based on the predictors
$\\{\mathbf{z}_{w}^{\textbf{y}}(t)\\}_{w\in\Sigma^{+}}$. Intuitively, (13)
could be viewed as a filter, i.e., a dynamical system driven by past values of
y and generating the best possible linear prediction $\hat{\textbf{y}}(t)$ of
$\textbf{y}(t)$ based on
$\\{\mathbf{z}_{w}^{\textbf{y}}(t)\\}_{w\in\Sigma^{+}}$. However, the solution
of (13) is defined on the whole time axis $\mathbb{Z}$ and hence cannot be
computed exactly. For stably invertable asLPV-SSA we can approximate
$\hat{\textbf{y}}(t)$ as follows.

###### Lemma 1

With the assumptions of Theorem 3, if $\mathcal{S}$ of the form (1) is a
stably invertable realization of $(\textbf{y},\bm{\mu})$, and we consider the
following dynamical system:

$\begin{split}&\bar{\textbf{x}}(t+1)=\sum_{i=1}^{n_{\mu}}(A_{i}-K_{i}C)\bar{\textbf{x}}(t)+K_{i}\textbf{y}(t))\bm{\mu}_{i}(t),\\\
&\bar{\textbf{y}}(t)=C\bar{x}(t),~{}\bar{\textbf{x}}(0)=0\end{split}$ (14)

then
$\underset{t\rightarrow\infty}{\lim}\left(\bar{\textbf{x}}(t)-\textbf{x}(t)\right)\\!\\!=\\!\\!0$,
and
$\underset{t\rightarrow\infty}{\lim}\left(\bar{\textbf{y}}(t)-\textbf{y}(t)\right)=0$,
where the limits are understood in the mean square sense.

The proof of Lemma 1 is found in Appendix -B. That is, the output
$\bar{\textbf{y}}(t)$ of the recursive filter (14) is an approximation of the
optimal prediction $\hat{\textbf{y}}(t)$ of $\textbf{y}(t)$ for large enough
$t$. That is, stably invertable asLPV-SSA not only result in asLPV-SSAs in
innovation form, but they represent a class of asLPV-SSAs for which recursive
filters of the form (14) exist.

Next, we present algebraic conditions for minimality of an asLPV-SSA in
innovation form.

######  Theorem 4 (Minimality condition in innovation form)

Assume that $\mathcal{S}$ is an asLPV-SSA of the form (1) and that
$\mathcal{S}$ is a realization of $(\textbf{y},\bm{\mu})$ in innovation form.
Assume that $(\textbf{y},\bm{\mu})$ is full rank and y is SII. Then
$\mathcal{S}$ is a minimal realization of $(\textbf{y},\bm{\mu})$, if and only
if the dLPV-SSA
$\mathcal{D}_{\mathcal{S}}=(\\{A_{i},K_{i}\\}_{i=0}^{n_{\mu}},C,I_{n_{y}})$ is
minimal.

The proof of Theorem 4 can be found in Appendix -B.

Theorem 4, in combination with Theorem 3, leads to the following corollary.

######  Corollary 1 (Minimality and innovation form)

With the assumptions of Theorem 4, if $\mathcal{D}_{\mathcal{S}}$ is minimal
and $\mathcal{S}$ if stably invertable, then $\mathcal{S}$ is a minimal asLPV-
SSA realization of $(\textbf{y},\bm{\mu})$ in innovation form.

######  Remark 2 (Checking minimality and innovation form)

We recall that $\mathcal{D}_{\mathcal{S}}$ is minimal, if and only if it
satisfies the rank conditions for the extended $n$-step reachability and
observability matrices [19, Theorem 2], which can easily be computed from the
matrices of $\mathcal{S}$. Checking that $\mathcal{S}$ is stably invertable
boils down to checking the eigenvalues of the matrix (12). That is, Corollary
1 provides effective procedure for verifying that an asLPV-SSA is minimal and
in innovation form. Note that in contrast to the rank condition of Theorem 2,
which required computing the limit of (10), the procedure above uses only the
matrices of the system.

######  Remark 3 (Parametrizations of asLPV-SSAs)

Below we will sketch some ideas for applying the above results to
parametrizations of asLPV-SSAs. A detailed study of these issues remains a
topic for future research.

For all the elements of a parametrization of asLPV-SSAs to be minimal and in
innovation form, by Corollary 1 it is necessary that (A) all elements of the
parametrization, when viewed as dLPV-SSA, are minimal, and that (B) they are
stably invertable and satisfy condition (3) of Definition 1. In order to deal
with (A), the techniques used in [1] or [18] 444In order to use [18] the
relationship between minimality of dLPV-SSA and that of switched systems [19]
should be epxloitedcould be used. The condition (B) is equivalent to stability
of a suitable parametrization of LTI systems, as we could view the matrices
(12) and $\sum_{q=1}^{n_{\mu}}p_{i}A_{i}\otimes A_{i}$ as matrices of an LTI
state-space representation. For the latter we can use standard techniques, see
[23] and the references therein.

Finally, note that extension of argument of [1, Theorem 2] would also lead to
identifiability conditions for parametrizations which satisfy the conditions
(A) and (B) described above.

Corollary 1 suggests the following minimization algorithm.

Algorithm 2 Minimization algorithm

Input : A stably invertable asLPV-SSA $\mathcal{S}$

  1. 1.

Transform the dLPV-SSA $\mathcal{D}_{\mathcal{S}}$ from Theorem 4 to a minimal
dLPV-SSA
$\mathcal{D}_{m}=(\\{A_{i}^{m},K_{i}^{m}\\}_{i=1}^{n_{\mu}},C^{m},I_{n_{y}})$
using [19, Corollary 1].

Output : asLPV-SSA
$\mathcal{S}_{m}=(\\{A_{i}^{m},K_{i}^{m}\\}_{i=1}^{n_{\mu}},C^{m},I_{n_{y}},\textbf{e})$.

######  Lemma 2 (Correctness of Algorithm 2)

The asLPV-SSA $\mathcal{S}_{m}$ is stably invertable and it is a minimal
asLPV-SSA realization of $(\textbf{y},\bm{\mu})$ in innovation form.

That is, Algorithm 2 is a simpler counterpart of Algorithm 1 which does not
require computing the limits (10) and (11), which can create difficulties (see
Remark 1). Lemma 2 implies that stable invertability is preserved by
minimization.

##  V Numerical examples

In this section, we present numerical examples in order to illustrate the main
results.

###### Example 3

Consider an asLPV-SSA of the form (1), where $n_{\mu}=2$ and

$\displaystyle A_{1}=\begin{bmatrix}0.4&0.4&0\\\ 0.2&0.1&0\\\ 0&0&0.2\\\
\end{bmatrix},\quad K_{1}=\begin{bmatrix}0\\\ 1\\\ 1\\\ \end{bmatrix},\quad
C=\begin{bmatrix}10&0&0\\\ \end{bmatrix}$ $\displaystyle
A_{2}=\begin{bmatrix}0.1&0.1&0\\\ 0.2&0.3&0\\\ 0&0&0.2\\\ \end{bmatrix},\quad
K_{2}=\begin{bmatrix}0\\\ 1\\\ 1\\\ \end{bmatrix},\quad\text{and}\quad F=1$

Note that this representation is not in innovation form. The scheduling signal
process is defined as $\bm{\mu}=[\bm{\mu}_{1}\quad\bm{\mu}_{2}]$ such that
$\bm{\mu}_{1}(t)=1$ and $\bm{\mu}_{2}(t)$ is a white-noise process with
uniform distribution $\mathbf{\mathcal{U}}(-1.5,1.5)$. This corresponds to the
parameters values $\\{p_{\sigma}\\}_{\sigma\in\\{1,2\\}}$ to be
$p_{1}=E\left[\mu_{1}^{2}(t)\right]=1$ and
$p_{2}=E\left[\mu_{2}^{2}(t)\right]=0.75$. The noise process is a white
Gaussian noise with a variance equal to 1, i.e.,
$\mathbf{v}\sim\mathbf{\mathcal{N}}(0,1)$. Using Algorithm 1, we can find a
minimal representation in innovation form with the following matrices:

$\displaystyle A_{1}^{m}=\begin{bmatrix}0.4007&0.3997\\\ 0.1997&0.0993\\\
\end{bmatrix},\quad K_{1}^{m}=\begin{bmatrix}-0.046\\\ -0.0541\\\
\end{bmatrix}$ $\displaystyle A_{2}^{m}=\begin{bmatrix}0.1003&0.1002\\\
0.2002&0.2997\\\ \end{bmatrix},\quad K_{2}^{m}=\begin{bmatrix}-0.0116\\\
-0.0578\\\ \end{bmatrix}$ $\displaystyle C^{m}=\begin{bmatrix}-10&0.0116\\\
\end{bmatrix},\quad\text{and}\quad F^{m}=1$

Note that the two systems
$\mathcal{S}=(\\{A_{\sigma},K_{\sigma}\\}_{\sigma=1}^{n_{\mu}},C,F,\mathbf{v})$
and
$\mathcal{S}^{m}=(\\{A^{m}_{\sigma},K^{m}_{\sigma}\\}_{\sigma=1}^{n_{\mu}},C^{m},F^{m},\mathbf{e})$
have the same output trajectory when the chosen scheduling is applied.
However, by changing the scheduling process to another process
$\bm{\mu}^{{}^{\prime}}$ with $\bm{\mu}_{1}^{{}^{\prime}}=\bm{\mu}_{1}$ and
$\bm{\mu}_{2}^{{}^{\prime}}$ is also a white-noise with a uniform distribution
$\mathbf{\mathcal{U}}(-\sqrt{3},\sqrt{3})$, we realize that the output
trajectories are not the same, in other words, the two systems are not input-
output equivalent. Subsequently, the two representations are not isomorphic.

###### Example 4

We use the same noise process $\mathbf{v}$ and scheduling process $\bm{\mu}$
as in Example 3. The system matrices are as follows.

$\displaystyle A_{1}=\begin{bmatrix}0.4&0.4\\\ 0.2&0.1\\\ \end{bmatrix},\quad
K_{1}=\begin{bmatrix}0\\\ 1\\\ \end{bmatrix},\quad C=\begin{bmatrix}10&0\\\
\end{bmatrix}$ $\displaystyle A_{2}=\begin{bmatrix}0.1&0.1\\\ 0.2&0.3\\\
\end{bmatrix},\quad K_{2}=\begin{bmatrix}0\\\ 1\\\
\end{bmatrix},\quad\text{and}\quad F=1$

The system above is not in innovation form, in fact, it is not stably
invertable. As before, we use Algorithm 1 to obtain a minimal asLPV-SSA
innovation form with the following matrices

$\displaystyle A_{1}^{m}=\begin{bmatrix}0.4007&-0.3997\\\ -0.1997&0.0993\\\
\end{bmatrix},\quad K_{1}^{m}=\begin{bmatrix}-0.046\\\ 0.0541\\\
\end{bmatrix},$ $\displaystyle A_{2}^{m}=\begin{bmatrix}0.1003&-0.1002\\\
-0.2002&0.2997\\\ \end{bmatrix},\quad K_{2}^{m}=\begin{bmatrix}-0.0116\\\
0.0578\\\ \end{bmatrix}$ $\displaystyle C^{m}=\begin{bmatrix}-10&-0.0116\\\
\end{bmatrix},\quad\text{and}\quad F^{m}=1$

The two asLPV-SSA systems
$\mathcal{S}=(\\{A_{\sigma},K_{\sigma}\\}_{\sigma=1}^{n_{\mu}},C,F,\mathbf{v})$
and
$\mathcal{S}^{m}=(\\{A^{m}_{\sigma},K^{m}_{\sigma}\\}_{\sigma=1}^{n_{\mu}},C^{m},F^{m},\mathbf{e})$
are not isomorphic. In fact, they have different output trajectories, when
using the scheduling process $\bm{\mu}^{{}^{\prime}}$ from Example 3. This is
due to the fact that the system $\mathcal{S}$ is not in innovation form.

From Examples 3 and 4, we can conclude that minimality alone does not provide
uniqueness.

###### Example 5

We present a minimal asLPV-SSA in innovation form, where the state dimension
$n_{x}=2$ and its matrices are as follows:

$\displaystyle A_{1}=\begin{bmatrix}0.4&0.4\\\ 0.2&0.1\\\ \end{bmatrix},\quad
A_{2}=\begin{bmatrix}0.1&0.1\\\ 0.2&0.3\\\ \end{bmatrix}$ $\displaystyle
K_{1}=\begin{bmatrix}0\\\ 1\\\ \end{bmatrix},\quad K_{2}=\begin{bmatrix}0\\\
1\\\ \end{bmatrix},\quad C=\begin{bmatrix}1&0\\\
\end{bmatrix},\quad\text{and}\quad F=1$

The asLPV-SSA above is stably invertable, and hence in innovation form. We use
the same scheduling process $\bm{\mu}$ and noise process $\mathbf{v}$ as
Example 3. If we apply Algorithm 1 to the system above, we get another asLPV-
SSA with the matrices

$\displaystyle A^{m}_{1}=\begin{bmatrix}0.4642&-0.3581\\\ -0.1581&0.0358\\\
\end{bmatrix},\quad K^{m}_{1}=\begin{bmatrix}-0.1143\\\ 0.9934\\\
\end{bmatrix}$ $\displaystyle A^{m}_{2}=\begin{bmatrix}0.1367&-0.1188\\\
-0.2188&0.2633\\\ \end{bmatrix},\quad K^{m}_{2}=\begin{bmatrix}-0.1143\\\
0.9934\\\ \end{bmatrix}$ $\displaystyle
C^{m}=\begin{bmatrix}-0.9934&-0.1143\\\ \end{bmatrix},\quad\text{and}\quad
F^{m}=1$

As expected, the two systems are isomorphic, the corresponding matrix is

$\footnotesize T=\begin{bmatrix}-0.9934&-0.1143\\\ -0.1143&0.9934\\\
\end{bmatrix}$

Finally, we realize that the output trajectories of both systems are indeed
the same not only for the chosen scheduling sequence, but also for _any_ other
scheduling process.

##  VI Conclusion

This paper formulates conditions for a LPV state-space representation to be
minimal and in innovation form. These conditions depend only the matrices of
the LPV representation. A minimization algorithm for transforming any LPV
representation to a minimal one in innovation form is formulated too. These
results are expected to be useful for system identification, in particular,
for making the identification problem mathematically well-posed. In the
future, we will explore the application of the proposed results to concrete
system identification algorithms.

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###  -A Technical definitions

######  Definition 4 (Admissible scheduling sequences)

A scheduling process $\bm{\mu}$ is called an _Admissible scheduling sequence_
if it satisfies the following properties:

  1. 1.

Denote by $\mathscr{F}^{\bm{\mu},-}_{t}$ the $\sigma$-algebra generated by the
random variables $\\{\bm{\mu}(k)\\}_{k<t}$. There exists positive numbers
$\\{p_{\sigma}\\}_{\sigma\in\Sigma}$ such that for any
$w,v\in\Sigma^{+},\sigma,\sigma^{{}^{\prime}}\in\Sigma$, $t\in\mathbb{Z}$:

$\begin{split}&E[\bm{\mu}_{w\sigma}(t)\bm{\mu}_{v\sigma^{{}^{\prime}}}(t)\mid\mathscr{F}^{\bm{\mu},-}_{t}]=\\\
&\left\\{\begin{array}[]{rl}p_{\sigma}\bm{\mu}_{w}(t-1)\bm{\mu}_{v}(t-1)&\sigma=\sigma^{{}^{\prime}}\\\
0&\mbox{otherwise}\end{array}\right.\\\
&E[\bm{\mu}_{w\sigma}(t)\bm{\mu}_{\sigma^{{}^{\prime}}}(t)\mid\mathscr{F}^{\bm{\mu},-}_{t}]=\\\
&\left\\{\begin{array}[]{rl}p_{\sigma}\bm{\mu}_{w}(t-1)&\sigma=\sigma^{{}^{\prime}}\mbox{
and }\\\ 0&\mbox{otherwise}\end{array}\right.\\\
&E[\bm{\mu}_{\sigma}(t)\bm{\mu}_{\sigma^{{}^{\prime}}}(t)\mid\mathscr{F}^{\bm{\mu},-}_{t}]=\begin{array}[]{rl}0&\mbox{if
}\sigma\neq\sigma^{{}^{\prime}}\end{array}\end{split}$ (15)

  2. 2.

There exist real numbers $\\{\alpha_{\sigma}\\}_{\sigma\in\Sigma}$ such that
$\sum_{\sigma\in\Sigma}\alpha_{\sigma}\bm{\mu}_{\sigma}(t)=1$ for all
$t\in\mathbb{Z}$.

  3. 3.

For each $w,v\in\Sigma^{+}$, the process
$\begin{bmatrix}\bm{\mu}_{w},\bm{\mu}_{v}\end{bmatrix}^{T}$ is wide-sense
stationary.

Definition 4 is a reformulation of [20, Definition 1], for
$S\subseteq\Sigma\times\Sigma$.

######  Definition 5 (ZMWSSI, [20, Definition 2])

A stochastic process $(\mathbf{r},\bm{\mu})$ is _Zero Mean Wide Sense
Stationary_ (ZMWSSI) if

  1. 1.

For $t\in\mathbb{Z}$, the $\sigma$-algebras generated by the variables
$\\{\mathbf{r}(k)\\}_{k\leq t}$,
$\\{\bm{\mu}_{\sigma}(k)\\}_{k<t,\sigma\in\Sigma}$ and
$\\{\bm{\mu}_{\sigma}(k)\\}_{k\geq t,\sigma\in\Sigma},$ denoted by
$\mathcal{F}_{t}^{\mathbf{r}}$, $\mathcal{F}_{t}^{\bm{\mu},-}$ and
$\mathcal{F}_{t}^{\bm{\mu},+}$ respectively, are such that
$\mathcal{F}_{t}^{\mathbf{r}}$ and $\mathcal{F}_{t}^{\bm{\mu},+}$ are
conditionally independent w.r.t. $\mathcal{F}_{t}^{\bm{\mu},-}$.

  2. 2.

The processes
$\\{\mathbf{r},\\{\mathbf{z}^{\mathbf{r}}_{w}\\}_{w\in\Sigma^{+}}\\}$ are zero
mean, square integrable and are jointly wide sense stationary, i.e., $\forall
t,s,k\in\mathbb{Z},w,v\in\Sigma^{+}$,

$\displaystyle
E\left[\mathbf{r}(t)\right]=0,~{}E\left[\mathbf{z}^{\mathbf{r}}_{w}(t)\right]=0$
$\displaystyle
E\left[\mathbf{r}(t+k)(\mathbf{z}^{\mathbf{r}}_{w}(s+k))^{T}\right]=E\left[\mathbf{r}(t)(\mathbf{z}^{\mathbf{r}}_{w}(s))^{T}\right],$
$\displaystyle
E\left[\mathbf{r}(t+k)(\mathbf{r}(s+k))^{T}\right]=E\left[\mathbf{r}(t)(\mathbf{r}(s))^{T}\right],$
$\displaystyle
E\left[\mathbf{z}^{\mathbf{r}}_{w}(t+k)(\mathbf{z}^{\mathbf{r}}_{v}(s+k))^{T}\right]=E\left[\mathbf{z}^{\mathbf{r}}_{w}(t)(\mathbf{z}^{\mathbf{r}}_{v}(s))^{T}\right].$

######  Definition 6 (SII process [20, Definition 5])

A process $\mathbf{r}$ is said to be _Square Integrable w.r.t. $\bm{\mu}$_
(abbreviated as _SII_ when $\bm{\mu}$ is clear from the context), for all
$w\in\Sigma^{*},t\in\mathbb{Z}$, the random variable
$\mathbf{z}^{\mathbf{r}+}_{w}(t)=\mathbf{r}(t+|w|)\bm{\mu}_{w}(t+|w|-1)\frac{1}{\sqrt{p_{w}}}$,
is square integrable.

###  -B Proofs of Theorems 3 and 4

First we show the following technical result, which states the following.

######  Lemma 3 (Mean-square stability block-diagonal matrices)

Consider matrices $n_{x}\times n$ $\\{B_{i=1}\\}_{i=1}^{n_{\mu}}$ and $n\times
n$ matrices $\\{F_{i}\\}_{i=1}^{n_{\mu}}$ and $n_{x}\times n_{x}$ matrices
$\\{A_{i}\\}_{i=1}^{n_{\mu}}$ such that that
$\sum_{i=1}^{n_{\mu}}p_{i}(F_{i}\otimes F_{i})$ and
$\sum_{i=1}^{n_{\mu}}p_{i}(A_{i}\otimes A_{i}$ are stable (all their
eiegenvalues are inside the unit disk). Conside the matrix

$\begin{split}&\tilde{F}_{i}=\begin{bmatrix}F_{i}&B_{i}\\\
0&A_{i}\end{bmatrix}\\\ \end{split}$

$\sum_{i=1}^{n_{\mu}}p_{i}(\tilde{F}_{i}\otimes\tilde{F}_{i})$ is stable.

###### Proof:

Recall from [21, Lemma 5] it follows that the stability of
$\sum_{i=1}^{n_{\mu}}p_{i}(\tilde{F}_{i}\otimes\tilde{F}_{i})$ is equivalent
to the existence of a matrix $P>0$ such that
$P-\sum_{i=1}^{n_{\mu}}p_{i}\tilde{F}_{i}P\tilde{F}_{i}^{T}>0$. We will
construct such a matrix $P$. To this end, from [21, Lemma 5] it follows that
there exist $P_{2}>0$ such that
$P_{2}-\sum_{i=1}^{n_{\mu}}p_{i}A_{i}P_{2}A_{i}^{T}>0$. Define

$S=\sum_{i=1}^{n_{\mu}}p_{i}(B_{i}P_{2}B_{i}^{T})+\mathscr{D}(P_{2}-\sum_{i=1}{n_{\mu}}p_{i}A_{i}P_{2}A_{i}^{T})^{-1}\mathscr{D}^{T}$

where $\mathscr{D}=(\sum_{i=1}^{n_{\mu}}p_{i}B_{i}P_{2}A_{i}^{T}$. It then
follows that $S\geq 0$, as
$(P_{2}-\sum_{i=1}{n_{\mu}}p_{i}A_{i}P_{2}A_{i}^{T})^{-1}$ is positive
definite and $B_{i}P_{2}B_{i}^{T}$ are postive semi-definite matrices for each
$i=1,\ldots,n_{\mu}$. From [21, Lemma 5] it follows that there exist $P_{1}>0$
such that

$P_{1}-\sum_{i=1}^{n_{\mu}}F_{i}P_{1}F_{i}^{T}-S>0$ (16)

Consider $P=\begin{bmatrix}P_{1}&0\\\ 0&P_{2}\end{bmatrix}$. Using standard
calculation, it follows that

$\begin{split}&P-\sum_{i=1}^{n_{\mu}}p_{i}\tilde{F}_{i}P\tilde{F}_{i}^{T}=M(P)\\\
&M(P)=\\\
&\begin{bmatrix}P_{1}-\sum_{i=1}^{n_{\mu}}p_{i}F_{i}P_{1}F_{i}^{T}-\mathcal{D}_{1}&-\mathscr{D}\\\
-\mathscr{D}^{T}&P_{2}-\sum_{i=1}{n_{\mu}}p_{i}A_{i}P_{2}A_{i}^{T}\end{bmatrix}\end{split}$

$\mathscr{D}=\sum_{i=1}^{n_{\mu}}p_{i}B_{i}P_{2}A_{i}^{T}$,
$\mathcal{D}_{1}=\sum_{i=1}^{n_{\mu}}p_{i}(B_{i}P_{2}B_{i}^{T})$. Note that
$P_{2}-\sum_{i=1}{n_{\mu}}p_{i}A_{i}P_{2}A_{i}^{T}>0$, hence, by using Schur
complement, $M(P)>0$ if (16) holds, which is true by the choice of $P_{1}$. ∎

######  Lemma 4 (Orthogonal noises)

Let $\mathbf{v}$ be a ZMWSII such that
$E[\mathbf{v}(t)\mathbf{z}^{\mathbf{v}}_{w}(t)]=0$, $w\in\Sigma^{+}$. Then
$\mathbf{z}_{w}(t)^{\mathbf{v}}(t),\mathbf{z}_{v}^{\mathbf{v}}(t)$ are
uncorrelated for all $w,v\in\Sigma^{+}$, $v\neq w$.

###### Proof:

Clearly
$E[\mathbf{z}_{w}^{\mathbf{v}}(t)(\mathbf{z}_{v}^{\mathbf{v}}(t))^{T}]$ is
either zero if $w$ is not a suffix of $v$, or if $v$ is not a suffix of $w$,
as it is ZMWSII. If say $v=sw$, $s\neq\epsilon$ then
$E[\mathbf{z}_{w}^{\mathbf{v}}(t)(\mathbf{z}_{v}^{\mathbf{v}}(t))^{T}]=E[\mathbf{v}(t)\mathbf{z}_{s}^{\mathbf{v}}(t)]=0$.
∎

######  Lemma 5 (Absolute convergence infinite sums)

Consider a $p\times n$ matrix $C$ and matrices $n\times n_{y}$
$\\{B_{i=1}\\}_{i=1}^{n_{\mu}}$ and $n\times n$ matrices
$\\{F_{i}\\}_{i=1}^{n_{\mu}}$ such that that
$\sum_{i=1}^{n_{y}}p_{i}(F_{i}\otimes F_{i})$ is stable. Let $\mathbf{v}$ be a
ZMWSII such that $E[\mathbf{v}(t)(\mathbf{z}^{\mathbf{v}}_{w}(t))^{T}]=0$,
$w\in\Sigma^{+}$. Then the infinite sum

$\mathbf{r}(t)=\sum_{w\in\Sigma^{*},\sigma\in\Sigma}\sqrt{p_{\sigma
w}}CF_{w}B_{\sigma}\mathbf{z}_{\sigma w}^{\mathbf{v}}(t)$ (17)

is absolutely convergent in the mean-square sense, i.e.,

$\sum_{w\in\Sigma^{*},\sigma\in\Sigma}E[\|CF_{w}B_{\sigma}\mathbf{z}_{\sigma
w}^{\mathbf{v}}(t)\|^{2}_{2}]$ (18)

and the process $\mathbf{r}(t)$ is ZMWSII and it the unique state process of
the asLPV-SSA $(\\{F_{i},B_{i}\\}_{i=1}^{n_{\mu}},C,I,\mathbf{v})$.

###### Proof:

From [20, Lemma 3] it follows that (17) is convergent in mean square sense and
$\mathbf{r}(t)$ is ZMWSII as it is the solution of the asLPV-SSA
$(\\{F_{i},B_{i}\\}_{i=1}^{n_{\mu}},C,I,\mathbf{v})$. From Lemma 4 it follows
that $\mathbf{z}_{w}(t)^{\mathbf{v}}(t),\mathbf{z}_{v}^{\mathbf{v}}(t)$ are
uncorrelated. Hence

$\begin{split}&E[\|\sum_{w\in\Sigma^{*},|w|\leq
N,\sigma\in\Sigma}CF_{w}B_{\sigma}\mathbf{z}_{\sigma
w}^{\mathbf{v}}(t)\|^{2}_{2}]=\\\
&trace\left(\sum_{w,v\in\Sigma^{*},|w|,|v|\leq
N,\sigma,\sigma_{1}\in\Sigma}CA_{w}B_{\sigma}E[\mathbf{z}_{w}^{\mathbf{v}}(t)(\mathbf{z}_{v}^{\mathbf{v}}(t))^{T}](CA_{v}B_{\sigma_{1}})^{T}\right)=\\\
&\sum_{w\in\Sigma^{*},|w|\leq
N,\sigma\in\Sigma}E[\|CF_{w}B_{\sigma}\mathbf{z}_{\sigma
w}^{\mathbf{v}}(t)\|^{2}_{2}]\end{split}$

and as $E[\|\sum_{w\in\Sigma^{*},|w|\leq
N,\sigma\in\Sigma}CF_{w}B_{\sigma}\mathbf{z}_{\sigma
w}^{\mathbf{v}}(t)\|^{2}_{2}]$ is convergent by mean-square convergence of
(17), it follows that (18) is convergent. ∎

###### Lemma 6

Assume that $(\textbf{y},\bm{\mu})$ has a asLPV-SSA realization. Consider
matrices $n\times n_{y}$ $\\{B_{i=1}\\}_{i=1}^{n_{\mu}}$ and $n\times n$
matrices $\\{F_{i}\\}_{i=1}^{n_{\mu}}$ such that that
$\sum_{i=1}^{n_{\mu}}p_{i}(F_{i}\otimes F_{i})$ is stable. Then the infinite
sum

$\sum_{w\in\Sigma^{*},\sigma\in\Sigma}F_{w}B_{\sigma}\mathbf{z}_{\sigma
w}^{\textbf{y}}(t)$ (19)

converges absolutely in the mean-square sense and
$\mathbf{r}(t)=\sum_{w\in\Sigma^{*},\sigma\in\Sigma}F_{w}B_{\sigma}\mathbf{z}_{\sigma
w}^{\textbf{y}}(t)$ is a ZMWSII process, and $\mathbf{r}$ is the unique
process $\bar{\mathbf{r}}$ which satisfies

$\bar{\mathbf{r}}(t+1)=\sum_{i=1}^{n_{\mu}}(F_{i}\bar{\mathbf{r}}(t)+B_{i}\textbf{y}(t))\bm{\mu}_{i}(t)$
(20)

and $\begin{bmatrix}\bar{\mathbf{r}}^{T}&\textbf{y}^{T}\end{bmatrix}$ is
ZMWSII and the components of $\bar{\mathbf{r}}(t)$ belong to the Hilbert-space
$\mathcal{H}^{\textbf{y}}_{t}$ generated by
$\\{\mathbf{z}_{w}^{\textbf{y}}(t)\\}_{w\in\Sigma^{+}}$.

###### Proof:

Assume that $\mathcal{S}=(\\{A_{i},K_{i}\\}_{i=1}^{n_{\mu}},C,D,\mathbf{v})$
is a minimal asLPV-SSA realization of $(\textbf{y},\bm{\mu})$ in innovation
form. Let us define the matrices

$\begin{split}&\tilde{F}_{i}=\begin{bmatrix}F_{i}&B_{i}C\\\
0&A_{i}\end{bmatrix},\\\ &\tilde{B}_{i}=\begin{bmatrix}B_{i}D\\\
K_{i}\end{bmatrix}\\\
&\tilde{C}=\begin{bmatrix}I_{n}&0\end{bmatrix}\end{split}$

From Lemma 3 it follows that
$\sum_{i=1}^{n_{\mu}}p_{i}(\tilde{F}_{i}\otimes\tilde{F}_{i})$ is stable, and
hence by Lemma 5

$\tilde{\textbf{x}}(t)=\sum_{w\in\Sigma^{*},\sigma\in\Sigma}\sqrt{p_{\sigma}w}\tilde{F}_{w}B_{\sigma}\mathbf{z}_{\sigma
w}^{\mathbf{v}}(t)$ (21)

is absolutely convergent in the mean square sense and it is the unique state
process of
$(\\{\tilde{F}_{i},\tilde{B}_{i}\\}_{i=1}^{n_{\mu}},\tilde{C},I,\mathbf{v})$.

From [20, Lemma 2 and Lemma 9] it follows that

$\mathbf{z}^{\textbf{y}}_{v}(t)=\sum_{w\in\Sigma^{*},\sigma\in\Sigma}CA_{w}K_{\sigma}\mathbf{z}_{\sigma
wv}^{\mathbf{v}}(t)+D\mathbf{z}^{\mathbf{v}}_{v}(t)$ (22)

Notice that

$\tilde{F}_{w}=\begin{bmatrix}F_{w}&\sum_{s_{1},s_{2}\in\Sigma^{*},\sigma_{1}\in\Sigma,w=s_{1}\sigma
s_{2}}F_{s_{2}}B_{\sigma_{1}}CA_{s_{1}}\\\ 0&A_{w}\end{bmatrix}$ (23)

and notice that $\hat{\textbf{x}}(t)=\begin{bmatrix}\bar{\textbf{x}}(t)\\\
\textbf{x}(t)\end{bmatrix}$, where x is the unique state process of the asLPV-
SSA $\mathcal{S}=(\\{A_{i},K_{i}\\}_{i=1}^{n_{\mu}},C,D,\mathbf{v})$ which
realizes $(\textbf{y},\bm{\mu})$. In particular, $\bar{\textbf{x}}(t)$

$\begin{split}&\bar{\textbf{x}}(t)=\sum_{w\in\Sigma^{*},\sigma\in\Sigma}\left(F_{w}B_{\sigma}D\mathbf{z}^{\mathbf{v}}_{w\sigma}(t)+\right.\\\
&\left.\sum_{s_{1},s_{2}\in\Sigma^{*},\sigma_{1}\in\Sigma,w=s_{1}\sigma
s_{2}}F_{s_{2}}B_{\sigma_{1}}CA_{s_{1}}K_{\sigma}\mathbf{z}_{\sigma
w}^{\mathbf{v}}(t)\right)\end{split}$ (24)

From Lemma 5 it follows that
$S_{3}=\sum_{w\in\Sigma^{*},\sigma\in\Sigma}F_{w}B_{\sigma}D\mathbf{z}^{\mathbf{v}}_{w\sigma}(t)$
is absolutely convergent in the mean-square sense. Hence,

$\begin{split}&S_{4}:=\bar{\textbf{x}}(t)-S_{3}=\\\
&=\sum_{w\in\Sigma^{*},\sigma_{1},\in\Sigma}\left(\sum_{s_{1},s_{2}\in\Sigma^{*},\sigma_{1}\in\Sigma,w=s_{1}\sigma
s_{2}}F_{s_{2}}B_{\sigma_{1}}CA_{s_{1}}K_{\sigma}\mathbf{z}_{\sigma
w}^{\mathbf{v}}(t)\right)\end{split}$ (25)

is absolutely convergent in the mean-square sense. It is known that for
absolutely convergent series of elements Hilbert-spaces can be rearranged
while preserving convergence, hence, by using (22)

$\begin{split}&S_{4}=\sum_{s_{1},s_{2}\in\Sigma^{*}\sigma_{1},\sigma\in\Sigma}\sqrt{p_{\sigma
s_{1}\sigma_{1}s_{2}}}F_{s_{2}}B_{\sigma_{1}}CA_{s_{1}}K_{\sigma}\mathbf{z}_{\sigma
w}^{\mathbf{v}}(t)=\\\
&\sum_{s_{2}\in\Sigma^{*},\sigma_{1}\in\Sigma}F_{s_{2}}B_{\sigma_{1}}\underbrace{\left(\sum_{s_{2}\in\Sigma^{*},\sigma\in\Sigma}CA_{s_{2}}K_{\sigma}\mathbf{z}_{\sigma
s_{2}\sigma_{1}s_{1}}^{\mathbf{v}}(t)\right)}_{\mathbf{z}^{\textbf{y}}_{\sigma_{1}s_{2}}(t)-D\mathbf{z}_{\sigma_{1}s_{2}}^{\mathbf{v}}(t)}\\\
&\sum_{s_{2}\in\Sigma^{*},\sigma_{1}\in\Sigma}F_{s_{2}}B_{\sigma_{1}}\mathbf{z}_{\sigma_{1}s_{1}}^{\textbf{y}}(t)-F_{s_{2}}B_{\sigma_{1}}D\mathbf{z}_{\sigma_{1}s_{2}}^{\mathbf{v}}(t)=\\\
&\sum_{s_{2}\in\Sigma^{*},\sigma_{1}\in\Sigma}F_{s_{2}}B_{\sigma_{1}}\mathbf{z}_{\sigma_{1}s_{1}}^{\textbf{y}}(t)-S_{3}\end{split}$
(26)

is absolutely convergent, and hence
$\mathbf{r}(t)=\bar{\textbf{x}}(t)=S_{4}+S_{3}$ is absolutely convergent.
Finally notice that $\mathbf{r}(t)=\bar{\textbf{x}}(t)$ is the component of
the unique state process $\tilde{\textbf{x}}(t)$ of the asLPV-SSA
$(\\{\tilde{F}_{i},\tilde{B}_{i}\\}_{i=1}^{n_{\mu}},\tilde{C},I,\mathbf{v})$
and hence $\tilde{\textbf{x}}$ is ZMWSII and hence so is $\mathbf{r}$.
Finally, from $\tilde{\textbf{x}}$ being the state process of
$(\\{\tilde{F}_{i},\tilde{B}_{i}\\}_{i=1}^{n_{\mu}},\tilde{C},I,\mathbf{v})$
it follows that $\mathbf{r}(t)$ satisfies (20). Moreover, if
$\bar{\mathbf{r}}(t)$ is a ZMWSII process which satisfies (20) such that
$\begin{bmatrix}\bar{\mathbf{r}}^{T}&\textbf{y}^{T}\end{bmatrix}^{T}$ is
ZMWSII and the components of $\bar{\mathbf{r}}(t)$ belong to
$\mathcal{H}_{t}^{\textbf{y}}$. Note that since $\mathcal{S}$ is a minimal
asLPV-SSA realization o$(\textbf{y},\bm{\mu})$ in innovation form,
$\mathbf{v}(t)=\textbf{y}(t)-C\textbf{x}(t)$ and the elements of $\mathbf{v}$
belong to the Hilbert-space $\mathcal{H}^{\textbf{y}}_{t}$. Notice that from
(22) it follows that $\mathcal{H}^{\textbf{y}}_{t}$ is a subspace of the
Hilbert-space $\mathcal{H}_{t}^{\mathbf{v}}$ generated by
$\\{\mathbf{z}^{\mathbf{v}}_{w}(t)\\}_{w\in\Sigma^{+}}$. That is,
$\mathcal{H}^{\textbf{y}}_{t}=\mathcal{H}_{t}^{\mathbf{v}}$. Then
$\hat{\textbf{x}}=\begin{bmatrix}\bar{\mathbf{r}}^{T}&\textbf{x}\end{bmatrix}^{T}$
is such that $\hat{\textbf{x}}(t)$ satisfies the conditions of [20, Lemma 10]
for $\mathbf{r}=\mathbf{v}$,
$\begin{bmatrix}\hat{\textbf{x}}^{T}&\mathbf{v}^{T}\end{bmatrix}^{T}$ is
ZMWSII. Moreover,
$E[\mathbf{v}(t)(\mathbf{z}_{w}^{\bar{\mathbf{r}}}(t))^{T}]=0$ as
$\mathbf{v}(t)$ is orthogonal to
$\mathcal{H}_{t}^{\mathbf{v}}=\mathcal{H}_{t}^{\textbf{y}}$ and the components
of $\mathbf{z}_{w}^{\bar{\mathbf{r}}}(t)$ belongs
$\mathcal{H}_{t}^{\textbf{y}}$ for all $w\in\Sigma^{*}$. Finally
$\hat{\textbf{x}}(t+1)=\sum_{i=1}^{n_{\mu}}(\tilde{F}_{i}\hat{\textbf{x}}(t)+\tilde{B}_{i}\mathbf{v}(t))\bm{\mu}_{i}(t)$.
Hence, $\hat{\textbf{x}}(t)$ is a state process of
$(\\{\tilde{F}_{i},\tilde{B}_{i}\\}_{i=1}^{n_{\mu}},\tilde{C},I,\mathbf{v})$
and hence it is unique and equals $\tilde{\textbf{x}}$ and
$\bar{\mathbf{r}}=\mathbf{r}$.

∎

###### Proof:

From Lemma 6 it follows that
$\tilde{\textbf{x}}(t)=\sum_{w\in\Sigma^{*},\sigma\in\Sigma}F_{w}K_{\sigma}\mathbf{z}_{\sigma
w}^{\textbf{y}}(t)$, where $F_{i}=(A_{i}-K_{i}C)$, $i=1,\ldots,n_{\mu}$ is
absolutely convergent in the mean-square sense, hence
$\bar{\textbf{x}}(t)-\tilde{\textbf{x}}(t)$ converges to zero in the mean
square sense. It remains to show that $\tilde{\textbf{x}}(t)=\textbf{x}(t)$.
To this end, notice that
$\begin{bmatrix}\tilde{\textbf{x}}^{T}&\textbf{y}^{T}\end{bmatrix}^{T}$ is
ZMWSII by Lemma 6, the elements of $\tilde{\textbf{x}}$ belong to
$\mathcal{H}^{\textbf{y}}_{t}$ and it satisfies

$\begin{split}&\tilde{\textbf{x}}(t+1)=\sum_{i=1}^{n_{\mu}}((A_{i}-K_{i}C)\tilde{\textbf{x}}(t)+K_{i}\textbf{y}(t))\bm{\mu}_{i}(t)\end{split}$

At the same time,
$\begin{bmatrix}\textbf{x}^{T}&\textbf{y}^{T}\end{bmatrix}^{T}$ is ZMWSII.
Note that the components of x belong to $\mathcal{H}^{\textbf{y}}$: as it was
pointed out in the proof of Lemma 6, $\mathcal{H}^{\textbf{y}}_{t}$ equals the
Hilbert-space generated by
$\\{\mathbf{z}_{w}^{\textbf{e}}(t)\\}_{w\in\Sigma^{+}}$ and the components of
$\textbf{x}(t)$ belong to the latter Hilbert-space. Hence, $\textbf{x}(t)$
satisfies (13), hence by Lemma 6 $\textbf{x}(t)=\bar{\textbf{x}}(t)$. ∎

###### Proof:

Note that we can write $\mathbf{v}(t)=\textbf{y}(t)-C\textbf{x}(t)$ and hence
the first equation of (1) holds, i.e.,
$\textbf{x}(t+1)=\sum_{i=1}^{n_{\mu}}(A_{i}-K_{i}C)\textbf{x}(t)+K_{i}\textbf{y}(t))\bm{\mu}_{i}(t)$.
Since the matrix
$\sum_{i=1}^{n_{\mu}}\bm{\mu}_{i}(A_{i}-K_{i}C)\otimes(A_{i}-K_{i}C)$ is
stable, then by repeating the steps of the proof of [20, Lemma1] it can be
shown that
$\textbf{x}(t)=\sum_{w\in\Sigma^{*},i\in\Sigma}\sqrt{p_{iw}}A_{w}K_{i}\mathbf{z}^{\textbf{y}}_{iw}(t)$,
and hence the elements of $\textbf{x}(t)$ belong to the Hilber-space generated
by $\\{\mathbf{z}^{\textbf{y}}_{w}(t)\\}_{w\in\Sigma^{+}}$. Note that
$E[\mathbf{v}(t)\mid\\{\mathbf{z}^{\textbf{y}}_{w}(t)\\}_{w\in\Sigma^{+}}]=0$,
see the proof of [20, eq. (37), proof of Theorem 4], hence,
$E[\textbf{y}(t)\mid\\{\mathbf{z}^{\textbf{y}}_{w}(t)\\}_{w\in\Sigma^{+}}]=C\textbf{x}(t)$
and therefore $\textbf{e}(t)=\mathbf{v}(t)$. ∎

###### Proof:

Note that $\mathcal{S}$ is minimal if and only if the observability and
reachability matrices satisfy the following rank conditions
$\mathrm{rank}~{}\mathscr{O}_{n-1}(\mathcal{S})=n$ and
$\mathrm{rank}~{}\mathscr{R}_{n-1}(\mathcal{S})=n$ . Note that the rows of the
extended observability matrix $\mathcal{O}_{n-1}$ of the associated dLPV-SSA
$\mathcal{D}_{\mathcal{S}}$ are either zero or they coincide with the rows of
the observability matrix $\mathscr{O}_{n-1}(\mathcal{S})$, i.e.,
$\mathrm{rank}~{}\mathscr{O}_{n-1}(\mathcal{S})=\mathrm{rank}~{}\mathcal{O}_{n-1}$.
That is, $\mathcal{S}$ satisfies the observability rank condition if and only
if the dLPV-SSA $\mathcal{D}_{\mathcal{S}}$ is observable. We will show that
$\bm{Im}\mathscr{R}_{n-1}(\mathcal{S})=\bm{Im}\mathcal{R}_{n-1}$, where
$\mathcal{R}_{n-1}$ is the extended controllability matrix of the dLPV-SSA
$\mathcal{D}_{\mathcal{S}}$. From this, it follows that $\mathcal{S}$
satisfies the reachability rank condition if and only if
$\mathcal{L}_{\mathcal{S}}$ is span-reachable.

Now we will show that
$\bm{Im}~{}\mathscr{R}_{n-1}(\mathcal{S})=\bm{Im}~{}\mathcal{R}_{n-1}$. To
this end, we recall that $\textbf{x}(t)$ belongs to the linear space generated
by the columns of $A_{w}K_{\sigma}$, $w\in\Sigma^{*}$, $\sigma\in\Sigma$.
Since $B_{\sigma}=E[\textbf{x}(t)(\mathbf{z}^{\textbf{y}}_{\sigma}(t))^{T}]$,
it then follows that the columns of $B_{\sigma}$ also belong to the linear
space generated by the columns of $A_{w}K_{\sigma}$,
$w\in\Sigma^{*},\sigma\in\Sigma$. Therefore, the columns of $A_{v}B_{\sigma}$,
$v\in\Sigma^{*},\sigma\in\Sigma$ also belong to the linear space generated by
the columns of $A_{w}K_{\sigma}$, $w\in\Sigma^{*},\sigma\in\Sigma$. In turn,
it is easy to see that latter subspace equals $\bm{Im}~{}\mathcal{R}_{n-1}$.
That is, $\bm{Im}~{}A_{v}B_{\sigma}$ is a subspace of
$\bm{Im}~{}\mathcal{R}_{n-1}$, and therefore
$\bm{Im}~{}\mathscr{R}_{n-1}(\mathcal{S})\subseteq\bm{Im}~{}\mathcal{R}_{n-1}$.
Conversely, from [20, eq. (37), proof of Theorem 4] it follows that
$E[\textbf{x}(t)\mathbf{z}^{\textbf{y}}_{\sigma
v}(t)]=\sqrt{p_{w}}A_{v}B_{\sigma}$, i.e., for every $w\in\Sigma^{+}$, the
columns $E[\textbf{x}(t)(\mathbf{z}^{\textbf{y}}_{w}(t))^{T}]$ belong to the
space generated by $A_{v}B_{\sigma}$, $\sigma\in\Sigma$, $v\in\Sigma^{*}$.
Notice that by [17, Theorem 2 and Remark 1] applied to the LSS
$\Sigma_{\mathcal{S}}$, the latter space equals
$\bm{Im}\mathscr{R}_{n-1}(\mathcal{S})$. Since the elements of
$\mathbf{z}^{\textbf{e}}_{\sigma}(t)$ are limits of finite linear combinations
of the rows of $\\{\mathbf{z}^{\textbf{y}}_{w}(t)\\}_{w\in\Sigma^{+}}$, it
then follows that the columns of
$E[\textbf{x}(t)(\mathbf{z}^{\textbf{e}}(t))^{T}]$ are the limits of finite
linear combinations of columns of
$E[\textbf{x}(t)(\mathbf{z}^{\textbf{y}}_{w}(t))^{T}]$, $w\in\Sigma^{+}$, and
hence the columns of $E[\textbf{x}(t)(\mathbf{z}^{\textbf{y}}_{w}(t))^{T}]$,
$w\in\Sigma^{+}$ also belong to $\bm{Im}\mathscr{R}_{n-1}(\mathcal{S})$. From
[20, Proof of Theorem 4] it follows that
$K_{\sigma}Q_{\sigma}=E[\textbf{x}(t)(\mathbf{z}^{\textbf{e}}_{\sigma}(t))^{T}]$,
where $Q_{\sigma}=E[\textbf{e}(t)\textbf{e}^{T}(t)\bm{\mu}_{\sigma}^{2}(t)]$,
and hence the columns of $K_{\sigma}Q_{\sigma}$ belong to
$\bm{Im}\mathscr{R}_{n-1}(\mathcal{S})$. Since $Q_{\sigma}$ is non-singular,
it then follows that the columns of $K_{\sigma}$ belong to
$\bm{Im}\mathscr{R}_{n-1}(\mathcal{S})$. Since
$\bm{Im}\mathscr{R}_{n-1}(\mathcal{S})$ is $A_{\sigma}$-invariant for all
$\sigma\in\Sigma$ and $A_{0}=0$, it then follows that
$\bm{Im}A_{v}K_{\sigma}\subseteq\bm{Im}\mathscr{R}_{n-1}$ for all
$v\in\mathbb{I}_{0,n_{p}}^{*}$, $\sigma\in\mathbb{I}_{0,n_{p}}$, and thus
$\bm{Im}\mathcal{R}_{n-1}\subseteq\bm{Im}\mathscr{R}_{n-1}(\mathcal{S})$. ∎

###  -C Proof of Lemma 2

######  Definition 7 (Necessary Maps)

Define, for the sequel, the following maps:

$\mathscr{Y}:(\mathbf{u},\bm{\mu})\longmapsto\mathbf{y}=\mathscr{Y}(\mathbf{u},\bm{\mu})$
$\mathscr{F}:\mathcal{D}\longmapsto\mathscr{F}(\mathcal{D})$
$\mathscr{D}:\mathcal{D}\longmapsto\mathscr{D}(\mathcal{D})$

where $\mathscr{Y}$ is the input-output map,
$\mathcal{D}=(\\{A_{i},K_{i}\\}_{i=1}^{n_{\mu}},C,I)$ is a dLPV-SSA and
$\mathscr{F}(\mathcal{D})$ and $\mathscr{D}(\mathcal{D})$ are two transformed
dLPV-SSAs with
$\mathscr{F}(\mathcal{D})=(\\{A_{i}-K_{i}C,K_{i}\\}_{i=1}^{n_{\mu}},C,I)$ and
$\mathscr{D}(\mathcal{D})=(\\{A_{i},K_{i}\\}_{i=1}^{n_{\mu}},-C,I)$

Note that, if $\mathcal{X}$ is a dLPV-SSA representation, then it is clear
that
$\mathscr{D}(\mathscr{F}(\mathscr{D}(\mathscr{F}(\mathcal{X}))))=\mathcal{X}$.

###### Lemma 7

Consider the following dLPV-SSA:
$\mathcal{D}=(\\{A_{i},K_{i}\\}_{i=1}^{n_{\mu}},C,I)$ where it is a
realization of the sub-Markov parameters $M_{\mathcal{D}}=CA_{w}K_{\sigma}$
where $w\in\Sigma^{*}$, $\sigma\in\Sigma$. Subsequently,
$\mathscr{F}(\mathcal{D})$ is a realization of
$M_{\mathscr{F}(\mathcal{D})}=C\tilde{A}_{w}K_{\sigma}$ with
$\tilde{A}_{i}=A_{i}K_{i}C$. If a dLPV-SSA
$\mathcal{D}^{\prime}=(\\{A^{\prime}_{i},K^{\prime}_{i}\\}_{i=1}^{n_{\mu}},C^{\prime},I)$
is a realization of $M_{\mathcal{D}}$, then the
$\mathscr{F}(\mathcal{D^{\prime}})$ dLPV-SSA is a realization of
$M_{\mathscr{F}(\mathcal{D})}$.

###### Proof:

Consider that $(\textbf{x},\mathbf{u},\bm{\mu},\textbf{y})$ is a solution of
the system $\mathcal{D}$ it then follows that

$\displaystyle\textbf{x}(t+1)=\sum_{i=1}^{n_{\mu}}(A_{i}\textbf{x}(t)+K_{i}\mathbf{u}(t))\bm{\mu}_{i}(t)$
$\displaystyle\textbf{y}(t)=C\textbf{x}(t)$

It can then be modified to

$\textbf{x}(t+1)=\sum_{i=1}^{n_{\mu}}(A_{i}\textbf{x}(t)+K_{i}(C\textbf{x}(t)-C\textbf{x}(t)+\mathbf{u}(t)))\bm{\mu}_{i}(t)$
$\textbf{x}(t+1)=\sum_{i=1}^{n_{\mu}}((A_{i}-K_{i}C)\textbf{x}(t)+K_{i}(\mathbf{y}(t)+\mathbf{u}(t)))\bm{\mu}_{i}(t)$

Subsequently, it is safe to say that
$(\textbf{x},\mathbf{v},\bm{\mu},\textbf{y})$ is a solution of the system
$\mathscr{F}(\mathcal{D})$ with $\mathbf{v}=\mathbf{y}+\mathbf{u}$. It can
also be shown, with the same demonstration, that if
$(\textbf{x},\mathbf{v},\bm{\mu},\mathbf{y})$ is a solution of
$\mathscr{F}(\mathcal{D})$, then $(\textbf{x},\mathbf{u},\bm{\mu},\mathbf{y})$
is a solution of $\mathcal{D}$, with $\mathbf{u}=\mathbf{v}-\mathbf{y}$. That
is, it can be said that
$\mathscr{Y}_{\mathcal{D},0}(\mathbf{u},\bm{\mu})=\mathbf{y}=\mathscr{Y}_{\mathscr{F}(\mathcal{D}),0}(\mathbf{v},\bm{\mu})$.
Consider now that $(\tilde{\mathbf{x}},\mathbf{u},\bm{\mu},\bar{\mathbf{y}})$
is a solution of $\mathcal{D}^{\prime}$, it is safe to say that
$(\tilde{\mathbf{x}},\bar{\mathbf{v}},\bm{\mu},\bar{\mathbf{y}})$ is a
solution of $\mathscr{F}(\mathcal{D}^{\prime})$, with
$\bar{\mathbf{v}}=u+\bar{\mathbf{y}})$. Lemma 7 assumes that $\mathcal{D}$ and
$\mathcal{D}^{\prime}$ are both realization of the sub-Markov parameters
$M_{\mathcal{D}}$. This leads to the following conclusion:

$\mathscr{Y}_{\mathscr{F}(\mathcal{D}),0}(\mathbf{v},\bm{\mu})=\mathscr{Y}_{\mathcal{D},0}(\mathbf{u},\bm{\mu})=\mathbf{y}=\bar{\mathbf{y}}=\mathscr{Y}_{\mathcal{D}^{\prime},0}(\mathbf{u},\bm{\mu})=\mathscr{Y}_{\mathscr{F}(\mathcal{D}^{\prime}),0}(\bar{\mathbf{v}},\bm{\mu})$

In other words, $\mathscr{F}(\mathcal{D}^{\prime})$ is a realization of
$M_{\mathscr{F}(\mathcal{D})}$. ∎

###### Corollary 2

An dLPV-SSA representation $\Sigma$ is a minimal realization of $M_{\Sigma}$
if and only if $\mathscr{F}(\Sigma)$ is a minimal realization of
$M_{\mathscr{F}(\Sigma)}$.

###### Proof:

Suppose that $\Sigma$ is minimal and $\mathscr{F}(\Sigma)$ is not. Which means
that, there exists a minimal dLPV-SSA representation $\Sigma^{m}$ such that
$dim\Sigma^{m}<dim\mathscr{F}(\Sigma)=dim\Sigma$. Define a dLPV-SSA
representation
$\hat{\Sigma}^{m}:=\mathscr{D}(\mathscr{F}(\mathscr{D}(\Sigma^{m})))$. It is
known that $\Sigma^{m}$ and $\mathscr{F}(\Sigma)$ are both a realization of
$M_{\Sigma^{m}}=M_{\mathscr{F}(\Sigma)}$, then, by applying
$\mathscr{D}(\mathscr{F}(\mathscr{D}(\cdot)))$ on both sides, we will get the
following:

$M_{\mathscr{D}(\mathscr{F}(\mathscr{D}(\Sigma^{m})))}=M_{\mathscr{D}(\mathscr{F}(\mathscr{D}(\mathscr{F}(\Sigma))))}\Longrightarrow
M_{\Sigma}=M_{\hat{\Sigma}^{m}}$

Which means that $\hat{\Sigma}^{m}$ is a realization of $M_{\Sigma}$ and
$dim\Sigma=dim\mathscr{F}(\Sigma)>dim\Sigma^{m}=dim\hat{\Sigma}^{m}$, which
implies that $\Sigma$ is not minimal, and results to a contradiction. ∎

###### Proof:

In order to prove Lemma 2, two steps are needed: (1) proving that
$\mathcal{S}_{m}$ is stably invertable, and (2) proving that $\mathcal{S}_{m}$
is a realization of $(\mathbf{y},\bm{\mu})$ in innovation form.
(1) From Lemma 7 it follows that $\mathscr{F}(\mathcal{D})$ and
$\mathscr{F}(\mathcal{D}_{m})$ are both a realization of
$M_{\mathcal{F}(\mathcal{D})}$, this is due to the fact that $\mathcal{D}$ and
$\mathcal{D}_{m}$ are both a realization of $M_{\mathcal{D}}$. Recall from
[21, Lemma 6]555The termenology of [21] refers to the switch systems
representations not the LPV-SSA representations that if the matrix
$\sum_{i=1}^{n_{\mu}}(\sqrt{p_{i}}\mathscr{A}_{i}^{T})^{T}\otimes(\sqrt{p_{i}}\mathscr{A}_{i}^{T})^{T}=\sum_{i=1}^{n_{\mu}}p_{i}\mathscr{A}_{i}\otimes\mathscr{A}_{i}$
is stable then the matrix
$\sum_{i=1}^{n_{\mu}}p_{i}\mathscr{A}_{i}^{m}\otimes\mathscr{A}_{i}^{m}$ is
also stable, where $\mathscr{A}_{i}$ are the state matrices of an dLPV-SSA
$\Sigma_{1}$ and $\mathscr{A}_{i}^{m}$ are the state matrices of the minimized
dLPV-SSA $\Sigma_{1}^{m}$. This said, recall that Algorithm 2 assumes that its
input
$\mathcal{S}=(\\{A_{\sigma},K_{\sigma}\\}_{\sigma=1}^{n_{\mu}},C,I_{n_{y}})$
is stably invertable. It follows that the matrix
$\sum_{i=1}^{n_{\mu}}p_{i}(A_{i}-K_{i}C)\otimes(A_{i}-K_{i}C)$ is stable,
where $(A_{i}-K_{i}C)$ are the state matrices of $\mathscr{F}(\mathcal{D})$.
Note that $\mathscr{F}(\mathcal{D}_{m})$ is minimal due to the minimality of
$\mathcal{D}_{m}$. It follows, from [21, Lemma 6], that the matrix
$\sum_{i=1}^{n_{\mu}}p_{i}(A_{i}^{m}-K_{i}^{m}C^{m})\otimes(A_{i}^{m}-K_{i}^{m}C^{m})$
is stable, where $(A_{i}^{m}-K_{i}^{m}C^{m})$ are the state matrices of
$\mathscr{F}(\mathcal{D}_{m})$. Therefore, the algorithm’s output
$\mathcal{S}_{m}$ is indeed stably invertable.
(2) Since $\mathcal{S}$ is a realization of $(\mathbf{y},\bm{\mu})$ in
innovation form, then there exists a process $\mathbf{x}$ such that (1) holds
with $\mathbf{v}=\mathbf{y}$. This said, let us apply the linear
transformation $T$ as described in [19, Corollary 1] to $\mathcal{S}$. Let
$\hat{\mathbf{x}}(t)=T\mathbf{x}(t)$, $\hat{A}_{i}=TA_{i}$,
$\hat{K}_{i}=TK_{i}$, for $i=1,\ldots,n_{\mu}$, and $\hat{C}=CT^{-1}$. Then
the asLPV-SSA
$\hat{\mathcal{S}}=(\\{\hat{A}_{i},\hat{K}_{i}\\}_{i=1}^{n_{\mu}},\hat{C},I,\mathbf{e})$
is also a realization of $(\mathbf{y},\bm{\mu})$ in innovation form. Now, in
order to get the Kalman decomposition, recall form [19, Corollary 1] that:

$\hat{A}_{i}=\begin{bmatrix}A_{i}^{m}&0&A_{i}^{\prime}\\\
A_{i}^{\prime\prime}&A_{i}^{\prime\prime\prime}&A_{i}^{\prime\prime\prime\prime}\\\
0&0&A_{i}^{uc}\\\ \end{bmatrix},\quad\hat{K}_{i}=\begin{bmatrix}K_{i}^{m}\\\
K_{i}^{\prime}\\\ 0\\\
\end{bmatrix},\quad\hat{C}=\begin{bmatrix}C^{m}&0&C^{\prime}\\\ \end{bmatrix}$

for $i=1,\ldots,n_{\mu}$, and for suitable block matrices. In particular, for
all $w\in\Sigma^{*}$ we found:

$\hat{A}_{w}=\begin{bmatrix}A_{w}^{m}&0&\ast\\\
\ast&A_{w}^{\prime\prime\prime}&\ast\\\ 0&0&A_{w}^{uc}\\\ \end{bmatrix}$ (27)

where $\ast$ refers to a block matrix. Indeed, it can be shown that for
$w=\epsilon$, $\hat{A}_{w}=I$. Now if (27) holds for $w=v$, then for
$w=v\sigma$:

$\hat{A}_{w}=\begin{bmatrix}A_{\sigma}^{m}&0&\ast\\\
\ast&A_{\sigma}^{\prime\prime\prime}&\ast\\\ 0&0&A_{\sigma}^{uc}\\\
\end{bmatrix}\begin{bmatrix}A_{v}^{m}&0&\ast\\\
\ast&A_{v}^{\prime\prime\prime}&\ast\\\ 0&0&A_{v}^{uc}\\\
\end{bmatrix}=\begin{bmatrix}A_{\sigma v}^{m}&0&\ast\\\ \ast&A_{\sigma
v}^{\prime\prime\prime}&\ast\\\ 0&0&A_{\sigma v}^{uc}\\\ \end{bmatrix}$

Hence, by induction of the length of $w$, (27) holds. It the follows that:

$\hat{A}_{w}\hat{K}_{\sigma}=\begin{bmatrix}A_{w}^{m}K_{\sigma}^{m}\\\ \ast\\\
0\\\ \end{bmatrix}$

Consider the following decomposition:

$\hat{\mathbf{x}}(t)=T\mathbf{x}(t)=\begin{bmatrix}\mathbf{x}^{m}(t)\\\
\mathbf{x}^{c}(t)\\\ \mathbf{x}^{uc}(t)\\\ \end{bmatrix}$

From Lemma 5, $\hat{\mathbf{x}}(t)$ can be expressed as:

$\hat{\textbf{x}}(t)=\sum_{w\in\Sigma^{*},\sigma\in\Sigma}\sqrt{p_{\sigma}w}\hat{A}_{w}\hat{K}_{\sigma}\mathbf{z}_{\sigma
w}^{\mathbf{e}}(t)$

It follows that:

$\hat{\textbf{x}}(t)=\sum_{w\in\Sigma^{*},\sigma\in\Sigma}\sqrt{p_{\sigma}w}\begin{bmatrix}A_{w}^{m}K^{m}_{\sigma}\mathbf{z}_{\sigma
w}^{\mathbf{e}}(t)\\\ \ast\\\ 0\\\
\end{bmatrix}=\begin{bmatrix}\sum_{w\in\Sigma^{*},\sigma\in\Sigma}\sqrt{p_{\sigma}w}A_{w}^{m}K^{m}_{\sigma}\mathbf{z}_{\sigma
w}^{\mathbf{e}}(t)\\\ \ast\\\ 0\\\ \end{bmatrix}$ (28)

From (28) it follows that
$\mathbf{x}^{m}(t)=\sum_{w\in\Sigma^{*},\sigma\in\Sigma}\sqrt{p_{\sigma}w}A_{w}^{m}K^{m}_{\sigma}\mathbf{z}_{\sigma
w}^{\mathbf{e}}(t)$. Then from [20, Lemma 3], it follows that
$\mathbf{x}^{m}(t)$ is the state process of $\mathcal{S}_{m}$. Finally, notice
that, as $\hat{\mathcal{S}}$ is a realization of $(\mathbf{y},\bm{\mu})$,
$\mathbf{y}(t)=C\mathbf{x}(t)+\mathbf{e}(t)=\hat{C}T\mathbf{x}(t)+\mathbf{e}(t)=\hat{C}\hat{\mathbf{x}}(t)+\mathbf{e}(t)$.
The output can then be expressed as:

$\mathbf{y}(t)=\begin{bmatrix}C^{m}&0&C^{\prime}\end{bmatrix}\begin{bmatrix}\mathbf{x}^{m}(t)\\\
\mathbf{x}^{c}(t)\\\ \mathbf{x}^{uc}(t)\\\ \end{bmatrix}+\mathbf{e}(t)$ (29)

From (28) it follows that $\mathbf{x}^{uc}(t)=0$. Hence, from (29) it follows
that $y(t)=C^{m}\mathbf{x}^{m}(t)+e(t)$. Since $\mathbf{x}^{m}$ is the state
process of $\mathcal{S}_{m}$, it then follows that $\mathcal{S}_{m}$ is a
realization of $(\mathbf{y},\bm{\mu})$. In addition, because $\mathcal{S}_{m}$
is stably invertable, as proven in (1), then $\mathcal{S}_{m}$ is a
realization of $(\mathbf{y},\bm{\mu})$ in innovation form. ∎