# Minimal realizations of input-output behaviors by LPV state-space
representations with affine dependency

Mihály Petreczky, Roland Tóth, and Guillaume Mercère Mihály Petreczky is with
Univ. Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France.
<EMAIL_ADDRESS>Guillaume Mercère is with the University of
Poitiers, Laboratoire d’Informatique et d’Automatique pour les Systèmes,
<EMAIL_ADDRESS>Roland Tóth is with the Control Systems
Group, Department of Electrical Engineering, Eindhoven University of
Technology, Eindhoven, The Netherlands and the Systems and Control Laboratory,
Institute for Computer Science and Control, Budapest, Hungary<EMAIL_ADDRESS>
###### Abstract

The paper makes the first steps towards a behavioral theory of LPV state-space
representations with an affine dependency on scheduling, by characterizing
minimality of such state-space representations. It is shown that minimality is
equivalent to observability, and that minimal realizations of the same
behavior are isomorphic. Finally, we establish a formal relationship between
minimality of LPV state-space representations with an affine dependence on
scheduling and minimality of LPV state-space representations with a dynamic
and meromorphic dependence on scheduling.

##  I Introduction

_Linear parameter-varying_ (LPV) [21, 22] systems represent a widely used
system class which is more general than _linear time-invariant_ (LTI) systems
and which can capture _nonlinear_ and _time-varying_ behavior. LPV systems are
modeled by linear time-varying difference or differential equations, where the
time varying coefficients are functions of a time-varying _scheduling_ signal.
LPV systems are widely used in control ([16, 11, 1, 17]) and in system
identification, ([10, 2, 5, 8, 23, 20, 24]).

Despite these advances, there are still gaps in theory of LPV systems, in
particular in their _realization theory_. Realization theory aims at
characterizing the relationship between the input-output behavior and certain
classes of state-space representations (linear time-invariant, bilinear,
etc.), see [9, 18]. Since realization theory is used in system identification
and model reduction, and data-driven control [25], filling this gap is
important for further development of those disciplines for LPV systems.

Prior work on realization theory and motivation Realization theory oi LPV-SS
was first addressed [21, 22], where behavioral theory was used to clarify
realization theory, the concepts of minimality and equivalence for so called
LPV state-space representation (_LPV-SS_ for short) with the so called
_meromorphic_ and _dynamical dependency_ of the model parameters on the
scheduling variable. We will refer to the latter class of LPV-SSs as
_meromorphic LPV-SS_. Recall from [21, 22] that meromorphic and dynamic
dependency means that the entries of the system matrices are fractions of
analytic functions of the current value and past values of the scheduling
variables (in discrete-time), or the high-order derivatives of the scheduling
variables (in continuous-time). A major drawback of meromorphic LPV-SSs is
that for practical applications it is often preferable to use LPV-SSs with a
static and affine dependency on the scheduling variable (_LPV-SSA_ for short),
i.e., LPV-SS whose system parameters are affine functions of the instantaneous
value of the scheduling variable. However, whereas LPV-SSAs are a subclass of
meromorphic LPV-SSs, the system-theoretic transformations (passing from input-
output to state-space representation, transforming a state-space
representation to a minimal one, etc.) of [21, 22] result in meromorphic LPV-
SSs, even if applied to LPV-SSAs, see [7].

In [4], realizability of LPV input-output equations by LPV-SSs with a general
(non affine) dependence on the scheduling variable was investigated. However,
it is not clear that all behaviors of interest admit the LPV input-output
representations from [4], and [4] does not address minimality.

In [14] Kalman-style realization theory for LPV-SSAs was developed. A drawback
of [14] lies in the use of input-output functions, which captures the input-
output behavior only from a certain fixed initial state. In contrast, for
control synthesis, the initial state is not fixed.

That is, [21, 22, 4] do not address behavioral realization theory for LPV-
SSAs.

Contribution In this paper we make a first step towards a behavioral approach
for LPV-SSAs. Similarly to [21], we use the concept of manifest behavior from
[15] to formalize the input-output behavior of LPV-SSAs. Then, under suitable
assumptions, the following counterparts of the well-known results for LTI
behaviors [15] hold:

  * •

A LPV-SSA is a minimal realization of a given behavior if and only if it is
observable, and all minimal realization of the same manifest behavior are
related by a linear (constant) isomorphism.

  * •

A behavior is controllable, if and only if its minimal LPV-SSAs is span-
reachable from the zero initial state.

We also formulate a computionally effective minimization procedure for LPV-
SSAs. Furthermore, we show that under some assumptions, a minimal LPV-SSA
realizatio of a behavior is also minimal if viewed as a meromorphic LPV-SS
[21]. The latter is interesting, as in contrast to meromorphic LPV-SSs, there
are computationally effective algorithms for minimization and checking
minimality of LPV-SSAs.

Outline In Section II, we present the necessary background on LPV-SSA
representations and then formalize several system theoretic concepts. In
Section III, the main results are introduced, Section IV gathers the proofs.

##  II Preliminaries

Let $\mathbb{T}=\mathbb{R}_{0}^{+}=[0,+\infty)$ be the time axis in the
_continuous-time_ (CT) case and $\mathbb{T}=\mathbb{N}$ in the _discrete-time_
(DT) case. Let $\xi$ be the differentiation operator $\frac{d}{dt}$ (in CT)
and the forward time-shift operator $q$ (in DT), _i.e._ , if
$z:\mathbb{T}\rightarrow\mathbb{R}^{n}$, then $(\xi z)(t)=\frac{d}{dt}z(t)$,
if CT, and $(\xi z)(t)=z(t+1)$, if DT.

Define a LPV state-space representations with _affine_ dependence on the
_scheduling variable_ (_LPV-SSA_) as

$\Sigma\ \left\\{\begin{array}[]{lcl}\xi x(t)&=&A(p(t))x(t)+B(p(t))u(t),\\\
y(t)&=&C(p(t))x(t)+D(p(t))u(t),\end{array}\right.$ (1)

where $x:\mathbb{T}\rightarrow\mathbb{R}^{n_{\mathrm{x}}}$ is the state
trajectory, $y:\mathbb{T}\rightarrow\mathbb{R}^{n_{\mathrm{y}}}$ is the
(measured) output trajectory,
$u:\mathbb{T}\rightarrow\mathbb{R}^{n_{\mathrm{u}}}$ is the (control) input
signal and
$p:\mathbb{T}\rightarrow\mathbb{P}\subseteq\mathbb{R}^{n_{\mathrm{p}}}$ is the
so called _scheduling signal_ of the system represented by $\Sigma$.
Moreover, $A,B,C,D$ are matrix valued affine functions defined on
$\mathbb{P}$, i.e. there exists matrices
$A_{i}\in\mathbb{R}^{n_{\mathrm{x}}\times n_{\mathrm{x}}}$,
$B_{i}\in\mathbb{R}^{n_{\mathrm{x}}\times n_{\mathrm{u}}}$,
$C_{i}\in\mathbb{R}^{n_{\mathrm{y}}\times n_{\mathrm{x}}}$ and
$D_{i}\in\mathbb{R}^{n_{\mathrm{y}}\times n_{\mathrm{u}}}$ for all
$i=0,1,\ldots,n_{\mathrm{p}}$, such that

$\begin{split}A(\mathbf{p})=A_{0}+\sum_{i=1}^{n_{\mathrm{p}}}A_{i}\mathbf{p}_{i},\leavevmode\nobreak\
B(\mathbf{p})=B_{0}+\sum_{i=1}^{n_{\mathrm{p}}}B_{i}\mathbf{p}_{i}\\\
C(\mathbf{p})=C_{0}+\sum_{i=1}^{n_{\mathrm{p}}}C_{i}\mathbf{p}_{i},\leavevmode\nobreak\
D(\mathbf{p})=D_{0}+\sum_{i=1}^{n_{\mathrm{p}}}D_{i}\mathbf{p}_{i}\end{split}$

for every
$\mathbf{p}=[\begin{array}[]{ccc}\mathbf{p}_{1}&\ldots&\mathbf{p}_{n_{\mathrm{p}}}\end{array}]^{\top}\in\mathbb{P}$
111Note that in the sequel we use italic letters to denote scheduling signals,
and boldface letters to denote elements of $\mathbb{P}$..

Note that in LPV systems, the input and scheduling signals play the role of
exogenous inputs. Note that it is often assumed that the scheduling signals
are bounded to ensure desirable properties, e.g., stability, hence in general
$\mathbb{P}\neq\mathbb{R}^{n_{\mathrm{p}}}$.

In the sequel, we use the shorthand notation

$\Sigma=(\mathbb{P},\left\\{A_{i},B_{i},C_{i},D_{i}\right\\}_{i=0}^{n_{\mathrm{p}}})$

to denote a LPV-SSA of the form (1) and use $\dim{(\Sigma)}=n_{\mathrm{x}}$ to
denote its state dimension.

For the purposes of this paper we need to formalize the solution concept for
LPV-SSAs. To this end, we define the sets
$\mathcal{X},\mathcal{Y},\mathcal{U},\mathcal{P}$ of respectively state and
output trajectories, and input and scheduling signals as follows. For a set
$X$ let $X^{\mathbb{N}}$ denote the set of all functions of the form
$f:\mathbb{N}\rightarrow X$. In DT, let
$\mathcal{X}=(\mathbb{R}^{n_{\mathrm{x}}})^{\mathbb{N}},\mathcal{Y}=(\mathbb{R}^{n_{\mathrm{y}}})^{\mathbb{N}},\mathcal{U}=(\mathbb{R}^{n_{\mathrm{u}}})^{\mathbb{N}},\mathcal{P}=\mathbb{P}^{\mathbb{N}}$.
In CT, let us denote by
$\mathcal{C}_{\mathrm{p}}(\mathbb{R}_{0}^{+},\mathbb{R}^{n})$ the set of all
functions of the form $f:\mathbb{R}_{0}^{+}\rightarrow\mathbb{R}^{n}$ which
are piecewise-continuous. In addition let
$\mathcal{C}_{\mathrm{a}}(\mathbb{R}_{0}^{+},\mathbb{R}^{n})$ be the set of
all absolutely continuous functions of the form
$f:\mathbb{R}_{0}^{+}\rightarrow\mathbb{R}^{n}$. Then, in CT let
$\mathcal{X}=\mathcal{C}_{\mathrm{a}}(\mathbb{R}_{0}^{+},\mathbb{R}^{n_{\mathrm{x}}}),\mathcal{Y}=\mathcal{C}_{\mathrm{p}}(\mathbb{R}_{0}^{+},\mathbb{R}^{n_{\mathrm{y}}}),\mathcal{U}=\mathcal{C}_{\mathrm{p}}(\mathbb{R}_{0}^{+},\mathbb{R}^{n_{\mathrm{u}}}),\mathcal{P}=\mathcal{C}_{\mathrm{p}}(\mathbb{R}_{0}^{+},\mathbb{P})$.

By a solution of $\Sigma$ we mean a tuple of trajectories
$(x,y,u,p)\in(\mathcal{X},\mathcal{Y},\mathcal{U},\mathcal{P})$ satisfying (1)
for almost all $t\in\mathbb{T}$ in CT case, and for all $t\in\mathbb{T}$ in
DT.

Notice that without loss of generality, the solution trajectories in CT can be
considered on the half line $\mathbb{R}_{0}^{+}$ with $t_{\mathrm{o}}=0$. Note
that for any input and scheduling signal
$(u,p)\in\mathcal{U}\times\mathcal{P}$ and any initial state
$x_{\mathrm{o}}\in\mathbb{R}^{n_{\mathrm{x}}}$, there exists a _unique_ pair
$(y,x)\in\mathcal{Y}\times\mathcal{X}$ such that $(x,y,u,p)$ is a solution of
(1) and $x(0)=x_{\mathrm{o}}$, see [21]. Next, inspired by [15, 21], we define
the notion of manifest behavior for LPV-SSAs.

###### Definition 1

A _manifest behavior_ is a subset
$\mathcal{B}\subseteq\mathcal{Y}\times\mathcal{U}\times\mathcal{P}$. The
_manifest behavior_ $\mathcal{B}(\Sigma)$ of a LPV-SSA $\Sigma$ is defined as

$\begin{split}{\color[rgb]{0,0,1}\mathcal{B}}(\Sigma)=\bigl{\\{}(y,u,p)\in\mathcal{Y}\times\mathcal{U}\times\mathcal{P}\mid\exists
x\in\mathcal{X}\\\ \mbox{ s.t. }(x,y,u,p)\mbox{ is a solution of
\eqref{equ:alpvss}}\bigr{\\}}\end{split}$

The LPV-SSA $\Sigma$ is a _realization_ of a manifest behavior
$\mathcal{B}\subseteq\mathcal{Y}\times\mathcal{U}\times\mathcal{P}$, if
$\mathcal{B}=\mathcal{B}(\Sigma)$.

That is, the manifest behavior of a LPV-SSA $\Sigma$ is the set of all tuples
$(y,u,p)$ such that $\Sigma$ generates the output $y$ for some initial state,
if $\Sigma$ is fed input $u$ and scheduling $p$. The corresponding definition
of minimality is then as follows.

###### Definition 2

A LPV-SSA $\Sigma$ is a _minimal realization_ of a manifest behavior
$\mathcal{B}$, if it is a realization of $\mathcal{B}$, and for any LPV-SSA
$\Sigma^{{}^{\prime}}$ such that $\Sigma^{{}^{\prime}}$ is a realization of
$\mathcal{B}$, $\dim\Sigma\leq\dim\Sigma^{{}^{\prime}}$. We say that $\Sigma$
is _minimal_ , if $\Sigma$ is a minimal realization of its own manifest
behavior $\mathcal{B}(\Sigma)$.

Manifest behaviors are a natural formalization of the intuition behind input-
output behaviors of LPV-SSAs. However, input-output behaviors can also be
formalized using input-output functions. The latter was used in [14] for
proposing a Kalman-style realization theory for LPV-SSAs. The principal
definitions are as follows:

###### Definition 3

Let $x_{\mathrm{o}}\in\mathbb{X}$ be an initial state of $\Sigma$. Define the
_input-output (i/o) function_
$\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}:\mathcal{U}\times\mathcal{P}\rightarrow\mathcal{Y}$,
induced by the initial state $x_{\mathrm{o}}$, as follows : for any
$(u,p)\in\mathcal{U}\times\mathcal{P}$,
$y=\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(u,p)$ holds if and only if there
exists a solution $(x,y,u,p)$ of (1) such that $x(0)=x_{\mathrm{o}}$.

###### Definition 4

A LPVS-SSA $\Sigma$ is a _realization_ of an i/o function
$\mathfrak{F}:\mathcal{U}\times\mathcal{P}\rightarrow\mathcal{Y}$ from the
initial state $x_{\mathrm{o}}\in\mathbb{X}$, if $\mathfrak{F}$ coincides with
the i/o function of $\Sigma$ induced by $x_{\mathrm{o}}$, i.e.
$\mathfrak{F}=\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}$. We say $\Sigma$ is a
_realization_ of $\mathfrak{F}$, if it is a realization of $\mathfrak{F}$ from
some initial state.

The LPV-SSA $\Sigma$ is a _minimal realization of $\mathfrak{F}$_ if it is a
realization of $\mathfrak{F}$, and for every LPV-SSA $\Sigma^{\prime}$ which
is a realization of $\mathfrak{F}$,
$\dim{(\Sigma)}\leq\dim{(\Sigma^{\prime})}$.

An drawback of using i/o functions instead of manifest behaviors is that the
former capture the input-output behavior for one choice of initial states.
However, we can account for all initial states by using families of i/o
functions:

###### Definition 5

If $\Sigma$ is a LPV-SSA with the state-space $\mathbb{R}^{n_{\mathrm{x}}}$,
then the set $\mathbb{F}(\Sigma)=\\{\mathcal{Y}_{\Sigma,x_{\mathrm{o}}}\mid
x_{\mathrm{o}}\in\mathbb{R}^{n_{\mathrm{x}}}\\}$ of all i/o functions of
$\Sigma$ induced by some initial state of $\Sigma$ is called the _family of
i/o functions_ of $\Sigma$. A family $\Phi$ of i/o functions of the form
$\mathfrak{F}:\mathcal{U}\times\mathcal{P}\rightarrow\mathcal{Y}$ is
_realized_ by $\Sigma$, if $\Phi=\mathbb{F}(\Sigma)$.

It is natural to ask if using families of i/o functions are equivalent to
using LPV manifest behaviors. Clearly, if
$\mathbb{F}(\Sigma)=\mathbb{F}(\hat{\Sigma})$, then
$\mathcal{B}(\Sigma)=\mathcal{B}(\hat{\Sigma})$ holds.  In fact, the example
below shows that the converse is not true.

###### Example 1

Consider the LPV-SSAs $\Sigma$ and $\Sigma^{{}^{\prime}}$

$\begin{split}&\Sigma\left\\{\begin{array}[]{l}x_{1}(t+1)=x_{1}(t)+p(t)x_{2}(t),\leavevmode\nobreak\
x_{2}(t+1)=0,\\\ y(t)=x_{1}(t)+p(t)x_{2}(t),\end{array}\right.\\\
&\Sigma^{{}^{\prime}}\left\\{\begin{array}[]{l}z(t+1)=0,\leavevmode\nobreak\
y(t)=z(t),\end{array}\right.\end{split}$

with the scheduling space $\mathbb{P}=\mathbb{R}$. A straightforward
calculation reveals that
$\mathcal{B}(\Sigma^{{}^{\prime}})=\mathcal{B}(\Sigma)$. Indeed, note that
$(x,y,u,p)$ is a solution of $\Sigma$, if and only if
$x_{1}(t)=x_{1}(0)+p(0)x_{2}(0)$ and $x_{2}(t)=0$ for $t>0$, hence
$y(t)=x_{1}(0)+p(0)x_{2}(0)$ for all $t\geq 0$. Therefore,
$\mathcal{B}(\Sigma)=\\{(y,u,p)\mid y(t)\mbox{ is constant}\\}$. It then
follows that $\mathcal{B}(\Sigma^{{}^{\prime}})=\mathcal{B}(\Sigma)$.

However, for $x_{\mathrm{o}}=\begin{bmatrix}1&1\end{bmatrix}^{\top}$ and
$x_{\mathrm{o}}^{{}^{\prime}}=1$,
$\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}\neq\mathfrak{Y}_{\Sigma^{{}^{\prime}},x_{\mathrm{o}}^{{}^{\prime}}}$.
To see this, it is enough to evaluate the i/o functions involved for any two
scheduling signals $p_{1}$ and $p_{2}$ such that $p_{1}(0)=0$ and
$p_{1}(1)=1$. Hence, $\mathbb{F}(\Sigma)\neq\mathbb{F}(\Sigma^{{}^{\prime}})$.
Indeed,
$\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(0,p)=1=\mathfrak{Y}_{\Sigma^{{}^{\prime}},x_{\mathrm{o}}^{{}^{\prime}}}(0,p)=x_{\mathrm{o}}^{{}^{\prime}}$.
However, if $p(0)=1$, then $\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(0,p)=2\neq
1=\mathfrak{Y}_{\Sigma^{{}^{\prime}},x_{\mathrm{o}}^{{}^{\prime}}}(0,p)$.

That is, realization theory of manifest behaviors is not equivalent to that of
i/o functions. In particular, the minimality results of [14] do not apply in
the behavioral setting. It is then a natural to ask if similarly to [14],
observability and span-reachability to characterize minimality in the
behaviroal setting.  The latter are recalled below.

###### Definition 6

Let $\Sigma$ be a LPV-SSA of the form (1). $\Sigma$ is _span -reachable_ from
an initial state $x_{\mathrm{o}}\in\mathbb{R}^{n_{\mathrm{x}}}$, if the linear
span of all states reachable from $x_{\mathrm{o}}$ equals the whole state-
space $\mathbb{R}^{n_{\mathrm{x}}}$, i.e.
$\mathrm{Span}\\{x(t)\mid(x,y,u,p)\in\mathcal{X}\times\mathcal{Y}\times\mathcal{U}\times\mathcal{P},t\in\mathbb{T},x(0)=x_{\mathrm{o}}\\}=\mathbb{R}^{n_{\mathrm{x}}}$.

The LPV-SSA $\Sigma$ is _observable_ if any two distinct initial states induce
distinct i/o functions, i.e. $\forall\leavevmode\nobreak\
x_{1},x_{2}\in\mathbb{R}^{n_{\mathrm{x}}}$: $x_{1}\neq
x_{2}\implies\mathfrak{Y}_{\Sigma,x_{1}}\neq\mathfrak{Y}_{\Sigma,x_{2}}$.

Observability and span-reachability can be characterized by rank conditions
[14]. Finally, similarly to [14], we we would like minimal realizations of the
same manifest behavior to be isomorphic. The latter notion is defined below

###### Definition 7

Let $\Sigma$ be of the form (1) and let and let
$\Sigma^{\prime}=(\mathbb{P},\\{A_{i}^{{}^{\prime}},B_{i}^{{}^{\prime}},C_{i}^{{}^{\prime}},D_{i}^{{}^{\prime}}\\}_{i=0}^{n_{\mathrm{p}}})$
be a LPV-SSA with $\dim(\Sigma)=\dim(\Sigma^{\prime})=n_{\mathrm{x}}$. A
nonsingular matrix $T\in\mathbb{R}^{n_{\mathrm{x}}\times n_{\mathrm{x}}}$ is
_isomorphism_ from $\Sigma$ to $\Sigma^{\prime}$, if

$A^{\prime}_{i}T=TA_{i}\leavevmode\nobreak\ \leavevmode\nobreak\
B^{\prime}_{i}=TB_{i}\leavevmode\nobreak\ \leavevmode\nobreak\
C^{\prime}_{i}T=C_{i}\leavevmode\nobreak\ \leavevmode\nobreak\
D^{\prime}_{i}=D_{i},\leavevmode\nobreak\ i=0,\ldots,n_{\mathrm{p}}$

Note that the matrix $T$ in the definition above does not depend on the
scheduling signal, and it acts only on the states of the LPV-SSAs involved. In
particular, the LPV-SSAs $\Sigma$ and $\Sigma^{\prime}$ have the same inputs
and outputs and are defined over the same set of scheduling signals.

Problem formulation: in this paper we will address the following questions:

(1) If two LPV-SSAs have the same manifest behavior, do they have the same set
of i/o functions ?

(2) Can we characterize minimal LPV-SSAsin terms of observability and span-
reachability ?

(3) Are minimal LPV-SSA realizations of the same manifest LPV behavior
isomorphic ?

(4) Is there an algorithm for transofrming an LPV-SSA to a minimal LPV-SSA
realization of its manifest behavior.

(5) Are minimal LPV-SSAs also minimal as meromorphic LPV-SS from [21, 22] ?

##  III Main results

In this section, we present the main results of the paper, which answer the
questions formulated above.

###  III-A Input-output functions vs. behaviors

We start by clarifying the relationship between manifest behaviors and i/o
functions of LPV-SSAs. To this end, we need the following definition.

###### Definition 8

A LPV-SSA of the form (1) is said to satisfy the _regularity certificate (RC)_
if  (1) $\mathbb{P}$ is convex with non-empty interior, and (2) in addition,
in DT case, the matrix $A(\mathbf{p})$ is invertible for all
$\mathbf{p}\in\mathbb{P}$.

In CT, the satisfaction of RC depends only on $\mathbb{P}$, and it is
satisfied if $\mathbb{P}$ is a Cartesian product of intervals, e.g.,
$\mathbb{P}=[a,b]^{n_{\mathrm{p}}}$, $a<b$. In DT, the RC condition is more
restrictive.

###### Theorem 1

Let $\Sigma$ and $\hat{\Sigma}$ be two LPV-SSAs which satisfy RC. Then
$\Sigma$ and $\hat{\Sigma}$ have the same family of i/o functions,i.e.
$\mathbb{F}(\Sigma)=\mathbb{F}(\hat{\Sigma})$, if and only if their manifest
behavior is the same, i.e., $\mathcal{B}(\Sigma_{1})=\mathcal{B}(\Sigma_{2})$.

The proof of Theorem 1 is presented in Section IV. The theorem above says that
manifest behaviors and families of i/o function are equivalent formalizations
of input-output behaviors of LPV-SSA satisfying RC.  Theorem 1 is no longer
true if we drop RC, see Example 1.

###  III-B Minimality

Theorem 1 and an extension of the results of [14] to families of i/o functions
leads to the following characterization of minimal realizations of manifest
behaviors.

###### Theorem 2

A LPV-SSA which satisfies RC is minimal, if and only if it is observable.
Furthermore, any two minimal LPV-SSAs which satisfy RC and which are
realizations of the same manifest behavior are isomorphic.

The proof of Theorem 2 is presented in Section IV. Note that minimality of
LPV-SSAs does not require span-reachability. This is in contrast with minimal
LPV-SSA realizations of i/o functions, but this is consistent with the
classical results for LTI systems [15].

Theorem 2 suggests a minimization procedure based on the observability
reduction procedure from [3, 14]. We recall the latter below. Let $\Sigma$ be
an LPV-SSA of the form (1), and recall from [14] the definition of extended
$n$-step observability matrices $\mathcal{O}_{n}$ of $\Sigma$,
$n\in\mathbb{N}$:

$\displaystyle\mathcal{O}_{0}$
$\displaystyle=\left[\begin{array}[]{cccc}C_{0}^{\top}&\cdots&C_{n_{\mathrm{p}}}^{\top}\end{array}\right]^{\top}$
$\displaystyle\mathcal{O}_{n+1}$
$\displaystyle=\left[\begin{array}[]{cccc}\mathcal{O}_{n}^{\top}&A_{0}^{\top}\mathcal{O}_{n}^{\top}&\cdots&A_{n_{\mathrm{p}}}^{\top}\mathcal{O}^{\top}_{n}\end{array}\right]^{\top}.$

By [14], $\Sigma$ is observable, if and only if
$\mathrm{rank}\leavevmode\nobreak\
(\mathcal{O}_{n_{\mathrm{x}}-1})=n_{\mathrm{x}}$.

######  Procedure 1 (Observability reduction)

Consider the matrix
$T=\begin{bmatrix}b_{1}&b_{2}&\ldots&b_{n_{\mathrm{x}}}\end{bmatrix}^{-1}$,
where $\\{b_{i}\\}_{i=1}^{n_{\mathrm{x}}}\subset\mathbb{R}^{n_{\mathrm{x}}}$
is a basis such that
$\mathrm{Span}\\{b_{o+1},\ldots,b_{n_{\mathrm{x}}}\\}=\mathrm{Ker}\\{\mathcal{O}_{n_{\mathrm{x}}-1}\\}$,

$\displaystyle TA_{i}T^{-1}=\begin{bmatrix}A_{i}^{\mathrm{O}}&0\\\
A^{\prime}_{i}&A^{\prime\prime}_{i}\end{bmatrix},\leavevmode\nobreak\
TB_{=}\begin{bmatrix}B_{i}^{\mathrm{O}}\\\
B_{i}^{\prime}\end{bmatrix},\leavevmode\nobreak\
C_{i}T^{-1}=\begin{bmatrix}C_{i}^{\mathrm{O}}&0\end{bmatrix},$

where $A^{\mathrm{O}}_{i}\in\mathbb{R}^{o\times
o},B_{i}^{\mathrm{O}}\in\mathbb{R}^{o\times n_{\mathrm{u}}}$ and
$C_{i}^{\mathrm{O}}\in\mathbb{R}^{n_{\mathrm{y}}\times o}$. Define
$\Sigma^{\mathrm{O}}=(\mathbb{P},\\{A_{i}^{\mathrm{O}},B_{i}^{\mathrm{O}},C_{i}^{\mathrm{O}},D_{i}\\}_{i=0}^{n_{\mathrm{p}}})$.

Procedure 1 is similar to the well-known observability reduction for
LTI/bilinear systems, and it can readily be implemented numerically, e.g., see
[3, Remark 2]

###### Remark 1

By [14], $\Sigma^{\mathrm{O}}$ is observable.  Let $\Pi\in\mathbb{R}^{o\times
n_{\mathrm{x}}}$ be such that $\Pi z$ is formed by the first $o$ elements of
$Tz$. Then $(x,y,u,p)$ is a solution of $\Sigma$, if and only if $(\Pi
x,y,u,p)$ is a solution of $\Sigma_{\mathrm{O}}$. Hence,
$\mathcal{B}(\Sigma)=\mathcal{B}(\Sigma_{\mathrm{O}})$. Moreover, for any
initial state $x_{\mathrm{o}}$ of $\Sigma$,
$\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}=\mathfrak{Y}_{\Sigma_{\mathrm{O}},\Pi(x_{\mathrm{o}})}$.
Furthermore, if $\Sigma$ satisfies RC, then so does $\Sigma_{\mathrm{O}}$. For
CT, there is nothing to show. For DT, notice that
$A_{\mathrm{O}}(\mathbf{p})$, $\mathbf{p}\in\mathbb{P}$ is the upper left
block of the triangular matrix $\hat{A}(\mathbf{p})=TA(\mathbf{p})T^{-1}$,
hence if $A(\mathbf{p})$ is invertible, then so is
$A_{\mathrm{O}}(\mathbf{p})$.

######  Corollary 1 (Minimization)

If $\Sigma$ satisfies RC, then $\Sigma_{\mathrm{O}}$ returned by Procedure 1
satisfies RC, it is minimal and it has the same manifest behavior as $\Sigma$.

As in the LTI case, span-reachability is necessary for minimality of LPV-SSA
realizations of _controllable_ behaviors, defined similarly to [15].

###### Definition 9

The manifest behavior $\mathcal{B}$ is _controllable_ , if for any two
elements $(y_{1},u_{1},p_{1})$, $(y_{2},u_{2},p_{2})$ of $\mathcal{B}$ and any
time instance $t\in\mathbb{T}$, there exists an element $(y,u,p)$ of
$\mathcal{B}$ and a time instance $\tau>0$, such that
$(y|_{[0,t]},u|_{[0,t]},p|_{[0,t]})=(y_{1}|_{[0,t]},u_{1}|_{[0,t]},p_{1}|_{[0,t]})$,
and for all $s\in\mathbb{T}$, $s\geq t+\tau$,
$(y(s),u(s),p(s))=(y_{2}(s-t-\tau),u_{2}(s-t-\tau),p_{2}(s-t-\tau))$.

Intuitively, a behavior is controllable, if any i/o trajectory generated by
the system up to some time can be continued by any other admissible i/o
trajectory.

###### Theorem 3

Let $\mathcal{B}$ is a manifest behavior, and let $\Sigma$ be a LPV-SSA which
satisfies RC. If $\mathcal{B}$ is a controllable, then $\Sigma$ is a minimal
realization of $\mathcal{B}$ if and only if $\Sigma$ is span-reachable from
zero and observable. Conversely, if $\Sigma$ is span-reachable from zero, then
its manifest behavior $\mathcal{B}(\Sigma)$ is controllable.

The proof of Theorem 3 is presented in Section IV. Recall from [14] that a
LPV-SSA is a minimal realization of an i/o function from the zero initial
state, if and only if it is observable and span-reachable from zero. Theorem 3
says that for LPV-SSAs which satisfy RC are minimal realizations of an i/o
function from the zero initial state, if and only if they minimal realizations
of their own manifest behaviors.

###  III-C Relationship with the prior results

Below we show that Theorem 2-3 are consistent with the results of [21]. To
this end, recall that LPV-SSAs are special cases of meromorphic LPV-SSs.
Recall from [21] the notions of structural state-observability and structural
state-reachability and state-trimness and minimality.

###### Theorem 4

If $\Sigma$ is a LPV-SSA which satisfies RC, then it is state-trim and the
following holds:

  * •

if it is observable, then it is structurally state-observable.

  * •

if is span-reachable from $x_{\mathrm{o}}=0$, then it is structurally state-
reachable.

  * •

if it is a minimal, then it is a minimal dimensional meromorphic LPV-SS in the
sense of [21].

The proof is presented in Section IV. Note that Theorem 4 ceases to be true,
if $\Sigma$ does not satisfy RC, see [7, Example 4.1] for a counter-example.
In general, there is a tradeoff between the dimensionality of LPV-SSs and the
dependence on the scheduling variable (meromorphic, affine), [7]. However, for
LPV-SSAs which satisfy RC, there is no such tradeoff, i.e. the algorithms of
[21, 22] will not result in smaller state-space representations when applied
to such LPV-SSAs. However, they may still introduce meromorphic dependencies
on the schedulung variable, see the example below.

###### Example 2

Consider in DT the LPV-SSA $\Sigma$ of the form (1), such that
$\mathbb{P}=[0,1]$ and

$\begin{split}&A_{0}=\begin{bmatrix}1&-2&-2\\\ 0&2&1\\\
-2&1&2\end{bmatrix},\leavevmode\nobreak\ A_{1}=\begin{bmatrix}1&-1&-1\\\
-1&2&0\\\ -1&0&2\end{bmatrix}\\\ &B_{0}=\begin{bmatrix}1\\\ -1\\\
-1\end{bmatrix},\leavevmode\nobreak\ B_{1}=\begin{bmatrix}2\\\ -2\\\
-2\end{bmatrix}\\\ &C_{0}=\begin{bmatrix}1\\\ 0\\\
0\end{bmatrix}^{\top},\leavevmode\nobreak\ C_{1}=\begin{bmatrix}0\\\ 1\\\
1\end{bmatrix}^{\top}\end{split}$

It follows that

$\begin{split}&A(\mathbf{p})=\begin{bmatrix}\mathbf{p}+1&-\mathbf{p}-2&-\mathbf{p}-2\\\
-\mathbf{p},2\mathbf{p}+2&1\\\ -\mathbf{p}-2&1&2\mathbf{p}+2\\\
\end{bmatrix},\\\ &B(\mathbf{p})=\begin{bmatrix}2\mathbf{p}+1\\\
-2\mathbf{p}-1\\\ -2\mathbf{p}-1\end{bmatrix},\leavevmode\nobreak\
C(\mathbf{p})=\begin{bmatrix}1\\\ \mathbf{p}\\\
\mathbf{p}\end{bmatrix}^{\top}\end{split}$

It follows that the determinant of $A(\mathbf{p})$ is
$-2\mathbf{p}^{2}-3\mathbf{p}-1$ and the latter polynomial is non-zero on
$[0,1]$. That is, in DT, $\Sigma$ satisfies RC. If we apply the observability
reduction procedure Procedure 1 to $\Sigma$, then we obtain the following
minimal LPV-SSA
$\Sigma_{m}=(\mathbb{P},\\{A_{i}^{m},B_{i}^{m},C_{i}^{m}\\}_{i=0}^{1})$,

$\begin{split}&A_{0}^{m}=\begin{bmatrix}1&-2\\\
-2&3\end{bmatrix},\leavevmode\nobreak\ A_{1}^{m}=\begin{bmatrix}1&-1\\\
-2&2\end{bmatrix},\leavevmode\nobreak\ \\\ &B_{0}^{m}=\begin{bmatrix}1\\\
2\end{bmatrix},\leavevmode\nobreak\ B_{1}^{m}=\begin{bmatrix}2\\\
-4\end{bmatrix},\leavevmode\nobreak\ C_{0}^{m}=\begin{bmatrix}1\\\
0\end{bmatrix}^{\top},\leavevmode\nobreak\ C_{1}^{m}=\begin{bmatrix}0\\\
1\end{bmatrix}^{\top}\end{split}$

which has the same manifest behavior as $\Sigma$.

Let us view $\Sigma$ as meromorphic LPV-SS and let us minimize $\Sigma$ using
[21, 22]. Recall the notation [21, 22], in particular, the $\diamond$ symbol
for DT. In particular, for a will consider matrix valued functions $S$ defined
on $\mathbb{P}^{k}$ and whos entries are meromoprhic functions, for any
scheduling signal $p\in\mathcal{P}$, $(S\diamond p)$ is a function defined on
the time axis $\mathbb{T}$, such that for any $t\in\mathbb{T}$, $(S\diamond
p)(t)=S(p(t),\xi p(t),\ldots,\xi^{k-1}p(t))=S(p(t),p(t+1),\ldots,p(t+k-1))$.
Recall that $\xi$ is the forward time shift. In particular, $(S\diamond p)$
can be viewed as an expression in $p,\xi p,\xi^{2}p,\ldots$, see [21].

Wit this notation, for any $p\in\mathcal{P}$ the $3$-step observability matrix
is

$(O\diamond p)=\begin{bmatrix}1&p&p\\\ (O\diamond p)_{2,1}&(O\diamond
p)_{2,2}&(O\diamond p)_{2,3}\\\ (O\diamond p)_{3,1}&(O\diamond
p)_{3,2}&(O\diamond p)_{3,3}\par\end{bmatrix}$

where

$\begin{split}&(O\diamond o)_{2,1}=-(2\xi p-1)(p+1)\\\ &(O\diamond
p)_{2,2}=3\xi p-p+2p\xi p-2\\\ &(O\diamond p)_{2,3}=3\xi p-p+2p\xi p-2\\\
&(O\diamond p)_{3,1}=(p+1)(3\xi p-8\xi^{2}p-6p\xi^{2}p+5)\\\ &(O\diamond
p)_{3,2}=(O\diamond p)_{3,3}=13\xi^{2}p-5\xi p-5p-3p\xi p+\\\
&+8p\xi^{2}p+10\xi p\xi^{2}p+6p\xi p\xi^{2}p-8\end{split}$

Recall that $p,\xi p,\xi^{2}p$ etc. are all viewed as formal variables when
defining the observability matrix. Over the field of meromoprhic functions,
then kernel of $(O\diamond p)$ is spanned by
$\begin{bmatrix}0&1&-1\end{bmatrix}^{\top}$. Consider the state-space
transformation

$(T\diamond p)=\begin{bmatrix}1&0&0\\\ 0&1&1\\\ (t_{31}\diamond
p)&(t_{32}\diamond p)&-1\end{bmatrix}^{-1}$

where $(t_{31}\diamond p)$ and $(t_{32}\diamond p)$ are defined in (2).

$\begin{split}&(t_{31}\diamond p)=\frac{((p+1)(2\xi p-3\xi^{2}p-4\xi
p\xi^{2}p-2\xi p^{2}\xi^{2}p+\xi p^{2}+2))}{(2p-3\xi p-4p\xi p-2p^{2}\xi
p+p^{2}+2)}\\\ &(t_{32}\diamond p)=\frac{(10p+5\xi p-13\xi^{2}p+6p\xi
p-16p\xi^{2}p-10\xi p\xi^{2}p+3p^{2}\xi p-8p^{2}\xi^{2}p+5p^{2}-6p^{2}\xi
p\xi^{2}p-12p\xi p\xi^{2}p+8}{(2p-3\xi p-4p\xi p-2p^{2}\xi
p+p^{2}+2)}\end{split}$ (2)

In the expression above, the inverse is understood as the inverse of a matrix
over the field of meromorphic functions of the variables $p,qp,\ldots$. The
expression $(T\diamond p)$ and $(O\diamond p)$ were computed using Matlab
symbolic toolbox. For any scheduling signal $p\in\mathcal{P}$, define

$(A\diamond p)=A_{0}+A_{1}p,\leavevmode\nobreak\ (B\diamond
p)=B_{0}+B_{1}p,\leavevmode\nobreak\ (C\diamond p)=C_{0}+C_{1}p,$

i.e., $(A\diamond p)(t)=A(p(t))$, $(B\diamond p)(t)=B(p(t))$ and $(C\diamond
p)(t)=C(p(t))$ for all $t\in\mathbb{T}$. Then the matrices of the transformed
system are of the form

$\begin{split}&\stackrel{{\scriptstyle\leftarrow}}{{(T\diamond p)}}(A\diamond
p)(T\diamond p)^{-1}=\begin{bmatrix}(A^{\mathrm{o}}\diamond p)&0\\\
\star&\star\end{bmatrix},\\\ &\stackrel{{\scriptstyle\leftarrow}}{{(T\diamond
p)}}(B\diamond p)=\begin{bmatrix}(B^{\mathrm{o}}\diamond p)\\\
\star\end{bmatrix},\leavevmode\nobreak\ \\\ &(C\diamond p)(T\diamond
p)^{-1}=\begin{bmatrix}(C^{\mathrm{o}}\diamond p)&0\end{bmatrix}\end{split}$

where $\stackrel{{\scriptstyle\leftarrow}}{{(T\diamond p)}}$ is obtained from
$(T\diamond p)$ by replacing each occurence of $p$ by $\xi p$ and each
occurence of $\xi^{i}p$ by $\xi^{i+1}p$, for all $i=1,2,\ldots$, and
$(A^{\mathrm{o}}\diamond p)$, $B^{\mathrm{o}}\diamond p)$ and
$(C^{\mathrm{o}}\diamond p)$ are $2\times 2$, $2\times 1$ and $1\times 2$
matrices with meromophic entries respectively, and $\star$ stands for arbitary
matrix. The matrices $(A^{\mathrm{o}}\diamond p)$, $B^{\mathrm{o}}\diamond p)$
and $(C^{\mathrm{o}}\diamond p)$ are $2\times 2$, $2\times 1$ and $1\times 2$
can be found in the .mat files in the supplementary material of this report.

These matrices depend on $p,\xi p,\xi^{2}p,\xi^{3}p$, hence, by a slight abuse
of notation, they can be viewed as matrices of $p,\xi p,\xi^{2}p,\xi^{3}p$.
According to [21, 22] LPV-SS

$\Sigma^{\mathrm{o}}\left\\{\begin{array}[]{lcl}x(t+1)&=&(A^{\mathrm{o}}\diamond
p)(t)x(t)+(B^{\mathrm{o}}\diamond p)(t)u(t),\\\ y(t)&=&(C^{\mathrm{o}}\diamond
p)(t)x(t),\end{array}\right.$

has the same manifest behavior as $\Sigma$ and it is a minimal one. Note that
unlike $\Sigma$, $\Sigma^{\mathrm{o}}$ has a non-linear and dynamic depency on
the scheduling variable. To illustrate this, we present
$(A^{\mathrm{o}}\diamond p)_{1,1}$ in (3).

$(A^{\mathrm{o}}\diamond p)_{1,1}=\frac{((p+1)(6\xi^{2}p-7\xi p-6p\xi
p+3p\xi^{2}p+8\xi p\xi^{2}p-p\xi p^{2}-2p^{2}\xi p+4\xi
p^{2}\xi^{2}p+p^{2}-2\xi p^{2}+2p\xi p^{2}\xi^{2}p+4p\xi
p\xi^{2}p-2))}{(2p-3\xi p-4p\xi p-2p^{2}\xi p+p^{2}+2)}$ (3)

The other entries of the $A^{\mathrm{o}}\diamond p$ are even more involved. In
accordance with Theorem 4, the LPV-SSA $\Sigma_{m}$ is also a minimal if
viewed as a meromorphic LPV-SS, and it has the same manifest behavior as
$\Sigma^{\mathrm{o}}$. Hence, by [21, 22] are related by a state-space
isomorphism which depends on the scheduling variable. We computed the
corresponding transformation, it can be found in the .mat file in the
supplementary material of this report.

The example above demonstrates that if the minimization procedure of [21, 22]
is applied to LPV-SSAs, it will result in a meromoprhic LPV-SS with a dynamic
depency. Moreover, even for a simple example, the dependency can get quite
involved. This is in contrast to the minimization procedure of this paper,
which preserves static and affine dependency.

##  IV Proofs of the result

###  IV-A Auxiliary results: observability revealing scheduling

The proofs rely on the following observation, which states that for any
observable LPV-SSA which satisfies RC, there exists a scheduling signal such
that the output response to that scheduling signal determines the initial
state uniquely.

###### Theorem 5

Let $\Sigma$ be an observable LPV-SSA which satisfies RC. There exists a
scheduling signal $p_{\mathrm{o}}\in\mathcal{P}$ and
$t_{\mathrm{o}}\in\mathbb{T}$ such that for any two initial states
$x_{1},x_{2}$ of $\Sigma$,

${\color[rgb]{0,0,1}\mathfrak{Y}_{\Sigma,x_{1}}(0,p_{\mathrm{o}})|_{[0,t_{\mathrm{o}}]}=\mathfrak{Y}_{\Sigma,x_{2}}(0,p_{\mathrm{o}})|_{[0,t_{\mathrm{o}}]}\implies\mathfrak{Y}_{\Sigma,x_{1}}=\mathfrak{Y}_{\Sigma,x_{2}}}$

In CT, $p_{\mathrm{o}}$ can be chosen to be analytic.

The proof relies on viewing LPV-SSAs with zero input as bilinear systems whose
inputs are the scheduling signals. Then the existence of $p_{\mathrm{o}}$
follows from the existence of a universal input for bilinear systems [26, 19].
The RC condition is necessary for using [26, 19].

###### Proof:

Let $\Sigma$ be of the form (1). since It is enough to show that there exists
$t_{\mathrm{o}}{\color[rgb]{0,0,1}\in\mathbb{T}}$, $p_{o}\in\mathcal{P}$, such
that if
$\mathfrak{Y}_{\Sigma,x_{1}}(0,p_{o})|_{[0,t_{\mathrm{o}}]}=\mathfrak{Y}_{\Sigma,x_{2}}(0,p_{o})|_{[0,t_{\mathrm{o}}]}$.
then $\mathfrak{Y}_{\Sigma,x_{1}}(0,p)=\mathfrak{Y}_{\Sigma,x_{2}}(0,p)$ for
all $p\in\mathcal{P}$. Indeed, the latter equality implies
$\mathfrak{Y}_{\Sigma,x_{1}}=\mathfrak{Y}_{\Sigma,x_{2}}$, as
$\mathfrak{Y}_{\Sigma,x_{i}}(u,p)=\mathfrak{Y}_{\Sigma,x_{i}}(0,p)+\mathfrak{Y}_{\Sigma,0}(u,p)$
for all $i=1,2$, $u\in\mathcal{U}$. To this end, consider the bilinear system

$\begin{split}&\xi\begin{bmatrix}x(t)\\\
z(t)\end{bmatrix}=\begin{bmatrix}A(p(t))&0\\\
C(p(t))&0\end{bmatrix}\begin{bmatrix}x(t)\\\
z(t)\end{bmatrix},\leavevmode\nobreak\ s(t)=z(t)\end{split}$ (4)

with input $p\in\mathcal{P}$ and output $s\in\mathcal{Y}$. For any
$p\in\mathcal{P}$, denote by $s((x_{\mathrm{o}},z_{\mathrm{o}}),p)$ the output
trajectory of (4) generated from the initial state
$(x^{T}_{\mathrm{o}},z^{T}_{\mathrm{o}})^{T}\in{\color[rgb]{0,0,1}\mathbb{R}^{n_{\mathrm{x}}}\times\mathbb{R}^{n_{\mathrm{y}}}}$
under input $p$.

In CT, le us take any $t_{\mathrm{o}}\in\mathbb{T}$, and let $p_{o}$ be the
analytic universal input from [19, Theorem 2.11] applied to (4).  Note that if
$\mathbb{P}$ is a convex set with a non-empty interior, then $\mathbb{P}$
satisfies [19, condition H4] by [27, Corollary 2.3.9]. For the DT case , let
$p_{\mathrm{o}}$ be the universal input applied to (4) from [26, Theorem 5.3].
Notice that (4) is observable in the sense of [26], if and only if $\Sigma$ is
observable. From the proof of [26, Theorem 5.3] it follows that there exists
$t_{\mathrm{o}}>0$ such for any two initial states, the corresponding output
trajectories of (4) under input $p_{\mathrm{o}}$ are different on
$[0,t_{\mathrm{o}}]$.

Then for any two initials states of (4), if the outputs from those initial
states are the same on $[0,t_{\mathrm{o}}]$ for the input $p_{\mathrm{o}}$,
then the outputs are the same for any input $p\in\mathcal{P}$ and any time
interval. Since $\xi
s((x_{\mathrm{o}},z_{\mathrm{o}})),p)=\mathfrak{Y}_{\Sigma,x_{i,\mathrm{o}}}(0,p)$
for any
$(x_{\mathrm{o}}^{\top},z^{\top}_{\mathrm{o}})^{\top}\in{\color[rgb]{0,0,1}\mathbb{R}^{n_{\mathrm{x}}}\times\mathbb{R}^{n_{\mathrm{y}}}}$
and $p\in\mathcal{P}$, it follows that $p_{\mathrm{o}}$ and $t_{\mathrm{o}}$
satisfy the statement of the theorem. ∎

We will also need the following corollary of Theorem 5.

###### Corollary 2

Assume that $\Sigma$, $\hat{\Sigma}$ satisfy RC. Then there exists a
scheduling signal $p_{\mathrm{o}}\in\mathcal{P}$ and a time instance
$t_{\mathrm{o}}\in\mathbb{T}$ such that for any two initial states
$x_{\mathrm{o}}$, $\hat{x}_{\mathrm{o}}$ of $\Sigma$ and $\hat{\Sigma}$
respectively, if
$\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(0,p_{\mathrm{o}})|_{[0,t_{\mathrm{o}}]}=\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}(0,p_{\mathrm{o}})|_{[0,t_{\mathrm{o}}]}$,
then $\left(\forall
p\in\mathcal{P}:\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(0,p)=\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}(0,p)\right)$.

###### Proof:

Let $\Sigma$ be as in (1) and
$\hat{\Sigma}=(\mathbb{P},\\{\hat{A}_{i},\hat{B}_{i},\hat{C}_{i},\hat{D}_{i}\\}_{i=0}^{\mathrm{n}_{p}})$.
Consider the LPV-SSA

$\Sigma_{c}=(\mathbb{P},\\{\begin{bmatrix}A_{i}&0\\\
0&\hat{A}_{i}\end{bmatrix},\begin{bmatrix}B_{i}\\\
\hat{B}_{i}\end{bmatrix},\begin{bmatrix}C_{i}&-\hat{C}_{i}\end{bmatrix},D_{i}-\hat{D}_{i}\\}_{i=0}^{\mathrm{n}_{p}}).$

Then $\Sigma_{c}$ satisfies RC, and the i/o functions of $\Sigma_{c}$ are the
difference between the i/o functions of $\Sigma$ and those of $\hat{\Sigma}$.
That is, for
$x_{\mathrm{o},c}=\begin{bmatrix}x^{\top}_{\mathrm{o}}&\hat{x}^{\top}_{\mathrm{o}}\end{bmatrix}^{\top}$,
$\mathfrak{Y}_{\Sigma_{c},x_{\mathrm{o},c}}=\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}-\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}$.
Let $\Sigma_{c,\mathrm{O}}$ be the result of applying Procedure 1 to
$\Sigma_{c}$. By Remark 1,
$\mathfrak{Y}_{\Sigma_{c},x_{\mathrm{o},c}}=\mathfrak{Y}_{\Sigma_{c,\mathrm{O}},\Pi
x_{\mathrm{O},c}}$ and $\Sigma_{c,\mathrm{O}}$ satisfies RC.  Consider the
scheduling signal $p_{\mathrm{o}}\in\mathcal{P}$ and time instance
$t_{\mathrm{o}}$ from Let us apply Theorem 5 to $\Sigma_{c,\mathrm{O}}$ and
let us We claim that $p_{\mathrm{o}}$, $t_{\mathrm{o}}$ satisfy the conclusion
of Theorem 2. Indeed, assume that
$\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(0,p_{\mathrm{o}})|_{[0,t_{\mathrm{o}}]}=\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}(0,p_{\mathrm{o}})|_{[0,t_{\mathrm{o}}]}$
holds. Then the output
$\mathfrak{Y}_{\Sigma_{c},x_{\mathrm{o},c}}(0,p_{\mathrm{o}})$ of the error
system $\Sigma_{c}$ is zero on $[0,t_{\mathrm{o}}]$, where
$x_{\mathrm{o},c}=\begin{bmatrix}x^{\top}_{\mathrm{o}}&\hat{x}^{\top}_{\mathrm{o}}\end{bmatrix}^{\top}$.
Hence, $\mathfrak{Y}_{\Sigma_{c,\mathrm{O}},\Pi
x_{\mathrm{o},c}}(0,p_{\mathrm{o}})|_{[0,t_{\mathrm{o}}]}=0=\mathfrak{Y}_{\Sigma_{c,\mathrm{O}},0}(0,p_{\mathrm{o}})|_{[0,t_{\mathrm{o}}]}$.
From Theorem 5 it then follows that $\mathfrak{Y}_{\Sigma_{c,\mathrm{O}},\Pi
x_{\mathrm{o},c}}=\mathfrak{Y}_{\Sigma_{c,\mathrm{O}},0}$. Hence,
$\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(0,p)-\mathfrak{Y}_{\Sigma,\hat{x}_{\mathrm{o}}}(0,p)=\mathfrak{Y}_{\Sigma_{c},x_{\mathrm{o},c}}(0,p)=\mathfrak{Y}_{\Sigma_{c,\mathrm{O}},\Pi
x_{\mathrm{o},c}}(0,p)=\mathfrak{Y}_{\Sigma_{c,\mathrm{O}},0}(0,p)=0$ for any
$p\in\mathcal{P}$.  ∎

###  IV-B Behaviors vs. i/o functions: proof of Theorem 1

The implication
$\mathbb{F}(\Sigma)=\mathbb{F}(\hat{\Sigma})\implies\mathcal{B}(\hat{\Sigma})=\mathcal{B}(\Sigma)$
is obvious. We show the reverse implication. Assume that
$\mathcal{B}(\hat{\Sigma})=\mathcal{B}(\Sigma)$. Consider the scheduling
signal $p_{\mathrm{o}}$ and the time instance $t_{\mathrm{o}}>0$ from
Corollary 2.

First, we show that $\mathfrak{Y}_{\Sigma,0}=\mathfrak{Y}_{\hat{\Sigma},0}$.
Consider any $p\in\mathcal{P}$, $u\in\mathcal{U}$ and $\mathbb{T}\ni\tau>0$
and define $\tilde{u}$ and $\tilde{p}$ such that for all $s\in\mathbb{T}$,
$\tilde{u}(s+t_{\mathrm{o}}+\tau)=u(s)$,
$\tilde{p}(s+t_{\mathrm{o}}+\tau)=p(s)$, and
$\tilde{u}|_{[0,t_{\mathrm{o}}+\tau}=0$ and
$\tilde{p}|_{[0,t_{\mathrm{o}}]}=p_{\mathrm{o}}$.  Let
$\tilde{y}=\mathfrak{Y}_{\Sigma,0}(\tilde{u},\tilde{p})$ be the output of
$\Sigma$ generated by the zero initial state for input $\tilde{u}$ and
scheduling signal $\tilde{p}$. Since $(\tilde{y},\tilde{u},\tilde{p})$ belongs
to $\mathcal{B}(\hat{\Sigma})=\mathcal{B}(\Sigma)$, there exists an initial
state $\hat{x}_{\mathrm{o}}$ such that
$\tilde{y}=\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}(\tilde{u},\tilde{p})$.
Since $\tilde{u}$ equals zero on $[0,t_{\mathrm{o}}]$, and $\hat{y}$ is the
output of $\Sigma$ generated from the zero initial state, $\hat{y}$ must be
zero on $[0,t_{\mathrm{o}}]$. Since $\hat{p}$ equals $p_{\mathrm{o}}$ on
$[0,t_{\mathrm{o}}]$,
$\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}(0,p_{\mathrm{o}})|_{[0,t_{\mathrm{o}}]}=\hat{y}|_{[0,t_{\mathrm{o}}]}=0$.
From Corollary 2, it follows that
$\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}(0,\bar{p})=\mathfrak{Y}_{\Sigma,0}(0,\bar{p})=0$
for all $\bar{p}\in\mathcal{P}$. Hence,
$\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}(\bar{u},\bar{p})=\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}(0,\bar{p})+\mathfrak{Y}_{\hat{\Sigma},0}(\bar{u},\bar{p})=\mathfrak{Y}_{\hat{\Sigma},0}(\bar{u},\bar{p})$
for all $\bar{u}\in\mathcal{U}$.

In particular,
$\mathfrak{Y}_{\hat{\Sigma},0}(\tilde{u},\tilde{p})=\mathfrak{Y}_{\Sigma,0}(\tilde{u},\tilde{p})=\hat{y}$.
Let $x_{f},\hat{x}_{f}$ be the states of $\Sigma$ respectively $\hat{\Sigma}$
reached from the zero initial state at time $t_{\mathrm{o}}+\tau$ under input
$\tilde{u}$ and scheduling signal $\tilde{p}$. Since
$\tilde{u}(s+\tau+t_{\mathrm{o}})=u(s)$,
$\tilde{p}=(s+\tau+t_{\mathrm{o}})=p(s)$, it follows that
$\mathfrak{Y}_{\Sigma,x_{f}}(u,p)(s)=\hat{y}(s+t_{\mathrm{o}}+\tau)=\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{f}}(u,p)(s)$
for all $s\in\mathbb{T}$. However, the restriction of $\tilde{u}$ to
$[0,t_{\mathrm{o}}+\tau)$ is zero, and hence $x_{f}$ and $\hat{x}_{f}$ are
zero. Hence,
$\mathfrak{Y}_{\hat{\Sigma},0}(u,p)=\mathfrak{Y}_{\Sigma,0}(u,p)$, and as $u$
and $p$ are arbitrary, $\mathfrak{Y}_{\hat{\Sigma},0}=\mathfrak{Y}_{\Sigma,0}$
follows.

Next we show that $\mathbb{F}(\Sigma)\subseteq\mathbb{F}(\hat{\Sigma})$. To
this end, let $x_{\mathrm{o}}$ be an initial state of $\Sigma$. If
$y=\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(0,p_{\mathrm{o}})$, then
$(y,0,p_{\mathrm{o}})\in\mathcal{B}(\Sigma)=\mathcal{B}(\hat{\Sigma})$, and
thus there exists an initial state $\hat{x}_{\mathrm{o}}$ of $\Sigma$ such
that $y=\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}(0,p_{\mathrm{o}})$.
From Corollary 2 it follows that
$\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(0,p)=\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}(0,p)$
for all $p\in\mathcal{P}$. Since
$\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(u,p)=\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(0,p)+\mathfrak{Y}_{\Sigma,0}(u,p)$,
$\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}(u,p)=\mathfrak{Y}_{\hat{\Sigma},x_{\mathrm{o}}}(0,p)+\mathfrak{Y}_{\hat{\Sigma},0}(u,p)$,
this implies
$\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}=\mathfrak{Y}_{\hat{\Sigma},\hat{x}_{\mathrm{o}}}$.
The inclusion $\mathbb{F}(\hat{\Sigma})\subseteq\mathbb{F}(\Sigma)$ can be
shown similarly.

###  IV-C Minimality results: proof of Theorem 2–3

###### Proof:

First, we show that if $\Sigma$ is a minimal realization of $\mathcal{B}$,
then it is observable. Assume that $\Sigma$ is not observable. Let us apply
Procedure 1 to $\Sigma$. Then the resulting LPV-SSA $\Sigma_{\mathrm{O}}$ has
a smaller state-space dimension than $\Sigma$. Remark 1 $\Sigma_{\mathrm{O}}$
is also realization of $\mathcal{B}$. This contradicts the minimality of
$\Sigma$.

Next, we prove that observability implies minimality.Consider two LPV-SSA
realizations $\Sigma$ and $\hat{\Sigma}$ of $\mathcal{B}$, such that $\Sigma$
and $\hat{\Sigma}$ both satisfy RC. Then by Theorem 1,
$\Phi=\mathbb{F}(\Sigma)=\mathbb{F}(\hat{\Sigma})$. Define
$n_{\mathrm{x}}=\dim(\Sigma)$ and $\hat{n_{\mathrm{x}}}=\dim(\hat{\Sigma})$.
Let $\mathfrak{S}(\Sigma)$ and $\mathfrak{S}(\hat{\Sigma})$ associated with
$\Sigma$ and $\hat{\Sigma}$ respectively, as defined in [14, Appendix,
Subsection B]. Recall from [14, Definition 2] the notion of a switched i/o
function $\mathfrak{S}(\mathfrak{F})$ associated with an i/o function
$\mathfrak{F}\in\Phi$. and recall that the mapping
$\mathfrak{F}\mapsto\mathfrak{S}(\mathfrak{F})$ is injective. Define
$\mathfrak{S}(\Phi)=\\{\mathfrak{S}(\mathfrak{F}\mid\mathfrak{F}\in\Phi\\}$
and let
$\mu:\mathfrak{S}(\Phi)\rightarrow{\color[rgb]{0,0,1}\mathbb{R}^{n_{\mathrm{x}}}}$
and
$\hat{\mu}:\mathfrak{S}(\Phi)\rightarrow{\color[rgb]{0,0,1}\mathbb{R}^{\hat{n_{\mathrm{x}}}}}$
be such that for any $\mathfrak{F}\in\Phi$, the i/o functions of $\Sigma$ and
$\hat{\Sigma}$ induced by the initial state $\mu(\mathfrak{S}(\mathfrak{F}))$
and $\hat{\mu}(\mathfrak{S}(\mathfrak{F}))$ respectively are both equal to
$\mathfrak{F}$. Then $(\mathfrak{S}(\Sigma),\mu)$,
$(\mathfrak{S}(\hat{\Sigma}),\hat{\mu})$ are both realizations of
$\mathfrak{S}(\Phi)$ in the sense of [12]222The results of [12] can readily be
extended to DT, by using the relationship from [13] between linear switched
systems in DT and rational representations.  Assume that $\Sigma$ is
observable. Then by [14, Theorem 4], $\mathfrak{S}(\Sigma)$ is observable.
Moreover, $\mu$ is surjective. Indeed, for any initial state $x_{\mathrm{o}}$,
the i/o function generated by
$\mu(\mathfrak{S}(\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}})$ and $x_{\mathrm{o}}$
are the same, and hence by observability of $\Sigma$,
$x_{\mathrm{o}}=\mu(\mathfrak{S}(\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}})$. Then
$(\mathfrak{S}(\Sigma),\mu)$ is span-reachable in the terminology of [12], and
by [12], $(\mathfrak{S}(\Sigma),\mu)$ is a minimal realization of
$\mathfrak{S}(\Phi)$. Hence,
$\dim\mathfrak{S}(\Sigma)=\dim\Sigma\leq\dim\mathfrak{S}(\hat{\Sigma})=\dim\hat{\Sigma}$,
i.e. $\Sigma$ is a minimal realization of $\mathcal{B}$.

Assume that $\Sigma$ and $\hat{\Sigma}$ are minimal realizations of
$\mathcal{B}$. By the first part of the theorem, they are observable, and
hence the linear switched systems $\mathfrak{S}(\Sigma)$ and
$\mathfrak{S}(\hat{\Sigma})$ are observable. From the argument of the previous
paragraph it follows that $(\mathfrak{S}(\hat{\Sigma}),\hat{\mu})$ and
$(\mathfrak{S}(\Sigma),\mu)$ are minimal realizations of $\mathfrak{S}(\Phi)$.
Hence, by [12, 13], they are isomorphic, and by [14, Theorem 4], $\Sigma$ and
$\hat{\Sigma}$ are isomorphic too. ∎

In order to prove Theorem 3, we will need to relate controllability and
observability of LPV-SSAs with with that of the linear-time varying state-
space representation (abbreviated as _LTV-SS_) obtained from the LPV-SSA by
fixing a a particular scheduling signal $p$. The latter LTV-SS is defined as

$\Sigma(p)\left\\{\begin{split}&\xi x(t)=A(t)x(t)+B(t)u(t)\\\
&y(t)=C(t)x(t)+D(t)\end{split}\right.$

where $A(t)=A(p(t))$, $B(t)=B(p(t))$, $C(t)=C(p(t))$, $D(t)=D(p(t))$. Note
that the LTV-SS $\Sigma(p)$ depends on the scheduling signal $p$. Recall for
[6] the notion of observability and controllability of LTV-SS on a given time-
interval.  Then Theorem 5 implies the following.

###### Corollary 3

With the assumptions and notations of Theorem 5, the LTV-SS
$\Sigma(p_{\mathrm{o}})$ is observable on $[0,t_{\mathrm{o}}]$.

###### Corollary 4

If $\Sigma$ is span-reachable from $x_{\mathrm{o}}=0$ and $\Sigma$ satisfies
RC, then there exists $p_{\mathrm{r}}\in\mathcal{P}$ and
$t_{\mathrm{r}}\in\mathbb{T}$, such that the LTV-SS $\Sigma(p_{\mathrm{r}})$
is controllable on $[0,t_{\mathrm{r}}]$.

###### Proof:

Consider the dual LPV-SSA
$\Sigma^{T}=(\mathbb{P},\\{(A^{T}_{q},C^{T}_{q},B_{q}^{T},D_{q}^{T})\\}_{q=0}^{n_{p}})$.
If $\Sigma$ is span-reachable from zero, then
$\mathrm{rank}\mathcal{R}_{n_{\mathrm{x}}-1}=n_{x}$, where
$\mathcal{R}_{n_{\mathrm{x}}-1}$ is the ($n_{\mathrm{x}}-1$)-step extended
reachability matrix of $\Sigma$ from $0$ as defined in [14]. Let
$\mathcal{O}_{n_{\mathrm{x}}-1}$ be the ($n_{\mathrm{x}}-1$)-step extended
observability matrix of $\Sigma^{T}$ as defined in [14]. It is clear that
$\mathcal{O}_{n_{\mathrm{x}}-1}=\mathcal{R}_{n_{\mathrm{x}}-1}^{T}$, and hence
$\mathrm{rank}\mathcal{O}_{n_{\mathrm{x}}-1}=n_{\mathrm{x}}$ and thus
$\Sigma^{T}$ is observable. From Corollary 3 it follows that there exist
$t_{\mathrm{o}}\in\mathbb{N}$, and $p_{\mathrm{o}}\in\mathcal{P}$ such that
the LTV-SS $\Sigma^{T}(p_{\mathrm{o}})$ is completely observable on
$[0,t_{\mathrm{o}}]$. This LTV system is given by matrices
$A(t)=A^{T}(p_{\mathrm{o}}(t))$, $B(t)=C^{T}(p_{\mathrm{o}}(t))$,
$C(t)=B^{T}(p_{\mathrm{o}}(t))$ The dual of $\Sigma^{T}(p_{\mathrm{o}})$ is
then completely controllable on $[0,t_{\mathrm{o}}]$, and it coincides with
$\Sigma(p_{\mathrm{o}})$. Hence, we can choose $p_{\mathrm{e}}=p_{\mathrm{o}}$
and $t_{\mathrm{e}}=t_{\mathrm{o}}$. ∎

###### Proof:

If $\Sigma$ is observable and span-reachable from zero, then it is a minimal
by Theorem 3. Conversely, assume that $\Sigma$ is a minimal realization of
$\mathcal{B}$. By Theorem 3 is $\Sigma$ is observable. Le $x_{\mathrm{o}}$ be
an initial state of $\Sigma$. Let $p_{\mathrm{o}}$, $t_{o}$ be the scheduling
signal and time instance from Theorem 5. Let
$y=\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(0,p_{\mathrm{o}})$. Then
$(y,0,p_{\mathrm{o}})\in\mathcal{B}$. Note that
$(0,0,p_{\mathrm{o}})\in\mathcal{B}$. From controllability of $\mathcal{B}$ it
follows that there exists a time instance $\tau>0$, and signals
$\tilde{u},\tilde{p},\tilde{y}$ such that
$(\tilde{y},\tilde{u},\tilde{p})\in\mathcal{B}$,
$(\tilde{y}|_{[0,t_{\mathrm{o}}]},\tilde{u}(t)|_{[0,t_{\mathrm{o}}]},\tilde{p}|_{[0,t_{\mathrm{o}}]})=(0,0,p_{\mathrm{o}}|_{[0,t_{\mathrm{o}}]})$,
and
$(\tilde{y}(s+t_{\mathrm{o}}+\tau),\tilde{u}(s+t_{\mathrm{o}}+\tau),\tilde{p}(s+t_{\mathrm{o}}+\tau))=(y(s),u(s),p(s))$
for all $s\in\mathbb{T}$. Since $\Sigma$ is a realization of $\mathcal{B}$,
there exists an initial state $\hat{x}_{\mathrm{o}}$ of $\Sigma$ such that
$\mathfrak{Y}_{\Sigma,\hat{x}_{\mathrm{o}}}(\tilde{u},\tilde{p})=\tilde{y}$.
It then follows that
$\mathfrak{Y}_{\Sigma,\hat{x}_{\mathrm{o}}}(0,p_{\mathrm{o}})|_{[0,t_{\mathrm{o}}]}=0=\mathfrak{Y}_{\Sigma,0}(0,p_{\mathrm{o}})|_{[0,t_{\mathrm{o}}]}$,
and hence
$\mathfrak{Y}_{\Sigma,0}=\mathfrak{Y}_{\Sigma,\hat{x}_{\mathrm{o}}}$. Since
$\Sigma$ is observable, it then follows that $\hat{x}_{\mathrm{o}}=0$.
Consider now the state $\hat{x}_{f}$ reached from the zero initial state using
$\tilde{u}$,$\tilde{p}$ at time $t_{\mathrm{o}}+\tau$. It then follows that
$\mathfrak{Y}_{\Sigma,\hat{x}_{f}}(0,p_{\mathrm{o}})(s+t_{\mathrm{o}}+\tau)=\mathfrak{Y}_{\Sigma,0}(\tilde{u},\tilde{p})=\mathfrak{Y}_{\Sigma,x_{\mathrm{o}}}(0,p_{\mathrm{o}})(s+t_{\mathrm{o}}+\tau)$,
$s\in\mathbb{T}$. Hence, by Theorem 5 and observability of $\Sigma$,
$x_{\mathrm{o}}=\hat{x}_{f}$, i.e., $x_{\mathrm{o}}$ is reachable from zero.
That is, $\Sigma$ is span-reachable.

Assume that $\Sigma$ a realization of $\mathcal{B}$ which is span-reachable
from zero. Consider $p_{c}\in\mathcal{P}$ and $t_{c}\in\mathbb{T}$ from
Corollary 4. Consider two elements $(u_{1},p_{1},y_{1})$ and
$(u_{2},p_{2},y_{2})$ of $\mathcal{B}$ and a time instance $t$. Then there
exist initial states $x_{\mathrm{o},1},x_{\mathrm{o},2}$ such that
$y_{i}=\mathfrak{Y}_{\Sigma,x_{\mathrm{o},i}}(u_{i},p_{i})$. Let
$x_{\mathrm{o}}$ be the state reached from $x_{\mathrm{o},1}$ at time $t^{+}$
under input $u_{1}$ and scheduling signal $p_{1}$, where $t^{+}=t$ in CT and
$t^{+}=t+1$ in DT. Let $u_{c}$ and $t_{c}>\tau\geq 0$ be such that
$x_{\mathrm{o},2}$ is the state reached from $x_{\mathrm{o}}$ at time
$\tau^{-}$ under input $u_{c}$ and scheduling signal $p_{c}$, where
$\tau^{-}=\tau$ in CT and $\tau^{-}=\tau-1$ in DT. Since $\Sigma(p_{c})$ is
controllable on $[0,t_{c}]$, such $u_{c}$ and $\tau$ exist. Define
$\tilde{u},\tilde{p}$ such that
$\tilde{u}|_{[0,t]}=u_{1}|_{[0,t]},\tilde{p}|_{[0,t]}=p_{1}|_{[0,t]}$,
$\tilde{u}(s)=u_{c}(s-t),\tilde{p}(s)=p_{c}(s-t)$, $s\in(t,t+\tau)$, and
$\tilde{u}(s+t+\tau)=u_{2}(s),\tilde{p}(s+t+\tau)=p_{2}(s)$, $s\in\mathbb{T}$.
Let $\tilde{y}=\mathfrak{Y}_{\Sigma,x_{\mathrm{o},1}}(\tilde{u},\tilde{p})$.
Then $(\tilde{u},\tilde{p},\tilde{y})\in\mathcal{B}$. That is, $\mathcal{B}$
is controllable. ∎

###  IV-D Relationship with prior work: proof of Theorem 4

###### Proof:

We prove the statement for observability, the statement on span-reachability
follows by duality. From Corollary 5 it follows that the LTV-SS
$\Sigma(p_{\mathrm{o}})$ is observable on $[0,t_{\mathrm{o}}]$. From [6] it
follows that there exists $k\geq 0$, such that the $k$ step observability
matrix of $\Sigma(p_{\mathrm{o}})$ is full column in DT, and it full column
rank for almost all $t$ on $(0,t_{\mathrm{o}})$ in the CT case. From [21,
Definition 3.34] it then follows that $\Sigma$ is structurally observable. The
last statement of the theorem follows from [21, Theorem 3.14]. ∎

##  V Conclusions

In this paper a characterization of minimal LPV-SSA realizations of LPV
behaviors in terms of observability has been presented. It has also been shown
that minimal LPV-SSA realizations of the same behavior are isomorphic. These
results represent the first steps towards a behavioral approach for LPV-SSAs.
Future work will be directed towards developing counterparts of i/o
partitioning, and kernel representations for manifest behaviors of LPV-SSAs.

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