# A hybrid systems framework for data-based adaptive control of linear time-
varying systems

Andrea Iannelli  Member, IEEE  Romain Postoyan  Senior Member, IEEE This work
was supported in part by the German Research Foundation (DFG) under Germany’s
Excellence Strategy - EXC 2075 - 390740016, and in part by the ANR under grant
OLYMPIA ANR-23-CE48-0006. Andrea Iannelli is with the Institute for Systems
Theory and Automatic Control, University of Stuttgart, Germany (e-mail:
andrea.iannelli@ist.uni-stuttgart.de).Romain Postoyan is with the Université
de Lorraine, CNRS, CRAN, F-54000 Nancy, France (e-mail: romain.postoyan@univ-
lorraine.fr).

###### Abstract

We consider the data-driven stabilization of discrete-time linear time-varying
systems. The controller is defined as a linear state-feedback law whose gain
is adapted to the plant changes through a data-based event-triggering rule. To
do so, we monitor the evolution of a data-based Lyapunov function along the
solution. When this Lyapunov function does not satisfy a designed desirable
condition, an episode is triggered to update the controller gain and the
corresponding Lyapunov function using the last collected data. The resulting
closed-loop dynamics hence exhibits both physical jumps, due to the system
dynamics, and episodic jumps, which naturally leads to a hybrid discrete-time
system. We leverage the inherent robustness of the controller and provide
general conditions under which various stability notions can be established
for the system. Two notable cases where these conditions are satisfied are
treated, and numerical results illustrating the relevance of the approach are
discussed.

Data-driven control; hybrid systems; Lyapunov stability; event-triggered
control; adaptive control; time-varying systems

##  1 Introduction

The problem of designing controllers for dynamical systems only using data
trajectories is a very active area of research. Motivations include the
increasing complexity of modern engineering and cyberphysical applications,
the large availability of data and the time and cost savings potentially
achieved by avoiding the modeling or system identification phase.
Additionally, for systems changing over time, updating models during operation
might not always be possible because by the time sufficiently many data for an
accurate identification have been collected, the system dynamics could have
substantially changed. In this work, we develop a framework for the synthesis
of data-based adaptive controllers for linear time-varying (LTV) systems with
unknown state and input matrices, undergoing unpredicted time-variations. This
problem poses two conceptual challenges. From a learning perspective, one
should answer the questions of _how_ and _when_ to update the controller on
the basis of data measured on-line. From a technical perspective, proving
closed-loop properties of the nonlinear time-varying interconnection between
the plant and such an adaptive controller requires careful modeling and
analysis steps. To address these challenges, we propose to cast this problem
as an instance of hybrid (discrete-time) dynamical systems [1, Section 4] and
blend concepts from the event-triggered control literature e.g., [2] with
recent results on data-based control [3]. Before elaborating more on the
proposed approach, we briefly review the related literature for linear time-
invariant (LTI) and LTV systems.

In the context of unknown LTI systems, the idea of designing a static state-
feedback controller such that the closed-loop dynamics enjoys a favourable
Lyapunov property can be traced back to [3]. Here, stabilizing controllers are
obtained by solving linear matrix inequalities (LMI) formulated in terms of
suitable data matrices containing state and input trajectories collected
offline, that is prior to the controller deployment. Different technical
solutions to achieve a similar goal were also proposed in other works
addressing the problems of noisy data [4] and integration of prior knowledge
in the form of a nominal model [5], to cite a few. Whereas most of the
approaches proposed in the literature use offline data and thus the resulting
controller is fixed during operation, a few works studied _adaptive_ data-
driven controllers, whereby the control law is modified during operation in
response to plant’s changes. In [6] the authors extended [3] to the class of
switched linear systems by using on-line collected data. The control update is
performed at every time instant and the stability guarantees rely on dwell-
time conditions. A similar approach was recently used in [7], where the more
general class of LTV is considered, and practical stability guarantees are
obtained by implementing a periodic update based on the assumed rate of change
of the system matrices. Data-driven control of LTV systems is also studied in
[8], where no on-line update is performed and finite horizon guarantees are
obtained by assuming that the data collected offline describe the behavior of
the system during operation. To the best of the authors’ knowledge, there is
no prior work that considers stabilizing data-driven adaptive controllers for
general LTV systems where on-line data are used not only to update
controllers, but also to decide on when the update should be performed.

We thus model the problem as a hybrid discrete-time system in the sense of [1,
Section 4], which exhibits two types of jumps that are either due to the
system dynamics, or to the triggering of a new episode. By episode, we mean
the update of the controller gain, chosen here as a linear state-feedback
gain, based on the last collected state and input data. We first present novel
results for this class of hybrid dynamical systems, whose applicability goes
beyond the scope of this work (Section 3). We then introduce all the variables
needed to describe the overall closed-loop model and thus to obtain a hybrid
discrete-time model, which allows us to formally state the stability problem
(Section 4). The objective is to design both the controller gain update rule
as well as the episode-triggering law for the closed-loop system to have
suitable stability properties. We also provide an explicit construction of the
controller update rule by revisiting the work in [3] adapted to the LTV
context (Section 5). Afterwards, we design the episode-triggering rule by
exploiting a data-driven Lyapunov function obtained in the controller gain
update step (Section 6). The triggering rule consists in monitoring whether
this Lyapunov function satisfies a desired decay rate along the solution. When
this is not the case, it means that the time-variations undergone by the plant
require a new controller to be designed and thus a new episode to be
triggered. Similar triggering rules have been proposed in e.g., [9, 10, 11,
12], where the set-up is different as a triggering instant refers there to a
sampling instant, and not to a controller and Lyapunov function update, which
leads to major modeling, analysis and design differences. In Section 7 we
exploit the previous constructive results to establish sufficient conditions
under which stability properties hold. A feature of these conditions is their
generality, which allows covering a range of scenarios. On the other hand,
these conditions may be difficult to check only based on data. We therefore
derive more specific conditions relating instances of the variations of the
system matrices to the stability guarantees. We finally provide two numerical
examples to illustrate the proposed methodology and compare it to time-
triggered controller updates strategies like in [7]. The proposed approach
show significant performance improvements and less episode triggering.

Compared to the related results mentioned above, the results do not explicitly
rely on conditions on the variations of the system matrices like in e.g., [7,
6] and the controller gain updates only occur when needed and not in a
prefixed manner. We also highlight that, although recent works have used
event-triggered concepts in the context of data-driven control e.g., [13, 14,
12], the set-up here is very different. In these references, the control law
is fixed once for all and the triggering rule is used to define the time at
which sensor or actuator information is updated. In this work, communication
between the plant and the controller occurs at every physical time step and
the triggering instant is the mechanism by which the controller gain is
adapted over time. These key differences require complete different
methodological tools.

##  2 Notation

$\mathbb{R}$ denotes the set of real numbers, $\mathbb{R}_{\geq
0}:=[0,\infty)$, $\mathbb{R}_{>0}:=(0,\infty)$, $\mathbb{Z}$ the set of
integers, $\mathbb{Z}_{\geq p}:=\\{p,p+1,\ldots\\}$ with $p\in\mathbb{Z}$ and
$\emptyset$ the empty set. Given $k_{1},k_{2}\in\mathbb{Z}_{\geq 0}$ with
$k_{1}<k_{2}$, we denote by $[k_{1},k_{2}]$ the finite set
$\\{k_{1},k_{1}+1,..,k_{2}-1,k_{2}\\}$. Given $n\in\mathbb{Z}_{\geq 1}$, we
denote by $\mathbb{S}^{n}$ the set of real symmetric matrices of dimension $n$
and by $\mathbb{S}^{n}_{\succeq 0}$ and $\mathbb{S}^{n}_{\succ 0}$ the set of
real symmetric positive semi-definite and positive definite matrices of
dimension $n$, respectively. When $S\in\mathbb{S}_{\succ 0}^{n}$ with
$n\in\mathbb{Z}_{\geq 1}$, its minimum and maximum eigenvalues are denoted
$\lambda_{\min}(S)$ and $\lambda_{\max}(S)$, respectively. The determinant of
a real square matrix is denoted $\text{det}(\cdot)$, and its spectrum norm by
$\|\cdot\|$. We use $\left[\begin{smallmatrix}A&B\\\
\star&C\end{smallmatrix}\right]$ in place of $\left[\begin{smallmatrix}A&B\\\
B^{\top}&C\end{smallmatrix}\right]$ for the sake of convenience. Given square
matrices $A_{1},\ldots,A_{n}\in\mathbb{R}^{n\times n}$ with
$n\in\mathbb{Z}_{\geq 1}$, we use $\text{diag}(A_{1},\ldots,A_{n})$ to denote
the block diagonal matrix of dimension $n^{2}\times n^{2}$ whose block
diagonal components are $A_{1},\ldots,A_{n}$. The symbol $\otimes$ stands for
the Kronecker product for matrices. Given $n,T\in\mathbb{Z}_{\geq 1}$,
$\mathcal{N}_{n,T}$ is the map from $\mathbb{R}^{n\times T}$ to
$\mathbb{R}^{nT\times T}$ such that, for any
$Z=[z_{1},\ldots,z_{T}]\in\mathbb{R}^{n\times T}$,
$\mathcal{N}_{n,T}(Z)=\left[\begin{smallmatrix}z_{0}&0&\ldots&0\\\
0&z_{1}&\ldots&\vdots\\\ \vdots&\ddots&\ddots&\vdots\\\
0&\cdots&\cdots&z_{T-1}\end{smallmatrix}\right]$, and we write
$\mathcal{N}(Z)$ when $n$ and $T$ are clear from the context. Given
$n,p,q\in\mathbb{Z}_{\geq 1}$ with $q\geq 2$ and a matrix
$M=(M_{ij})_{(i,j)\in[1,n]\times[1,p]}\in\mathbb{R}^{n\times p}$, $M_{q:p}$
stands for the matrix
$(M_{ij})_{(i,j)\in[1,n]\times[q,p]}\in\mathbb{R}^{n\times(p-q+1)}$ that is,
matrix $M$ truncated from its first $q-1$ columns. We use $I_{N}$ to denote
the identity matrix of size $N\in\mathbb{Z}_{\geq 1}$, and $\mathbf{1}_{N}$
for the unitary vector of $\mathbb{R}^{N}$. The notation $(x,y)$ stands for
$[x^{\top},\,y^{\top}]^{\top}$, where
$(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{m}$ and $n,m\in\mathbb{Z}_{\geq 1}$.
The symbols $\lor$ and $\land$ stand for the “or” and “and” logic operators,
respectively.

When we write $G:\mathcal{S}_{1}\rightrightarrows\mathcal{S}_{2}$ for some
sets $\mathcal{S}_{1}$, $\mathcal{S}_{2}$, it means that $G$ is a set-valued
map from $\mathcal{S}_{1}$ to $\mathcal{S}_{2}$. We consider $\mathcal{K}$,
$\mathcal{K}_{\infty}$ and $\mathcal{KL}$ functions as defined in [15, Section
3.5], and we say that a continuous function from $\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ is of class $\mathcal{L}$ if it is non-increasing
and converges to $0$ at infinity. We also write $\beta\in\exp-\mathcal{KL}$
when $\beta:\mathbb{R}_{\geq 0}\times\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq
0}$ and there exist $c_{1}\geq 1$ and $c_{2}>0$ such that
$\beta(s,t)=c_{1}e^{-c_{2}t}s$ for any $(s,t)\in\mathbb{R}_{\geq
0}\times\mathbb{R}_{\geq 0}$. Given a function $\alpha:\mathbb{R}_{\geq
0}\times\mathcal{S}\to\mathbb{R}_{\geq 0}$ with
$\mathcal{S}\subset\mathbb{R}^{n}$ for some $n\in\mathbb{Z}_{\geq 1}$, that is
invertible in its first argument, we denote the corresponding inverse (with
respect to the first argument) as $\alpha^{-1}(\cdot,z)$ for any
$z\in\mathcal{S}$. Finally, for a (set-valued) map $f$ and
$N\in\mathbb{Z}_{\geq 1}$, $f^{N}$ stands for the $N^{\text{th}}$ composition
of $f$ with itself when it is well-defined.

##  3 Background

###  3.1 Discrete-time hybrid systems

We consider dynamical systems given by

$\left\\{\begin{array}[]{rlllll}q^{+}=f(q)&&q\in\mathcal{C}\\\ q^{+}\in
G(q)&&q\in\mathcal{D},\end{array}\right.$ (1)

where $\mathcal{C}\subset\mathbb{R}^{n_{q}}$,
$\mathcal{D}\subset\mathbb{R}^{n_{q}}$, $f:\mathcal{C}\to\mathbb{R}^{n_{q}}$
and $G:\mathcal{D}\rightrightarrows\mathbb{R}^{n_{q}}$ a set-valued map with
$n_{q}\in\mathbb{Z}_{\geq 1}$; see [1, Section 4]. Solutions to (1) exhibit
two types of jumps depending on whether they belong to set $\mathcal{C}$ or to
set $\mathcal{D}$ as formalized below. In the setting of this paper, $q$ is
the concatenation of the plant state vector and some auxiliary variables,
$\mathcal{C}$ is the region of the state space where the system is operated,
and $\mathcal{D}$ is the region of state space where a new controller is
learned and, using the terminology adopted in the rest of the work, a new
episode is triggered; see Section 4. We therefore parameterize solutions to
(1) using two different times as in [1, Section 4] to distinguish jumps that
are due to the dynamics on $\mathcal{C}$ and jumps that are due to the
dynamics defined on $\mathcal{D}$. This is the reason why we call system (1) a
_hybrid discrete-time system_.

###  3.2 Solution concept

We adopt the notion of solution advocated in [1, Section 4]. A subset
$E\subset\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{\geq 0}$ is a _compact discrete
time domain_ if $E=\bigcup_{j=0}^{J-1}\bigcup_{k=k_{j}}^{k_{j+1}}(k,j)$ for
some finite sequence $0=k_{0}\leq k_{1}\leq\ldots\leq k_{J}$,
$k_{j}\in\mathbb{Z}_{\geq 0}$ for every $j\in[0,J]$ with $J\in\mathbb{Z}_{\geq
1}$, and it is a _discrete time domain_ if for any $(k,j)\in E$,
$E\cap([0,k]\times[0,j])$ is a compact discrete time domain [1, Definition
4.1]. A function $q:\text{dom}\,q\to\mathbb{R}^{n_{q}}$ is a _discrete arc_ if
$\text{dom}\,q$ is a discrete time domain [1, Definition 4.2]. We are ready to
define solutions to (1).

######  Definition 1 ([1, Definition 4.3])

A discrete arc $q:\text{dom}\,q\to\mathbb{R}^{n_{q}}$ is a _solution_ to
system (1) if

  1. (i)

for all $k,j\in\mathbb{Z}_{\geq 0}$ such that $(k,j),(k+1,j)\in\text{dom}\,q$,
$q(k,j)\in\mathcal{C}$ and $q(k+1,j)=f(q(k,j))$,

  2. (ii)

for all $k,j\in\mathbb{Z}_{\geq 0}$ such that $(k,j),(k,j+1)\in\text{dom}\,q$,
$q(k,j)\in\mathcal{D}$ and $q(k,j+1)\in G(q(k,j))$. $\Box$

Definition 1 essentially means that a solution to (1) jumps according to: (i)
the map $f$ when it lies in $\mathcal{C}$; (ii) the set-valued map $G$ when it
lies in $\mathcal{D}$; (iii) either $f$ or $G$ when it lies in
$\mathcal{C}\cap\mathcal{D}$. We note that, as mentioned above, solutions to
(1) are parameterized by two times: $k$ and $j$, which in our setting
respectively correspond to the physical time and the episodic time that is,
the number of episodes experienced so far by the solution. We call a pair
$(k,j)$ a _hybrid time_.

We give a few other definitions relevant to our work.

###### Definition 2

A solution $q$ to (1) is:

  * •

_maximal_ if there does not exist another solution $q^{\prime}$ to (1) such
that $\text{dom}\,q$ is a proper subset of $\text{dom}\,q^{\prime}$ and
$q(k,j)=q^{\prime}(k,j)$ for any $(k,j)\in\text{dom}\,q$;

  * •

_complete_ if $\text{dom}\,q$ is unbounded;

  * •

$k$-_complete_ if $\sup\\{k\in\mathbb{Z}_{\geq 0}\,:\,\exists
j\in\mathbb{Z}_{\geq 0}\,\,(k,j)\in\text{dom}\,q\\}=\infty$. $\Box$

We provide below conditions for any maximal solution to (1) to be
($k$-)complete.

###### Proposition 1

Consider system (1), the following holds.

  1. (i)

Any maximal solution is complete if and only if $f(\mathcal{C})\cup
G(\mathcal{D})\subset\mathcal{C}\cup\mathcal{D}$.

  2. (ii)

Any maximal solution is $k$-complete if $f(\mathcal{C})\cup
G(\mathcal{D})\subset\mathcal{C}\cup\mathcal{D}$ and there exists
$N\in\mathbb{Z}_{\geq 1}$ such that
$G^{N}(\mathcal{D})\cap\mathcal{D}=\emptyset$. $\Box$

Proof: (i) Suppose that any maximal solution to (1) is complete. This implies
that for any point $q$ in $\mathcal{C}$, $f(q)$ belongs to
$\mathcal{C}\cup\mathcal{D}$, and from any point $q$ in $\mathcal{D}$ and any
$g\in G(q)$, $g\in\mathcal{C}\cup\mathcal{D}$, otherwise we would have the
existence of a solution, which would leave $\mathcal{C}\cup\mathcal{D}$ after
one jump and would thus not be complete but this contradicts the made
assumption. We have established that when any maximal solution to (1) is
complete, $f(\mathcal{C})\cup
G(\mathcal{D})\subset\mathcal{C}\cup\mathcal{D}$. On the other hand, when
$f(\mathcal{C})\cup G(\mathcal{D})\subset\mathcal{C}\cup\mathcal{D}$, this
implies the forward invariance of $\mathcal{C}\cup\mathcal{D}$ for system (1)
in the sense that any solution to (1) initialized in
$\mathcal{C}\cup\mathcal{D}$ remains in this set for all future times. This
property implies that any maximal solution to (1) remains in
$\mathcal{C}\cup\mathcal{D}$ and is thus complete: item (i) of Proposition 1
holds.

(ii) Since $f(\mathcal{C})\cup
G(\mathcal{D})\subset\mathcal{C}\cup\mathcal{D}$, any maximal solution to (1)
is complete by item (i) of Proposition 1. Suppose that there exists a maximal
solution $q$ to (1) that is not $k$-complete. Since $q$ is complete that means
there exists $(k,j)\in\text{dom}\,q$ such that
$(k,j^{\prime})\in\text{dom}\,q$ for any $j^{\prime}\geq j$. Consequently, for
any $N\in\mathbb{Z}_{\geq 1}$,
$G^{N}(\\{q(k,j)\\})\cap\mathcal{D}\neq\emptyset$. This implies, as
$q(k,j)\in\mathcal{D}$, that $G^{N}(\mathcal{D})\cap\mathcal{D}\neq\emptyset$
for any $N\in\mathbb{Z}_{\geq 1}$: this contradicts the fact that there exists
$N\in\mathbb{Z}_{\geq 1}$ such that
$G^{N}(\mathcal{D})\cap\mathcal{D}=\emptyset$. We have obtained the desired
result by contradiction. $\blacksquare$

Lastly, we associate to any hybrid arc $q$ to (1) and any hybrid times
$(k,j)\in\text{dom}\,q$, the sequence $k_{i}\in\mathbb{Z}_{\geq 0}$,
$i\in\\{0,j\\}$ such that: $0=k_{0}\leq k_{1}\leq\ldots\leq k_{j}\leq k$; and
$\text{dom}\,q\cap([0,k]\times[0,j])=\big{(}\bigcup_{i\in[0,j-1]}[k_{i},k_{i+1}]\times\\{i\\}\big{)}$
$\bigcup\big{(}[k_{j},k]\times\\{j\\}\big{)}$. In words, the $k_{i}$’s will be
the physical times at which a new episode is triggered for solution $q$ in the
setting of this work.

###  3.3 Stability definitions

We will investigate stability of closed, unbounded sets of the form

$\mathcal{A}=\\{q=(q_{1},q_{2})\in\mathcal{S}_{1}\times\mathcal{S}_{2}\,:\,q_{1}=0\\},$
(2)

where $\mathcal{S}_{1}\subset\mathbb{R}^{n_{1}}$,
$\mathcal{S}_{2}\subset\mathbb{R}^{n_{2}}$,
$\mathcal{S}_{1}\cup\mathcal{S}_{2}=\mathcal{C}\cup\mathcal{D}$,
$n_{1},n_{2}\in\mathbb{Z}_{\geq 1}$ and $n_{1}+n_{2}=n_{q}$, for system (1).

###### Definition 3

Consider system (1) and set $\mathcal{A}$ in (2). We say that:

  * •

$\mathcal{A}$ is _stable_ if for any $\varepsilon\in\mathbb{R}_{>0}$, for any
$q_{2,0}\in\mathcal{S}_{2}$, there exists
$\delta(\varepsilon,q_{2,0})\in\mathbb{R}_{>0}$ such that any solution $q$
with $|q_{1}(0,0)|\leq\delta(\varepsilon,q_{2,0})$ and $q_{2}(0,0)=q_{2,0}$
verifies $|q_{1}(k,j)|\leq\varepsilon$ for any $(k,j)\in\text{dom}\,q$;

  * •

$\mathcal{A}$ is _uniformly stable_ if for any
$\varepsilon\in\mathbb{R}_{>0}$, there exists
$\delta(\varepsilon)\in\mathbb{R}_{>0}$ such that any solution $q$ with
$|q_{1}(0,0)|\leq\delta(\varepsilon)$ verifies $|q_{1}(k,j)|\leq\varepsilon$
for any $(k,j)\in\text{dom}\,q$;

  * •

$\mathcal{A}$ is _globally attractive_ if any complete solution $q$ verifies
$|q_{1}(k,j)|\to 0$ as $k+j\to\infty$;

  * •

$\mathcal{A}$ is _globally asymptotically stable (GAS)_ if it is stable and
globally attractive;

  * •

$\mathcal{A}$ is _uniformly globally asymptotically stable (UGAS)_ if there
exists $\beta\in\mathcal{KL}$ such that for any solution $q$,
$|q_{1}(k,j)|\leq\beta(|q_{1}(0,0)|,k+j)$ for any $(k,j)\in\text{dom}\,q$;

  * •

$\mathcal{A}$ is _uniformly globally exponentially stable (UGES)_ if there
exists $\beta\in\exp-\mathcal{KL}$ such that for any solution $q$,
$|q_{1}(k,j)|\leq\beta(|q_{1}(0,0)|,k+j)$ for any $(k,j)\in\text{dom}\,q$.
$\Box$

Definition 3 formalizes various stability notions and distinguishes whether
the stability property is uniform or not. This distinction is justified in the
context of this work because we will be dealing with a set $\mathcal{A}$ of
the form of (2), also called attractor, that is closed but not bounded.

We present next relaxed Lyapunov conditions to ensure the stability notions of
Definition 3 for $\mathcal{A}$ in (2) for system (1). By relaxed conditions,
we mean that the considered Lyapunov function candidate does not need to
strictly decrease along solutions to (1) at every point of the state space,
consistently with e.g., [15, Chapter 3.3]. This type of Lyapunov properties is
very natural for the data-driven control problem we address as this will be
shown in Section 7.

###  3.4 Lyapunov conditions

To prove the properties of Definition 3, we will be constructing a Lyapunov
function candidate $\mathcal{U}:\mathcal{C}\cup\mathcal{D}\cup
f(\mathcal{C})\cup G(\mathcal{D})\to\mathbb{R}_{\geq 0}$ verifying the next
properties.

  1. (P1)

There exist $\underline{\alpha},\overline{\alpha}:\mathbb{R}_{\geq
0}\times\mathcal{S}_{2}\to\mathbb{R}_{\geq 0}$ of class-$\mathcal{K}_{\infty}$
in their first argument such that
$\underline{\alpha}(|q_{1}|,q_{2})\leq\mathcal{U}(q)\leq\overline{\alpha}(|q_{1}|,q_{2})$
for any $q=(q_{1},q_{2})\in\mathcal{C}\cup\mathcal{D}$.

  2. (P2)

There exists $\nu_{c}:\mathcal{C}\to\mathbb{R}_{\geq 0}$ such that
$\mathcal{U}(f(q))\leq\nu_{c}(q)\mathcal{U}(q)$ for any $q\in\mathcal{C}$.

  3. (P3)

There exists $\nu_{d}:\mathcal{D}\to\mathbb{R}_{\geq 0}$ such that
$\mathcal{U}(g)\leq\nu_{d}(q)\mathcal{U}(q)$ for any $q\in\mathcal{D}$ and any
$g\in G(q)$.

Given (P2)-(P3), we have the next property for function $\mathcal{U}$ along
any solution to (1).

###### Lemma 1

Consider system (1) and suppose (P2)-(P3) hold. Any solution $q$ to (1)
satisfies

$\begin{array}[]{rlllll}\mathcal{U}(q(k,j))\leq\displaystyle\pi(q,k,j)\mathcal{U}(q(0,0))&&\forall(k,j)\in\text{dom}\,q,\end{array}$
(3)

with

$\begin{array}[]{rlllll}\overbrace{\prod_{j^{\prime}=0}^{j-1}\prod_{k^{\prime}=k_{j^{\prime}}}^{k_{j^{\prime}+1}-1}\nu_{c}(q(k^{\prime},j^{\prime}))\nu_{d}(q_{2}(k_{j^{\prime}+1},j^{\prime}))\prod_{k^{\prime\prime}=k_{j}}^{k-1}\nu_{c}(q(k^{\prime\prime},j))}^{\text{{\normalsize$\pi(q,k,j):=$}}},\end{array}$
(4)

and $k_{0},k_{1},\ldots,k_{j}$ as defined at the end of Section 3.2. $\Box$

Proof: Let $q$ be solution to (1), $(k,j)\in\text{dom}\,q$. By (P2), we have

$\mathcal{U}(q(k_{1},0))\leq\prod_{k^{\prime}=0}^{k_{1}-1}\nu_{c}(q(k^{\prime},0))\mathcal{U}(q(0,0)).$
(5)

At $(k_{1},1)$,

$\mathcal{U}(q(k_{1},1))\leq\nu_{d}(q(k_{1},0))\mathcal{U}(q(k_{1},0)),$ (6)

therefore
$\mathcal{U}(q(k_{1},1))\leq\prod_{k^{\prime}=0}^{k_{1}-1}\nu_{c}(q(k^{\prime},0))\nu_{d}(q(k_{1},0))$
$\times\mathcal{U}(q(0,0))$. We obtain the desired result by iterating the
same reasoning until $(k,j)$. $\blacksquare$

Property (3) is not enough to conclude about stability properties for set
$\mathcal{A}$ for system (1), extra conditions are required that are
formalized next.

###### Theorem 1

Consider system (1) and suppose (P1)-(P3) are satisfied. The following holds.

  1. (i)

If there exists $\vartheta:\mathbb{R}_{\geq
0}\times\mathcal{S}_{2}\to\mathbb{R}_{\geq 0}$ of class-$\mathcal{K}_{\infty}$
in its first argument such that for any solution $q$,

$\begin{array}[]{rlllll}\overline{\pi}(q,k,j)\leq\vartheta(|q_{1}(0,0)|,q_{2}(0,0))&\forall(k,j)\in\text{dom}\,q,\end{array}$
(7)

with
$\overline{\pi}(q,k,j):=\underline{\alpha}^{-1}\Big{(}\pi(q,k,j)\overline{\alpha}(|q_{1}(0,0)|,q_{2}(0,0)),$
$q_{2}(k,j)\Big{)}$, $\underline{\alpha},\overline{\alpha}$ in (P1) and $\pi$
in (4), then $\mathcal{A}$ is stable. In addition,

    1. (i-a)

if $\vartheta(s,\cdot)$ is constant for any $s\in\mathbb{R}_{\geq 0}$, then
$\mathcal{A}$ is uniformly stable,

    2. (i-b)

if for any complete solution $q$, $\overline{\pi}(q,k,j)\to 0$ as
$k+j\to\infty$, then $\mathcal{A}$ is GAS.

  2. (ii)

If there exists $\beta\in\mathcal{KL}$ (respectively,
$\beta\in\exp-\mathcal{KL}$) such that for any solution $q$,

$\begin{array}[]{rlllll}\overline{\pi}(q,k,j)\leq\beta(|q_{1}(0,0)|,k+j)&\forall(k,j)\in\text{dom}\,q,\end{array}$
(8)

then $\mathcal{A}$ is UGAS (respectively, UGES). $\Box$

Proof: (i) Let $q$ be a solution to (1). By Lemma 1,
$\mathcal{U}(q(k,j))\leq\pi(q,k,j)\mathcal{U}(q(0,0))$. Consequently, in view
of (P1), $|q_{1}(k,j)|\leq\overline{\pi}(q,k,j)$ with $\overline{\pi}$ defined
in item (i), and thus $|q_{1}(k,j)|\leq\vartheta(|q_{1}(0,0)|,$ $q_{2}(0,0))$
by (7). Set $\mathcal{A}$ is stable according to Definition 3 as the required
property holds by taking $\delta=\vartheta^{-1}(\varepsilon,q_{2,0})$ for any
$\varepsilon>0$ and any $q_{2,0}\in\mathcal{S}_{2}$. Item (i-a) then follows.
When, in addition to $\mathcal{A}$ being stable, $\overline{\pi}(q,k,j)\to 0$
for $k+j\to\infty$ when $q$ is complete, we have $|q_{1}(k,j)|\to 0$ as
$k+j\to\infty$, as $|q_{1}(k,j)|\leq\overline{\pi}(q,k,j)$ for any
$(k,j)\in\text{dom}\,q$. This means $\mathcal{A}$ is globally attractive and
thus GAS. (ii) The desired result is obtained by exploiting similar bounds as
above together with (8). $\blacksquare$

We will exploit the conditions of Theorem 1 to establish properties for the
data-driven control problem presented in the next section.

##  4 Problem statement

###  4.1 Plant

Consider the discrete-time LTV system

$\begin{array}[]{rlll}x(k+1)&=&A(k)x(k)+B(k)u(k),\end{array}$ (9)

where $x(k)\in\mathbb{R}^{n_{x}}$ is the state and $u(k)\in\mathbb{R}^{n_{u}}$
is the control input at time $k\in\mathbb{Z}_{\geq 0}$, with
$n_{x},n_{u}\in\mathbb{Z}_{\geq 1}$. Time-varying matrices $A(k)$ and $B(k)$
are _unknown_ and take values in $\mathbb{R}^{n_{x}\times n_{x}}$ and
$\mathbb{R}^{n_{x}\times n_{u}}$ for any $k\in\mathbb{Z}_{\geq 0}$,
respectively.

Our goal is to stabilize the origin of system (9) despite the fact that
$A(\cdot)$ and $B(\cdot)$ are unknown and time-varying. To address these two
challenges in a direct data-driven fashion whereby only measured data are
used, we propose using an event-triggered approach. Specifically, we apply the
last constructed control law until it is outdated in the sense that a state-
dependent condition related to some appropriately designed Lyapunov property
is _not_ fulfilled. The violation of this condition triggers a new episode
where, whenever this is possible, a new feedback law is computed based on the
last collected data, and then applied to plant (9). We then keep repeating
these steps, and provide a closed-loop stability analysis of this nonlinear
adaptive feedback interconnection by framing it as an instance of the hybrid
system class described in Section 3. We introduce next additional variables
needed to proceed with the design and the analysis of the feedback gains and
the episode-triggering condition.

###  4.2 Auxiliary variables for hybrid system modeling

####  4.2.1 Counter

We first introduce a counter $\kappa\in\mathbb{Z}_{\geq 0}$ of the physical
time on which $A$ and $B$ depend in (9); see Remark 4 below for a technical
justification. Hence, when no new episode is triggered,

$\begin{array}[]{rlllll}\kappa^{+}&=&\kappa+1,\end{array}$ (10)

and when a new episode is triggered, $\kappa$ keeps the same value,

$\begin{array}[]{rlllll}\kappa^{+}&=&\kappa.\end{array}$ (11)

As a result, by (9), when no episode is triggered,

$\begin{array}[]{rlllll}x^{+}&=&A(\kappa)x+B(\kappa)u,\end{array}$ (12)

and, when a new episode is triggered,

$\begin{array}[]{rlllll}x^{+}&=&x.\end{array}$ (13)

####  4.2.2 Data variables

To design a feedback controller with some desired properties for system
(12)-(13), we use at each episode-triggering time instant the values of the
plant state $x$ and of the input $u$ over the last $T$ physical time steps,
where $T\in\mathbb{Z}_{\geq 1}$ is a design parameter. We elaborate below on
the choice of $T$ in Remark 5 and on episodes that may be triggered before $T$
physical steps have past in Remark 1.

To model the data collected process, we introduce variables
$\widehat{X},X\in\mathbb{R}^{n_{x}\times T}$ and $U\in\mathbb{R}^{n_{u}\times
T}$, whose dynamics are, when no episode is triggered,

$\begin{array}[]{rlllll}\widehat{X}^{+}&=&[\widehat{X}_{2:T},x]\\\
X^{+}&=&[X_{2:T},A(\kappa)x+B(\kappa)u]\\\ U^{+}&=&[U_{2:T},u],\end{array}$
(14)

and, when a new episode is triggered,

$\begin{array}[]{rlllll}(\widehat{X}^{+},X^{+},U^{+})&=&(\widehat{X},X,U).\end{array}$
(15)

Equations (14)-(15) mean that, at time $(k,j)$ with $k\geq T$, $\widehat{X}$
and $U$ collect the values of $x$ and $u$ from physical time $k-T$ to $k-1$,
respectively, and $X$ collects the value of $x$ from $k-T+1$ to $k$ as they
remain unchanged when a new episode is triggered. The next lemma formalizes
these claims considering system (10)-(15) as a hybrid system of the form of
(1) where set $\mathcal{C}$ corresponds to no episode triggering and
$\mathcal{D}$ to episode triggering.

###### Lemma 2

Given $T\in\mathbb{Z}_{\geq 1}$, for any solution $(x,\kappa,\widehat{X},X,U)$
to (10)-(15) and any discrete arc111Although we did not define solutions to
hybrid discrete-time systems with external inputs in Section 3, in the
following $u$ will be a function of the state variables so that the adopted
notion of solution in Definition 1 will apply. $u$, for any
$(k,j)\in\text{dom}\,(x,\kappa,\widehat{X},X,U,u)$ with $k\geq T$,

$\begin{array}[]{rllll}\widehat{X}(k,j)&=&\left[x(k-T,j_{k-T})\,\ldots\,x(k-1,j_{k-1})\right]\in\mathbb{R}^{n_{x}\times
T}\\\
X(k,j)&=&\left[x(k-T+1,j_{k-T+1})\,\ldots\,x(k,j_{k})\right]\in\mathbb{R}^{n_{x}\times
T}\\\
\mathcal{U}(k,j)&=&\left[u(k-T,j_{k-T})\,\ldots\,u(k-1,j_{k-1})\right]\in\mathbb{R}^{n_{u}\times
T},\end{array}$ (16)

where $j_{k-i}\in\mathbb{Z}_{\geq 0}$ is such that
$(k-i,j_{k-i})\in\text{dom}\,(x,\kappa,\widehat{X},X,U,u)$ for any
$i\in\\{0,\ldots,T\\}$. $\Box$

Proof: Let $(x,\kappa,\widehat{X},X,U)$ be a solution to (10)-(15) with
discrete arc $u$. By (12)-(15),

$\begin{array}[]{rclll}\widehat{X}(1,j_{1})&=&[\widehat{X}_{2:T}(0,0),x(0,0)],\end{array}$
(17)

where $\widehat{X}_{2:T}(0,0)$ corresponds to the initialization of
$\widehat{X}$. Then,

$\begin{array}[]{rclll}\widehat{X}(2,j_{2})&=&[\widehat{X}_{2:T}(1,j_{1}),x(1,j_{1})]\\\
&\vdots&\\\
\widehat{X}(T,j_{2})&=&[\widehat{X}_{2:T}(T-1,j_{T-1}),x(T-1,j_{T-1})]\\\
&=&[x(0,0),x(1,j_{j}),\ldots,x(T-1,j_{T-1})].\end{array}$ (18)

By repeating these steps, we obtain for any
$(k,j)\in\text{dom}\,(x,\kappa,\widehat{X},X,U,u)$ with $k\geq T$,
$\widehat{X}(k,j)=[x(k-T,j_{k-T}),\ldots,x(k-1,j_{k-1})]$, which corresponds
to the first line of (16). We similarly derive the other equations of (16).
$\blacksquare$

Lemma 2 implies that variables $\widehat{X}$, $X$ and $U$ are consistent with
the data matrices commonly encountered for the data-driven control of LTI
systems (see, e.g., [3, 16, 17]) after $T$ physical time steps have elapsed.
Contrary to the offline LTI system setting where these have been previously
employed, it is essential here to include these data-based matrices in the
state vector as they evolve with time and play a key role in the design of the
adaptive controller.

###### Remark 1

Matrices $\widehat{X}$, $X$ and $U$ can be initialized with any real matrix of
the appropriate dimensions. When $k\leq T$, all the columns of $\widehat{X}$,
$X$ and $U$ may not be related to the plant states and input, respectively.
Our analysis covers this situation and we will be able to guarantee stability
properties as desired despite arbitrary initializations of $\widehat{X}$, $X$
and $U$. Nevertheless, in practice, it is reasonable to first run an
experiment over a “physical” time interval of length $T$ on the system, using
possibly an open-loop sequence of inputs, just before $(0,0)$ so that
$\widehat{X}$, $X$ and $U$ are initialized with relevant data at the initial
time. $\Box$

####  4.2.3 Controller variables

In this work we restrict our attention to the policy class of linear state-
feedback controllers, i.e., $u=Kx$, where $K\in\mathbb{R}^{n_{u}\times n_{x}}$
is the controller gain to be determined. Because this controller gain will
vary at each episode (only), it is natural to model it as a state variable,
whose dynamics depend on the most recently collected data, namely
$\widehat{X}$, $X$ and $U$ in (14)-(15). We also introduce the associated
Lyapunov-like matrix $S\in\mathcal{S}_{S}\subset\mathbb{S}_{\succ 0}^{n_{x}}$
with $\mathcal{S}_{S}$ bounded, as well as other related auxiliary variables
$a_{1}\in[0,1]$, $a_{2}\in\mathbb{R}_{\geq 0}$ and
$F\in\mathbb{S}^{n_{x}}_{\succ 0}$, which will be presented in Section 5. As
these variables are only updated at each new episode, the dynamics of
$K,S,F,a_{1},a_{2}$ when no episode is triggered is,

$\begin{array}[]{rlllll}(K^{+},S^{+},F^{+},a_{1}^{+},a_{2}^{+})&=&(K,S,F,,a_{1},a_{2}).\end{array}$
(19)

On the other hand, when a new episode is triggered $K,S,F,a_{1}$ and $a_{2}$
are updated according to

$\begin{array}[]{rlllll}(K^{+},S^{+},F^{+},a_{1}^{+},a_{2}^{+})&\in&L(\widehat{X},X,U),\end{array}$
(20)

where $L:\mathbb{R}^{n_{x}\times T}\times\mathbb{R}^{n_{x}\times
T}\times\mathbb{R}^{n_{u}\times T}\rightrightarrows\mathbb{R}^{n_{u}\times
n_{x}}\times\mathcal{S}_{S}\times\mathbb{S}^{n_{x}}_{\succ
0}\times[0,1]\times\mathbb{R}_{\geq 0}$ is a set-valued map to be designed.
Notice that $\widehat{X},X,U$ are only used at the episode triggering instant
to define the controller gain $K$ (and the associated variables $S$, $F$,
$a_{1}$, $a_{2}$).

###### Remark 2

Since $u=Kx$, (12) reads $x^{+}=A(\kappa)x+B(\kappa)Kx$, and the update of $U$
in (14) between two successive episode-triggering instants is given by
$U^{+}=[U_{2:T},Kx]$. This will be reflected in the definition of map $f$ for
system (1), see (24) below. $\Box$

####  4.2.4 Episode-triggering variables

We will finally need some auxiliary variables to design the episode-triggering
condition, which we denote by
$\eta\in\mathcal{S}_{\eta}\subset\mathbb{R}^{n_{\eta}}$ with
$n_{\eta}\in\mathbb{Z}_{\geq 1}$. We write the dynamics of the $\eta$-system
as, when no episode is being triggered,

$\begin{array}[]{rlllll}\eta^{+}&=&h(x,\kappa,\eta)\end{array}$ (21)

and when an episode is triggered,

$\begin{array}[]{rlllll}\eta^{+}&=&\ell(x,\kappa,\eta).\end{array}$ (22)

We will see in Section 6 how to define $\eta$ and its dynamics to ensure the
desired stability properties. Adding extra variables to define triggering
conditions is well-known in the event-triggered control literature, see, e.g.,
[18, 11].

###  4.3 Hybrid model

We collect the variables introduced so far to form the state vector

$q:=(x,\kappa,\widehat{X},X,U,K,S,F,a_{1},a_{2},\eta)\in\mathcal{S}_{q}$ (23)

where $\mathcal{S}_{q}:=\mathbb{R}^{n_{x}}\times\mathbb{Z}_{\geq
0}\times\mathbb{R}^{n_{x}\times T}\times\mathbb{R}^{n_{x}\times
T}\times\mathbb{R}^{n_{u}\times T}\times\mathbb{R}^{n_{u}\times
n_{x}}\times\mathcal{S}_{S}\times\mathbb{S}^{n_{x}}_{\succ
0}\times[0,1]\times\mathbb{R}_{\geq 0}\times\mathcal{S}_{\eta}$. We can then
model the overall closed-loop system as (1) with, in view of Section 4.2,

$\begin{array}[]{rlllll}{\small\begin{aligned}
f(q)\\!:=\\!\left(\begin{matrix}A(\kappa)x+B(\kappa)Kx\\\ \kappa+1\\\
[\widehat{X}_{2:T},x]\\\ [X_{2:T},A(\kappa)x+B(\kappa)Kx]\\\ [U_{2:T},Kx]\\\
K\\\ S\\\ F\\\ a_{1}\\\ a_{2}\\\
h(q)\end{matrix}\right),\,G(q)\\!:=\\!\left(\begin{matrix}x\\\ \kappa\\\
\widehat{X}\\\ X\\\ U\\\ L(\widehat{X},X,U)\\\
\ell(q)\end{matrix}\right),\end{aligned}}\end{array}$ (24)

where $f$ is defined over $\mathcal{C}$ and $g$ over $\mathcal{D}$, both sets
that will be defined in the following in Section 6.2.

###### Remark 3

Variables $\widehat{X},X,U,K,S,F$ are matrix-valued while $q$ in Section 3 is
given as a vector. This inconsistency can easily be overcome by vectorizing
these matrix variables. We have chosen not to do so in (24) to not over
complicate the notation and keep the exposition as simple as possible. $\Box$

###### Remark 4

The reason why we have introduced $\kappa$ is to obtain an autonomous hybrid
model of the form of (1), like in [15, Example 3.3] and [19]. When a solution
$q$ to (1) with (24) is such that $\kappa(0,0)=0$, then $\kappa(k,j)=k$ for
any $k\in\mathbb{Z}_{\geq 0}$ such that there exists $j\in\mathbb{Z}_{\geq 0}$
with $(k,j)\in\text{dom}\,q$, in other words $\kappa$ corresponds to the
physical time $k$. On the other hand, by initializing $\kappa$ to an integer
value different from zero, we allow the initial physical time on which $A$ and
$B$ depend in the original plant equation (9) to be non-zero, which is
important when dealing with such time-varying systems. In that way, while any
solution to $(\ref{eq:sys-background})$ has for initial time $(0,0)$ by
Definition 1, the matrices $A$ and $B$ are allowed to initially depend on
$\kappa(0,0)\neq 0$. $\Box$

###  4.4 Design objective

Consider system (1) with (24), the objective is to design set-valued map $L$,
the triggering condition, i.e., the sets $\mathcal{C}$ and $\mathcal{D}$ as
well as $\eta$ and its dynamics, to ensure stability properties for set
$\mathcal{A}$ defined as

$\mathcal{A}:=\left\\{q\in\mathcal{S}_{q}\,:\,x=0\right\\}.$ (25)

With the notation of Section 3.3, this corresponds to set $\mathcal{A}$ in (2)
with

$\displaystyle q_{1}$ $\displaystyle=x,\quad
q_{2}=(\kappa,\widehat{X},X,U,K,S,F,a_{1},a_{2},\eta),\quad\mathcal{S}_{1}=\mathbb{R}^{n_{x}},$
(26) $\displaystyle\mathcal{S}_{2}$ $\displaystyle=\mathbb{Z}_{\geq
0}\times\mathbb{R}^{n_{x}\times T}\times\mathbb{R}^{n_{x}\times
T}\times\mathbb{R}^{n_{u}\times T}\times\mathbb{R}^{n_{u}\times
n_{x}}\times...$ $\displaystyle\mathcal{S}_{S}\times\mathbb{S}^{n_{x}}_{\succ
0}\times[0,1]\times\mathbb{R}_{\geq 0}\times\mathcal{S}_{\eta}.$

We provide for this purpose conditions under which (P1)-(P3) in Section 3
hold. In particular, we give in Section 5 conditions allowing to design $L$
and thus the adaptation law for the feedback gain $K$ such that desirable
Lyapunov properties are guaranteed on $\mathcal{C}$. In Section 6, we derive
Lyapunov properties for system (1), (24) on $\mathcal{D}$. We then merge the
results of Sections 5 and 6 and show that the established Lyapunov properties
imply the satisfaction of (P1)-(P3), and finally derive conditions allowing
the application of Theorem 1 in Section 7.

##  5 Controller design

We present the method employed to design a new controller when an episode is
triggered, i.e., a method to design the set-valued map $L$ in (20).

###  5.1 Data-based modeling

For any $\kappa\in\mathbb{Z}_{\geq 0}$, we define

$\begin{array}[]{rllll}\mathcal{A}(\kappa)&:=&\left[A(\kappa-T)\,\ldots\,A(\kappa-1)\right]\in\mathbb{R}^{n_{x}\times
n_{x}T}\\\
\mathcal{B}(\kappa)&:=&\left[B(\kappa-T)\,\ldots\,B(\kappa-1)\right]\in\mathbb{R}^{n_{x}\times
n_{u}T}.\end{array}$ (27)

These matrices are _not_ used for design but only for analysis purpose. The
next result is a direct consequence of Lemma 2 and (27), its proof is
therefore omitted.

###### Lemma 3

For any solution $q$ to (1), (24) and any $(k,j)\in\text{dom}\,q$ with $k\geq
T$,

$\begin{array}[]{rlllll}X(k,j)=\mathcal{A}(\kappa(k,j))\mathcal{N}(\widehat{X}(k,j))+\mathcal{B}(\kappa(k,j))\mathcal{N}(\mathcal{U}(k,j)),\end{array}$
(28)

where the map $\mathcal{N}$ is defined in the notation part. $\Box$

Next, we define variable

$\begin{array}[]{rlllll}\mathrm{d}:=(\kappa,\widehat{X},X,U)\in\mathcal{S}_{\mathrm{d}},\end{array}$
(29)

with $\mathcal{S}_{\mathrm{d}}:=\mathbb{Z}_{\geq
0}\times\mathbb{R}^{n_{x}\times T}\times\mathbb{R}^{n_{x}\times
T}\times\mathbb{R}^{n_{u}\times T}$ and, for any
$\mathrm{d}\in\mathcal{S}_{\mathrm{d}}$ and
$\kappa^{\prime}\in\mathbb{Z}_{\geq 0}$,

$D(\mathrm{d},\kappa^{\prime}):=A(\kappa^{\prime})\widehat{X}-\mathcal{A}(\kappa)\mathcal{N}(\widehat{X})+B(\kappa^{\prime})U-\mathcal{B}(\kappa)\mathcal{N}(U),$
(30)

which belongs to $\mathbb{R}^{n_{x}\times T}$. We can equivalently rewrite
(30) as

$\begin{array}[]{rlllll}D(\mathrm{d},\kappa^{\prime})&=&(A(\kappa^{\prime})\otimes\mathbf{1}_{T}^{\top}-\mathcal{A}(\kappa))\mathcal{N}(\widehat{X})\\\
&&+(B(\kappa^{\prime})\otimes\mathbf{1}_{T}^{\top}-\mathcal{B}(\kappa))\mathcal{N}(U).\end{array}$
(31)

We can interpret $D(\mathrm{d},\kappa^{\prime})$ as a matrix “measure” of the
distance between the matrices $A(\kappa^{\prime})$ and $B(\kappa^{\prime})$
evaluated at a given $\kappa^{\prime}\in\mathbb{Z}_{\geq 0}$, typically
greater than or equal to $\kappa$, and the corresponding matrices in
$\mathcal{A}(\kappa)$ and $\mathcal{B}(\kappa)$ in (27). Equation (31) shows
indeed that, if system (12) is time-invariant, then
$D(\mathrm{d},\kappa^{\prime})\equiv 0$ for any
$\kappa^{\prime}\in\mathbb{Z}_{\geq 0}$; otherwise, there exists
$\kappa^{\prime}\in\mathbb{Z}_{\geq 0}$ such that
$D(\mathrm{d},\kappa^{\prime})$ might be non-zero compatibly with the data
collected in $\mathcal{N}(\widehat{X})$ and $\mathcal{N}(U)$.

###  5.2 Matrix ellipsoids for LTV systems

Because gain $K$ is designed based on the values of $\widehat{X}$, $X$ and $U$
at the last episode-triggering instant in view of (24) and that $A$ and $B$
are time-varying, these data only carry partial information on the true system
matrices at the current time. We characterize here the uncertainty associated
with using the data-based representation (28) given $\widehat{X}$, $X$ and $U$
in place of the true plant model (12).

Motivated by the interpretation of $D$ above, we define the following matrix
function for any $\mathrm{d}\in\mathcal{S}_{\mathrm{d}}$,
$M_{A}\in\mathbb{R}^{n_{x}\times n_{x}}$ and $M_{B}\in\mathbb{R}^{n_{u}\times
n_{x}}$,

$\widetilde{D}(\mathrm{d},M_{A},M_{B}):=M_{A}\widehat{X}-\mathcal{A}(\kappa)\mathcal{N}(\widehat{X})+M_{B}U-\mathcal{B}(\kappa)\mathcal{N}(U).$
(32)

Clearly $\widetilde{D}(\mathrm{d},M_{A},M_{B})=D(\mathrm{d},\kappa^{\prime})$
when $M_{A}=A(\kappa^{\prime})$ and $M_{B}=B(\kappa^{\prime})$ for some
$\kappa^{\prime}\in\mathbb{Z}_{\geq 0}$ in view of (31). Using (32), for any
$\mathrm{d}\in\mathcal{S}_{\mathrm{d}}$ and $F\in\mathbb{S}^{n_{x}}_{\succ
0}$, we introduce the matrix set

$\displaystyle\mathcal{E}(\mathrm{d},F):=\big{\\{}$
$\displaystyle\left[M_{B}\,\,M_{A}\right]^{\top}\in\mathbb{R}^{(n_{u}+n_{x})\times
n_{x}}\,:\,$ (33)
$\displaystyle\widetilde{D}(\mathrm{d},M_{A},M_{B})\widetilde{D}(\mathrm{d},M_{A},M_{B})^{\top}\preceq
F\big{\\}},$

defining the set of matrices that are _close_ to the data-based representation
(28). It is useful to introduce this set because, given
$\kappa^{\prime}\in\mathbb{Z}_{\geq 0}$, the closeness condition for the
system matrices
$\left[B(\kappa^{\prime})\,\,A(\kappa^{\prime})\right]^{\top}\in\mathcal{E}(\mathrm{d},F)$
is equivalent to
$D(\mathrm{d},\kappa^{\prime})D(\mathrm{d},\kappa^{\prime})^{\top}\preceq F$,
which is a condition that can be enforced using robust control tools. We will
sometimes omit in the reminder the arguments of $\mathcal{E}$, and of related
sets, whenever this is clear from the context.

The definition of set $\mathcal{E}$ raises two important questions. First,
despite the quadratic dependence on $M_{A}$ and $M_{B}$, it is unclear by
simple inspection of (32) under which conditions $\mathcal{E}$ is non-empty
and whether it is an ellipsoidal set akin to those encountered in the recent
literature on data-driven control of LTI systems. Second, the set in (33)
depends on $\mathcal{A}(\kappa)$ and $\mathcal{B}(\kappa)$ and thus on the
unknown time-dependent matrix-valued maps $A(\cdot)$ and $B(\cdot)$ in view of
(27), and therefore cannot be explicitly computed. The next result, which
builds on the concept of matrix ellipsoid [20] and some related results shown
in [21], addresses both questions. Its proof is given in the appendix.

###### Lemma 4

Given $T\in\mathbb{Z}_{\geq 1}$, let $\mathrm{d}\in\mathcal{S}_{\mathrm{d}}$
be such that

$X=\mathcal{A}(\kappa)\mathcal{N}(\widehat{X})+\mathcal{B}(\kappa)\mathcal{N}(U)$
(34)

with $\mathcal{A},\mathcal{B}$ in (27). Define
$Z:=\left[\begin{smallmatrix}U\\\ \widehat{X}\end{smallmatrix}\right]$ and
$M:=ZZ^{\top}$. Given $F\in\mathbb{S}^{n_{x}}_{\succ 0}$ and the associated
set $\mathcal{E}(\mathrm{d},F)$ in (33), the following holds. When $Z$ has
full row rank,

  1. (i)

$\mathcal{E}(\mathrm{d},F)$ is non-empty if and only if

$\begin{array}[]{rlllll}\Delta:=XZ^{\top}M^{-1}ZX^{\top}-XX^{\top}+F\succeq
0,\end{array}$ (35)

  2. (ii)

for any $M_{A}\in\mathbb{R}^{n_{x}\times n_{x}}$ and
$M_{B}\in\mathbb{R}^{n_{u}\times n_{x}}$,

$\left[M_{B}\,\,M_{A}\right]^{\top}\in\mathcal{E}(\mathrm{d},F)\Leftrightarrow\left[M_{B}\,\,M_{A}\right]^{\top}\in\widehat{\mathcal{E}}(\mathrm{d},F),$
(36)

where $\widehat{\mathcal{E}}(\mathrm{d},F)$ is a data-based matrix ellipsoid
defined as

$\displaystyle\widehat{\mathcal{E}}(\mathrm{d},F):=\big{\\{}\widehat{Z}:(\widehat{Z}-Z_{c})^{\top}M(\widehat{Z}-Z_{c})\preceq\Delta\big{\\}}$
(37)

with
$Z_{c}:=\left(ZZ^{\top}\right)^{-1}ZX^{\top}\in\mathbb{R}^{(n_{u}+n_{x})\times
n_{x}}$.

When $Z$ is not full row rank, let us define $m\in[1,n_{u}+n_{x}-1]$ the
dimension222The case $m=0$ is not of interest since it corresponds to the
system at equilibrium at $x=0$. of the image of $Z$. Then

  1. (iii)

$\mathcal{E}(\mathrm{d},F)$ is non-empty if and only if

$\begin{array}[]{rlllll}\widetilde{\Delta}:=XZ^{\top}\widetilde{V}\widetilde{M}^{-1}\widetilde{V}^{\top}ZX^{\top}-XX^{\top}+F\succeq
0,\end{array}$ (38)

where $\widetilde{M}:=\widetilde{V}^{\top}M\widetilde{V}$,
$\widetilde{V}^{\top}\in\mathbb{R}^{m\times n_{x}+n_{u}}$ and
$\widetilde{W}^{\top}\in\mathbb{R}^{(n_{x}+n_{u}-m)\times n_{x}+n_{u}}$ are
the matrices built with a basis of the image and the left kernel of $Z$,
respectively,

  2. (iv)

if $\mathcal{E}(\mathrm{d},F)$ is non-empty, then it is unbounded. $\Box$

Under the full row rank assumption of $Z$, set $\mathcal{E}$ from (33) is an
ellipsoid in the space of real system matrices of dimensions
$(n_{u}+n_{x})\times n_{x}$ and $\widehat{\mathcal{E}}$ in (37) is an
equivalent data-based description (item (ii)). When this assumption does not
hold, an equivalent geometric characterization as an unbounded ellipsoidal-
type set is given in the proof of item (iv) in the appendix (cf. (81)). In
both cases, Lemma 4 provides a data-based test to check non-emptiness of
$\mathcal{E}$ (items (i) and (iii)).

###  5.3 Lyapunov property

We exploit set $\mathcal{E}$ in (33) to define the next central closed-loop
property guiding the control and episode-triggering condition design.

###### Property 1

Given $T\in\mathbb{Z}_{\geq 1}$ and $\mathrm{d}\in\mathcal{S}_{\mathrm{d}}$,
there exist $a_{1}\in[0,1]$, $a_{2}\in\mathbb{R}_{\geq 0}$,
$F\in\mathbb{S}^{n_{x}}_{\succ 0}$, $S\in\mathcal{S}_{S}$ and
$K\in\mathbb{R}^{n_{u}\times n_{x}}$ such that for any
$\varepsilon\in\mathbb{R}_{\geq 0}$,

$\begin{array}[]{rlllll}\begin{aligned}
&\left[M_{B}\,\,M_{A}\right]^{\top}\in\mathcal{E}(\mathrm{d},F+\varepsilon
S^{-1})\Rightarrow\\\
&V((M_{A}+M_{B}K)x,S)\leq(a_{1}+a_{2}\varepsilon)V(x,S)\quad\forall
x\in\mathbb{R}^{n_{x}},\end{aligned}\end{array}$
($\mathcal{P}_{T}(\mathrm{d})$)

with $V(x,S):=x^{\top}Sx$ and $\mathcal{E}$ defined in (33). $\Box$

Property 1 implies the existence of a piecewise quadratic function $V$ that
characterizes the closed-loop response of _any_ plant model described by state
and input matrices included in the set $\mathcal{E}(\mathrm{d},F+\varepsilon
S^{-1})$. This property is used to define the variables $S,F,a_{1},a_{2}$
mentioned in Section 4.2.3. We see that matrix $S$ plays the role of Lyapunov-
like matrix as already hinted at. Matrix $F$ characterizes the set of matrices
$\mathcal{E}(\mathrm{d},F)$ for which the function $V$ decays with rate333We
call $a_{1}$ decay rate although it can be equal to $1$ with some slight abuse
as $a_{1}$ will typically belong to $[0,1)$. $a_{1}$ along the corresponding
solutions. On the other hand, matrix $\varepsilon S^{-1}$ allows enlarging the
set for which the right hand-side of ($\mathcal{P}_{T}(\mathrm{d})$) holds
from $\mathcal{E}(\mathrm{d},F)$ to $\mathcal{E}(\mathrm{d},F+\varepsilon
S^{-1})$, which may result in the loss of the non-increasing property of $V$
along the corresponding solutions. Indeed, we see that when $\varepsilon\geq
0$ is big enough, $a_{1}+a_{2}\varepsilon$ becomes bigger than $1$. Note
finally that the choice to take $\varepsilon S^{-1}$ to inflate set
$\mathcal{E}(\mathrm{d},F)$ in place of a generic $E\in\mathbb{S}_{\succeq
0}^{n_{x}}$ is not restrictive as, given any $E\in\mathbb{S}^{n_{x}}_{\succeq
0}$, we can always take
$\varepsilon:=\min\left\\{\varepsilon^{\prime}\in\mathbb{R}_{\geq
0}\,:\,\varepsilon^{\prime}S^{-1}\succeq E\right\\}$, which exists as
$E\in\mathbb{S}^{n_{x}}_{\succeq 0}$ and $S\in\mathbb{S}^{n_{x}}_{\succ 0}$.

###  5.4 Intra-episodic control design

Given Property 1, we can define the map $L$ in (20) as any set-valued map from
$\mathbb{R}^{n_{x}\times T}\times\mathbb{R}^{n_{x}\times
T}\times\mathbb{R}^{n_{u}\times T}$ to $\mathbb{R}^{n_{u}\times
n_{x}}\times\mathcal{S}_{S}\times\mathbb{S}^{n_{x}}_{\succ
0}\times[0,1]\times\mathbb{R}_{\geq 0}$ such that, for any
$\mathrm{d}\in\mathcal{S}_{\mathrm{d}}$,

$\begin{array}[]{rlllll}L(\widehat{X},X,U)\\!\subset\\!\Big{\\{}(K^{\prime},S^{\prime},F^{\prime},a_{1}^{\prime},a_{2}^{\prime})\,:\mathcal{P}_{T}(\mathrm{d})\text{
holds}\Big{\\}}.\end{array}$ (39)

We present below a possible construction of $L$. The key observation is that,
due to the set inclusion nature of Property 1, $K$ should possess some
inherent robustness. We therefore take inspiration from the results in [3]
valid for LTI systems, and extend them to our setting. We emphasize that
Property 1 can also be achieved via alternative robust data-based approaches,
e.g., Petersen’s lemma [21] or S-lemma [4].

###### Proposition 2

Given $T\in\mathbb{Z}_{\geq 1}$ consider any
$\mathrm{d}\in\mathcal{S}_{\mathrm{d}}$ such that

$\begin{array}[]{rlllll}X&=&\mathcal{A}(\kappa)\mathcal{N}(\widehat{X})+\mathcal{B}(\kappa)\mathcal{N}(U).\end{array}$
(40)

If there exist $\varsigma\in\mathbb{R}_{>0}$, $Y\in\mathbb{R}^{T\times
n_{x}},H\in\mathbb{S}^{n_{x}}_{\succ 0}$ such that the following semi-definite
program has a feasible solution

$\left[\begin{matrix}\widehat{X}Y-\varsigma XX^{\top}-H&XY\\\
\star&\widehat{X}Y\end{matrix}\right]\succ
0,\quad\left[\begin{matrix}I_{T}&Y\\\
\star&\widehat{X}Y\end{matrix}\right]\succ 0.$ (41)

Then Property 1 holds with any $F\in\mathbb{S}^{n_{x}}_{\succ 0}$ satisfying
$F\prec\frac{\varsigma}{1+\varsigma}H$, $a_{1}=1-a$,
$a=\max\big{\\{}a^{\prime}\in\mathbb{R}_{\geq 0}\,:\,a^{\prime}S^{-1}\preceq
H-(1+\varsigma^{-1})F\big{\\}}$, $a_{2}=1+\varsigma^{-1}$,
$K:=UY(\widehat{X}Y)^{-1}$ and $S=(\widehat{X}Y)^{-1}\succ 0$. $\Box$

The proof of Proposition 2 is given in the appendix. Proposition 2 provides
data-based LMI conditions that can be tested on-line to ensure that Property 1
is guaranteed. Contrary to standard results in the recent data-based
literature [3, 21], Proposition 2 does not assume persistence of excitation of
the collected data sequence, more precisely that
$\big{[}\begin{matrix}U^{\top}\;\widehat{X}^{\top}\end{matrix}\big{]}^{\top}$
has full row rank (as in items (i)-(ii) of Lemma 4). This choice is made due
to the on-line nature of the algorithm, whereby it may be difficult to
guarantee a priori that such a condition will be satisfied without adding
exploratory (e.g. random) signals. The drawback is that the search for the
gain $K$ is restricted to a smaller space as shown in the proof of Proposition
2 in the appendix (cf. (82)). Note however that, due to the time-varying
dynamics, even if persistence of excitation was verified we would not obtain
in general necessary and sufficient control design statements as it is instead
the case for LTI systems [3].

In view of Proposition 2, a possible definition of $L$ in (39) is, for any
$\widehat{X},X\in\mathbb{R}^{n_{x}\times T}$ and $U\in\mathbb{R}^{n_{u}\times
T}$,

$\begin{array}[]{rlllll}\begin{aligned}
&L(\widehat{X},X,U):=\Big{\\{}(K,S,F,a_{1},a_{2})\,:\\\
&\exists(\varsigma,Y,H)\in\mathbb{R}_{>0}\times\mathbb{R}^{T\times
n_{x}}\times\mathbb{S}^{n_{x}}_{\succ 0}\text{ s.t. }(\ref{eq:KDesign})\text{
holds},&\\\ &K,S,F,a_{1},a_{2}\text{ as in Proposition \ref{prop:initial-
learning-young}}\Big{\\}}.\end{aligned}\end{array}$ (42)

###### Remark 5

The value of $T$ defines the number of columns in the data matrices
$\widehat{X},X,U$ and in data-driven control works is typically chosen large
enough to guarantee that
$\big{[}\begin{matrix}U^{\top}\;\widehat{X}^{\top}\end{matrix}\big{]}^{\top}$
has full row rank. Because this property is not used here, there is no strict
lower bound on its value for the statements of our work to hold. However, the
choice of $T$ has an important effect on the map $L$ (42) and can be seen as
navigating the important trade-off between using more of the past information
to design the new controller and increasing the set of matrices
$\mathcal{E}(\mathrm{d},F)$ with guaranteed decaying $V$ along the
corresponding solutions. Suggesting a systematic selection of this parameter
is thus an important topic of future research. $\Box$

##  6 episode-triggering

We explain in this section how to define the episode-triggering condition,
i.e., how to design sets $\mathcal{C}$ and $\mathcal{D}$ as well as $\eta$ and
its dynamics for system (1), (24).

###  6.1 Main idea

Assuming Property 1 holds, the idea is to monitor the Lyapunov function $V$
along the on-going solution. As long as this function strictly decreases with
a certain decay rate there is no need to trigger a new episode as this means a
desirable stability property holds. If we detect that this is not the case,
this means we need to update the controller gain and a new episode is
triggered, but this can only occur when $L(\widehat{X},X,U)$ is non-empty,
which will be reflected in the definition of set $\mathcal{C}$. Similar
triggering techniques have been developed in different contexts, see e.g., [9,
10, 11, 12].

To formalize this idea, we need to introduce a couple of auxiliary variables,
which will form the vector $\eta$ in Section 4.3. We first introduce
$\hat{x}$, which is essentially the value of $x$ at the last physical time.
Hence, the dynamics of the $\hat{x}$-system is given by, when no episode is
triggered,

$\begin{array}[]{rlllll}\hat{x}^{+}&=&x\end{array}$ (43)

and, when an episode is triggered and thus the physical time is “frozen”,

$\begin{array}[]{rlllll}\hat{x}^{+}&=&\hat{x}.\end{array}$ (44)

Thanks to $\hat{x}$, we can now compare the value of $V$ at the current plant
state, namely $V(x,S)$, with its value at the previous physical time, namely
$V(\hat{x},S)$. We can thus monitor on-line whether

$\begin{array}[]{rlllll}V(x,S)&\leq&\sigma(a_{1})V(\hat{x},S),\end{array}$
(45)

where $\sigma(a_{1})\in[0,1]$ is a desired decay rate of function $V$ along
the solutions to (1), (24). In particular, we would take
$\sigma(a_{1})\in\big{[}a_{1},1]$ so that $\sigma(a_{1})$ is greater than or
equal to the nominal decay rate of $V$ in Property 1, namely $a_{1}$. When
$\sigma(a_{1})<1$, (45) guarantees the strict decrease of $V$ along the
considered solution. On the other hand, when
$V(x,S)\geq\sigma(a_{1})V(\hat{x})$, a new episode is triggered if possible,
i.e., if $L(\widehat{X},X,U)\neq\emptyset$. Because the values of $x$ and
$\hat{x}$ do not change at each jump corresponding to a new episode in view of
(24) and (44), the condition $V(x,S)\geq\sigma(a_{1})V(\hat{x},S)$ may still
hold after triggering a new episode. If such a situation occurs, this would
lead to a Zeno-like behavior in the sense that infinitely many episodes will
be triggered in finite physical time. To avoid this shortcoming, we introduce
a toggle variable $\tau\in\mathbb{Z}_{\geq 0}$. The dynamics of $\tau$ is

$\begin{array}[]{rlllll}\tau^{+}=1\quad q\in\mathcal{C},&\tau^{+}=0\quad
q\in\mathcal{D}.\end{array}$ (46)

We only allow a new episode to be triggered when $\tau=1$, which enforces that
at least one physical time step has elapsed since the last episode-triggering
instant.

Having introduced all the auxiliary variables, we can define

$\begin{array}[]{rlllll}\eta:=(\hat{x},\tau)\in\mathcal{S}_{\eta}\text{ with
}\mathcal{S}_{\eta}:=\mathbb{R}^{n}\times\\{0,1\\},\end{array}$ (47)

whose dynamics is given by (21) and (22) with maps

$\begin{array}[]{rlllll}h(q):=(x,1)\,\,\forall
q\in\mathcal{C},&&\ell(q):=(\hat{x},0)\,\,\forall q\in\mathcal{D}.\end{array}$
(48)

$\begin{array}[]{rllll}\mathcal{C}&:=&\left\\{q:V(x,S)\leq\sigma(a_{1})V(\hat{x},S)\lor\tau=0\lor
L(\widehat{X},X,U)=\emptyset\right\\}\\\
\mathcal{D}&:=&\left\\{q:V(x,S)\geq\sigma(a_{1})V(\hat{x},S)\land\tau=1\land
L(\widehat{X},X,U)\neq\emptyset\right\\}.\end{array}\vspace{-0.8cm}$ (49)

###  6.2 Sets $C$ and $D$ and overall model

We define the sets $\mathcal{C}$ and $\mathcal{D}$ of system (1), (24), as in
(49), where $V(x,S)=x^{\top}Sx$ as in Property 1. These sets capture the
information description in Section 6.1. Indeed, set $\mathcal{C}$ defines the
region of the state space where the Lyapunov function $V$ decays as desired or
where an episode has just been triggered (i.e., $\tau=0$) or, lastly, where it
is not possible to design a new feedback gain (i.e.,
$L(\widehat{X},X,U)=\emptyset$). Set $\mathcal{D}$ is defined similarly to
enforce the triggering of an episode.

To summarize, the overall model is denoted by $\mathcal{H}$ and is defined by

$\begin{array}[]{rlllll}\mathcal{H}&:&(\ref{eq:sys-background}),(\ref{eq:sys-
f-g}),(\ref{eq:L}),(\ref{eq:eta}),(\ref{eq:h-ell}),(\ref{eq:C-D})\end{array}$

The next result establishes the $k$-completeness of any maximal solution to
$\mathcal{H}$ thereby ruling out Zeno-like behavior as already observed above.

###### Proposition 3

Any maximal solution to $\mathcal{H}$ is $k$-complete. $\Box$

Proof: We first show that any maximal solution is complete. In view of
Proposition 1, we have to prove that $f(\mathcal{C})\cup
G(\mathcal{D})\subset\mathcal{C}\cup\mathcal{D}$. Let $q\in\mathcal{C}$, and
denote for the sake of convenience $q^{+}=f(q)$. If $q^{+}\notin\mathcal{C}$,
this means, as $\tau^{+}=1$, that
$V(x^{+},S^{+})>\sigma(a_{1}^{+})V(\hat{x}^{+})$ and
$L(\widehat{X}^{+},X^{+},U^{+})\neq\emptyset$. Hence $q^{+}\in\mathcal{D}$. We
have proved that $f(q)\in\mathcal{C}\cup\mathcal{D}$ for any
$q\in\mathcal{C}$, i.e., $f(\mathcal{C})\subset\mathcal{C}\cup\mathcal{D}$.
Let $q\in\mathcal{D}$, $g\in G(q)$ and we write $q^{+}=g$. As $\tau^{+}=0$,
$q^{+}\in\mathcal{C}\backslash\mathcal{D}$. We deduce that for any
$q\in\mathcal{D}$ and any $g\in G(q)$, $g\in\mathcal{C}\backslash\mathcal{D}$.
Hence $G(\mathcal{D})\subset\mathcal{C}\backslash\mathcal{D}$. We have proved
that $f(\mathcal{C})\cup G(\mathcal{D})\subset\mathcal{C}\cup\mathcal{D}$,
consequently any maximal solution to $\mathcal{H}$ is complete by Proposition
1. In addition, since $G(\mathcal{D})\subset\mathcal{C}\backslash\mathcal{D}$,
it holds that $G(\mathcal{D})\cap\mathcal{D}=\emptyset$ and we derive that any
maximal solution to $\mathcal{H}$ is $k$-complete by item (ii) of Proposition
1. $\blacksquare$

###### Remark 6

Other triggering conditions could very well be considered. We could for
instance enforce a certain number of physical time instants before checking a
state-dependent criterion and not just one as in (49), like in time-
regularized event-triggered control, see, e.g., [22, 23, 24]. We could also
envision dynamic triggering rule, which only triggers a transmission when a
static state-dependent criterion is violated for a certain amount of time,
inspired by [18, 25]. We leave these extensions for future work. $\Box$

###  6.3 Lyapunov property

To conclude this section, we derive properties of function $V$ at jumps due to
the triggering of a new episode. This property will be exploited in Section 7
to ensure the satisfaction of (P3) for the overall model.

###### Proposition 4

For any $q\in\mathcal{D}$ and any444We acknowledge that we write
$K^{+},S^{+},F^{+},a_{1}^{+},a_{2}^{+}$ with some abuse of notation, this is
done only to simplify the exposition.
$g=(x,\kappa,\widehat{X},X,U,K^{+},S^{+},F^{+},a_{1}^{+},a_{2}^{+},\ell(q))\in
G(q)$,

$\begin{array}[]{rlllll}V(x,S^{+})&\leq&\nu_{d}(q_{2})V(x,S)\end{array}$ (50)

with $\nu_{d}(q_{2})\in\mathbb{R}_{\geq 0}$ verifying
$S^{\prime}\preceq\nu_{d}(q_{2})S$ for any
$(K^{\prime},S^{\prime},F^{\prime},a_{1}^{\prime},a_{2}^{\prime})\in
L(\widehat{X},X,U)$. $\Box$

Proof: Let $q\in\mathcal{D}$ and
$g=(x,\kappa,\widehat{X},X,U,K^{+},S^{+},a^{+},\ell(q))\in G(q)$. As
$q\in\mathcal{D}$, $L(\widehat{X},X,U)$ is non-empty. We have
$V(x,S^{+})=x^{\top}(S^{+})^{\top}x\leq\nu_{d}(q_{2})x^{\top}Sx=\nu_{d}(q_{2})V(x,S)$;
note that $\nu_{d}$ is well-defined as $\mathcal{S}_{S}$ is bounded and
$S\in\mathbb{S}^{n_{x}}_{\succ 0}$. We have obtained the desired result as $q$
and $g$ have been arbitrarily selected. $\blacksquare$

We can now establish stability properties for system $\mathcal{H}$.

##  7 Stability guarantees

We exploit the results of Sections 5 and 6 to show that (P1)-(P3) in Section
3.4 are satisfied for system $\mathcal{H}$ with set $\mathcal{A}$ in (25). We
then apply Theorem 1 to derive general conditions under which stability
properties hold for system $\mathcal{H}$. Afterwards, we focus on case studies
for which we can derive more interpretable stability conditions.

###  7.1 Ensuring (P1)-(P3) for $\mathcal{H}$

We introduce for the sake of convenience the next subset of $\mathcal{C}$ as
defined in (49)

$\begin{array}[]{rlllll}\mathcal{C}_{1}&:=&\left\\{q\in\mathcal{S}_{q}\,:\,V(x,S)\leq\sigma(a_{1})V(\hat{x},S)\right\\}.\end{array}$
(51)

We establish in the following result that (P1)-(P3), as stated in Section 3.4,
hold for system $\mathcal{H}$.

###### Proposition 5

Given $T\in\mathbb{Z}_{\geq 1}$, consider system $\mathcal{H}$. Then (P1)-(P3)
hold with

$\begin{array}[]{rlllll}\mathcal{U}(q)&:=&V(x,S)&\forall
q\in\mathcal{S}_{q}\\\
\underline{\alpha}(s,q_{2})&=&\lambda_{\min}(S)s^{2}&\forall(s,q_{2})\in\mathbb{R}_{\geq
0}\times\mathcal{S}_{2}\\\
\overline{\alpha}(s,q_{2})&=&\lambda_{\max}(S)s^{2}&\forall(s,q_{2})\in\mathbb{R}_{\geq
0}\times\mathcal{S}_{2}\\\
\nu_{c}(q)&=&\left\\{\begin{array}[]{lllll}\sigma(a_{1})&f(q)\in\mathcal{C}_{1}\\\
\theta(q_{2})&f(q)\notin\mathcal{C}_{1}\end{array}\right.&\forall
q\in\mathcal{C},\end{array}$ (52)

$\nu_{d}$ in Proposition 4, $V$ as in Property 1,
$\theta(q_{2})\in[\underline{\theta}(q_{2}),\infty)$,
$\underline{\theta}(q_{2}):=\min\big{\\{}\theta^{\prime}\in\mathbb{R}_{\geq
0}\,:\,(A(\kappa)+B(\kappa)K)^{\top}S(A(\kappa)+B(\kappa)K)\leq\theta^{\prime}S\big{\\}}$,
$q_{2}$ as in (26). $\Box$

Proof: Let $q\in\mathcal{C}\cup\mathcal{D}$, by definition of $\mathcal{U}$ in
(52), $\mathcal{U}(q)=V(x,S)=x^{\top}Sx$. As
$S\in\mathcal{S}_{S}\subset\mathbb{S}^{n_{x}}_{\succ 0}$,
$\lambda_{\min}(S)|x|^{2}\leq\mathcal{U}(q)\leq\lambda_{\max}(S)|x|^{2}$ with
$0<\lambda_{\min}(S)\leq\lambda_{\max}(S)$. This proves (P1) holds with
$\underline{\alpha}$ and $\overline{\alpha}$ in (52).

Let $q\in\mathcal{C}$, $\mathcal{U}(f(q))=V(A(\kappa)x+B(\kappa)Kx,S)$. When
$f(q)\in\mathcal{C}_{1}$, by definition of $f$ in (24) and of
$\mathcal{C}_{1}$ in (51),
$V(A(\kappa)x+B(\kappa)Kx,S)\leq\sigma(a_{1})V(x,S)$. Consequently
$\mathcal{U}(f(q))\leq\sigma(a_{1})\mathcal{U}(q)$. On the other hand, when
$f(q)\notin\mathcal{C}_{1}$, by definition of $\theta(q)$,
$\mathcal{U}(f(q))\leq\theta(q_{2})\mathcal{U}(q)$. Note that
$\underline{\theta}(q_{2})$ in Proposition 5 is well-defined as
$(A(\kappa)+B(\kappa)K)^{\top}S(A(\kappa)+B(\kappa)K)\succeq 0$ and $S\succ
0$. We have proved that (P2) holds with $\nu_{c}$ in (52). Finally, the
satisfaction of (P3) with $\nu_{d}$ as in (52) follows by Proposition 4.
$\blacksquare$

Note that $\theta(q_{2})$ in Proposition 5 is greater than $\sigma(a_{1})$
when $f(q)\notin\mathcal{C}_{1}$, as $f(q)$ would be in $\mathcal{C}_{1}$
otherwise. We show below how to relate $\theta$ to parameters of Property 1 in
Section 5.2.

###  7.2 Bound on $\mathcal{U}$ along the solutions to $\mathcal{H}$

The next lemma directly follows from Proposition 5, by application of Lemma 1
and noting that $a_{1}$ is constant on $\mathcal{C}$, see (24). Its proof is
therefore omitted.

###### Lemma 5

Given $T\in\mathbb{Z}_{\geq 1}$, consider system $\mathcal{H}$, for any
solution $q$ it holds that

$\begin{array}[]{rlllll}\mathcal{U}(q(k,j))&\leq\pi(q,k,j)\mathcal{U}(q(0,0))&&\forall(k,j)\in\text{dom}\,q,\end{array}$
(53)

with $\mathcal{U}$ as in Lemma 5,

$\begin{array}[]{rlllll}{\small\begin{aligned}
&\pi(q,k,j)=\displaystyle\prod_{j^{\prime}=0}^{j-1}\underbrace{\prod_{\begin{subarray}{c}k^{\prime}=k_{j^{\prime}}\\\
k^{\prime}\in\mathcal{T}_{1}(q)\end{subarray}}^{k_{j^{\prime}+1}-1}\\!\\!\sigma(a_{1}(k_{j}^{\prime},j^{\prime}))\nu_{d}(q_{2}(k_{j^{\prime}+1},j^{\prime}))}_{\text{(A)}}\\\
&\displaystyle\times\\!\\!\underbrace{\prod_{\begin{subarray}{c}k^{\prime}=k_{j^{\prime}}\\\
k^{\prime}\notin\mathcal{T}_{1}(q)\end{subarray}}^{k_{j^{\prime}+1}-1}\\!\\!\theta(q_{2}(k^{\prime},j^{\prime}))}_{\text{(B)}}\\!\\!\underbrace{\prod_{\begin{subarray}{c}k^{\prime\prime}=k_{j}\\\
k^{\prime\prime}\in\mathcal{T}_{1}(q)\end{subarray}}^{k-1}\sigma(a_{1}(k_{j},j))\\!\\!\prod_{\begin{subarray}{c}k^{\prime\prime}=k_{j}\\\
k^{\prime\prime}\notin\mathcal{T}_{1}(q)\end{subarray}}^{k-1}\\!\\!\\!\\!\theta(q_{2}(k^{\prime\prime},j))}_{\text{(C)}}\end{aligned}}\end{array}$
(54)

$\mathcal{T}_{1}(q):=\big{\\{}k\in\mathbb{Z}_{\geq 0}\,:\exists
j\in\mathbb{Z}_{\geq
0},\,(k,j),(k+1,j)\in\text{dom}\,q,q(k+1,j)\in\mathcal{C}_{1}\big{\\}}$,
$k_{0},k_{1},\ldots,k_{j}$ as defined at the end of Section 3.2,
$\mathcal{C}_{1}$ in (51) and $\theta$ in Proposition 5. $\Box$

Despite its apparent complexity, (53)-(54) admit an intuitive interpretation.
The term (A) is made of the product of the desired decay rate $\sigma(a_{1})$
with $\nu_{d}(q_{2})$, which is due to the potential change of value of $S$ at
episode triggering, along solutions. The term (B) characterizes the potential
growth of $\mathcal{U}$ when the desired decay rate $\sigma(a_{1})$ is not
met. Finally, the term (C) is the product of $\sigma(a_{1})$ and of the
potential growth quantified by $\theta$ when $k\in[k_{j},k_{j+1}]$.

As long as the solution remains in $\mathcal{C}_{1}$, the growth rate of
$\pi(q,\cdot,\cdot)$ is upper-bounded by the desired rate $\sigma(a_{1})$ and
we will be able to derive that $\pi$ and thus $\mathcal{U}$ converges to the
origin along the considered solution (under mild extra conditions). The
potential obstacle for the convergence of $\mathcal{U}(q)$ to $0$ as $k+j$
goes to infinity is if the solution wanders too often and for too long periods
of physical times outside $\mathcal{C}_{1}$. This may be due to the fact that
the matrices $A$ and $B$ are changing significantly too often, so that after a
new episode the solution does not remain in $\mathcal{C}_{1}$ for long enough
periods of physical time, or that the lastly collected data $\widehat{X}$,
$X$, $U$ do not yield a new controller gain $K$, i.e.,
$L(\widehat{X},X,U)=\emptyset$. Later we will capture analytically these
effects by providing sufficient conditions for stabilization that can be
related to the system’s time-variations.

Before proceeding further, we show how to derive an expression of $\theta$ in
(54) based on the parameters of the matrix sets in Property 1.

###### Lemma 6

Given $T\in\mathbb{Z}_{\geq 1}$, consider system $\mathcal{H}$. For any
solution $q$ and any $(k,j)\in\text{dom}\,q$ with $k\geq T$ and $j\geq
j_{T}+1$, there exists
$\varepsilon:\mathcal{S}_{\mathrm{d}}\to\mathbb{R}_{\geq 0}$ such that
$[B(\kappa(k,j))\,\,A(\kappa(k,j))]^{\top}\in\mathcal{E}\Big{(}\mathrm{d}(k,j),F(k,j)+\varepsilon(\mathrm{d}(k,j))S^{-1}(k,j)\Big{)}$,
with $\mathcal{E}$ in (33), $\mathrm{d}$ in (29), and Lemma 5 holds with
$\theta(q_{2}(k,j))=a_{1}(k,j)+a_{2}(k,j)\varepsilon(\mathrm{d}(k,j))$ $\Box$

Proof: Let $q$ be a solution to $\mathcal{H}$ and $(k,j)\in\text{dom}\,q$ with
$k\geq T$ and $j\geq j_{T+1}$. By Lemma 3 and item (iii) of Lemma 4, the set
$\mathcal{E}(\mathrm{d}(k,j),F(k,j)+\varepsilon S^{-1}(k,j))$ is non-empty for
$\varepsilon\in\mathbb{R}_{>0}$ sufficiently big as
$S(k,j)\in\mathbb{S}_{\succ 0}^{n_{x}}$. Furthermore, in view of the
definition of $\mathcal{E}$ in (33), we can always find
$\varepsilon(\mathrm{d}(k,j))\in\mathbb{R}_{\geq 0}$ sufficiently big such
that
$[B(\kappa(k,j))\,\,A(\kappa(k,j))]^{\top}\in\mathcal{E}\Big{(}\mathrm{d}(k,j),F(k,j)+\varepsilon(\mathrm{d}(k,j))S^{-1}(k,j)\Big{)}$.
Consequently, as $\mathcal{P}_{T}(\mathrm{d}(q(k_{j},j)))$ holds by definition
of $L$ in (39),
$V((M_{A}+M_{B}K(k_{j},j))x(k,j),S(k_{j},j))\leq\big{(}a_{1}(k_{j},j)+a_{2}(k_{j},j)\epsilon\big{)}V(x(k,j),S(k_{j},j))$.
Therefore,

$\begin{array}[]{l}V((A(\kappa(k,j))+B(\kappa(k,j))K(k,j))x(k,j),S(k,j))\\\
\quad\quad\quad\leq\big{(}a_{1}(k,j)+a_{2}(k,j)\varepsilon(\mathrm{d}(k,j)\big{)}V(x(k,j),S(k,j)).\end{array}$

In view of the definition of $\underline{\theta}$ in Proposition 5,
$\underline{\theta}(q_{2}(k,j))\leq
a_{1}(k,j)+a_{2}(k,j)\varepsilon(\mathrm{d}(k,j))$, which means we can select
$\theta(q_{2}(k,j))=a_{1}(k,j)+a_{2}(k,j)\varepsilon(\mathrm{d}(k,j))$ in
Lemma 5. $\blacksquare$

Lemma 6 allows to derive a more insightful expression for $\pi$ in (54) by
replacing the terms (B) and (C) as below

$\begin{array}[]{lllll}\text{(B)}=\\!\\!\\!\\!\underbrace{\prod_{\begin{subarray}{c}k^{\prime}=k_{j^{\prime}}\\\
k^{\prime}\notin\mathcal{T}_{1}(q)\\\ j^{\prime}\leq
j_{T}+1\end{subarray}}^{k_{j^{\prime}+1}-1}\\!\\!\\!\\!\\!\\!\theta(q_{2}(k^{\prime},j^{\prime}))}_{\text{(B1)}}\\!\\!\\!\\!\underbrace{\displaystyle\prod_{\begin{subarray}{c}k^{\prime}=k_{j^{\prime}}\\\
k^{\prime}\notin\mathcal{T}_{1}(q)\\\ j^{\prime}\geq
j_{T}+1\end{subarray}}^{k_{j^{\prime}+1}-1}\\!\\!\\!\\!(a_{1}(k^{\prime},j^{\prime})+a_{2}(k^{\prime},j^{\prime})\varepsilon(q_{2}(k^{\prime},j^{\prime})))}_{\text{(B2)}}\\\
\text{(C)}=\prod_{\begin{subarray}{c}k^{\prime\prime}=k_{j}\\\
k^{\prime\prime}\in\mathcal{T}_{1}(q)\end{subarray}}^{k-1}\\!\\!\sigma(a_{1}(k_{j},j))\\\
\quad\quad\times\prod_{\begin{subarray}{c}k^{\prime\prime}=k_{j}\\\
k^{\prime\prime}\notin\mathcal{T}_{1}(q)\end{subarray}}^{k-1}\\!\\!\\!\\!(a_{1}(k^{\prime\prime},j)+a_{2}(k^{\prime\prime},j)\varepsilon(q_{2}(k^{\prime\prime},j))).\end{array}$
(55)

The term (B1) is due to what happens before the $j_{T}+1^{\text{th}}$ episode
has occurred, during which we do not have much control of the Lyapunov-like
function $V$ unless we know a stabilizing policy valid for the first physical
time steps for instance. This is reasonable when we have a good knowledge of
the initial values of $A$ and $B$, but these may significantly vary afterwards
thereby justifying the proposed approach. The term (B2) characterizes the
potential growth of $\mathcal{U}$ when the desired decay rate $\sigma(a_{1})$
is not verified. We see that the corresponding terms depend on $\varepsilon$
as in Lemma 6, which somehow characterizes the “distance” of
$[B(\kappa)\,\,A(\kappa)]^{\top}$ to the nominal set
$\mathcal{E}(\mathrm{d},F)$ as discussed in Section 5.3. Similar
interpretations hold for term (C).

###  7.3 General stability conditions

The next theorem follows by application of Theorem 1 to system $\mathcal{H}$
with $\pi$ in (54).

###### Theorem 2

Given $T\in\mathbb{Z}_{\geq 1}$, consider system $\mathcal{H}$. The following
holds.

  1. (i)

If there exists $\mu:\mathbb{R}_{\geq
0}\times\mathcal{S}_{2}\to\mathbb{R}_{\geq 0}$ continuous and non-decreasing
in its first argument such that for any solution $q$ and any
$(k,j)\in\text{dom}\,q$,
$\pi(q,k,j)\frac{\lambda_{\max}(S(0,0))}{\lambda_{\min}(S(k,j))}\leq\mu(|q_{1}(0,0)|,q_{2}(0,0))$
with $q_{1},q_{2},\mathcal{S}_{2}$ in (26), $\pi$ in (54), then $\mathcal{A}$
is stable. In addition,

    1. (i-a)

if $\mu(s,\cdot)$ is constant for any $s\in\mathbb{R}_{\geq 0}$, then
$\mathcal{A}$ is uniformly stable,

    2. (i-b)

if for any complete solution $q$, $\pi(q,k,j)\lambda_{\min}(S(k,j))^{-1}\to 0$
as $k+j\to\infty$ then $\mathcal{A}$ is GAS.

  2. (ii)

If there exists $\beta\in\mathcal{L}$ such that for any solution $q$,
$\pi(q,k,j)\frac{\lambda_{\max}(S(0,0))}{\lambda_{\min}(S(k,j))}\leq\beta(k+j)$
for any $(k,j)\in\text{dom}\,q$, then $\mathcal{A}$ is UGAS.

  3. (iii)

If there exist $c_{1}\geq 1$, $c_{2}>0$ such that for any solution $q$,
$\pi(q,k,j)\frac{\lambda_{\max}(S(0,0))}{\lambda_{\min}(S(k,j))}\leq
c_{1}e^{-c_{2}(k+j)}$ for any $(k,j)\in\text{dom}\,q$, then $\mathcal{A}$ is
UGES. $\Box$

Proof: (i) Let $q$ be a solution to $\mathcal{H}$. By Proposition 5 and Lemma
5, for any $(k,j)\in\text{dom}\,q$,

$\begin{array}[]{rlllll}\begin{aligned}
&|q_{1}(k,j)|\leq\big{(}\tfrac{\lambda_{\max}(S(0,0))}{\lambda_{\min}(S(k,j))}\pi(q,k,j)\big{)}^{\tfrac{1}{2}}|q_{1}(0,0)|\\\
&\leq\mu(|q_{1}(0,0)|,q_{2}(0,0))^{\tfrac{1}{2}}|q_{1}(0,0)|=:\vartheta(|q_{1}(0,0)|,q_{2}(0,0)).\end{aligned}\end{array}$
(56)

The map555If $\mu\equiv 0$, we can take any $\vartheta:\mathbb{R}_{\geq
0}\times\mathcal{S}_{2}\to\mathbb{R}_{\geq 0}$ that is $\mathcal{K}_{\infty}$
in its first argument and the result holds. $\vartheta$ is of
class-$\mathcal{K}_{\infty}$ in its first argument. As $q$ has been
arbitrarily chosen, we derive by item (i) of Theorem 1 that $\mathcal{A}$ is
stable for system $\mathcal{H}$. Items (i-a) and (i-b) of Theorem 2 follow by
application of items (i-a) and (i-b) of Theorem 1, respectively.

(ii) Let $q$ be a solution to $\mathcal{H}$. By (56), for any
$(k,j)\in\text{dom}\,q$,
$|q_{1}(k,j)|\leq\beta(k+j)^{\tfrac{1}{2}}|q_{1}(0,0)|$. As the map from
$\mathbb{R}_{\geq 0}\times\mathbb{R}_{\geq 0}$ to $\mathbb{R}_{\geq 0}$
defined as $(s,t)\mapsto\beta(t)^{\tfrac{1}{2}}s$ is of class-$\mathcal{KL}$
and $q$ has been arbitrarily chosen, we deduce from item (ii) of Theorem 1
that $\mathcal{A}$ is UGAS. Item (iii) of Theorem 2 is proved in a similar
manner. $\blacksquare$

A strength of Theorem 2 lies in the generality of the conditions it proposes
under which set $\mathcal{A}$ in (25) exhibits stability properties. However,
these conditions are difficult to check a priori as these involve _any_
solution to system $\mathcal{H}$. We provide in the following easier-to-check
conditions, which can be related to the matrices $A$ and $B$ notably via the
matrix ellipsoidal characterization in Section 5.2 and Property 1. We
emphasize that these are only a sample of conditions ensuring satisfaction of
Theorem 2.

###  7.4 Case studies

####  7.4.1 Solutions eventually always in $\mathcal{C}_{1}$

The next result provides sufficient conditions to derive stability properties
for $\mathcal{A}$ for system $\mathcal{H}$ in the case where all solutions
eventually always lie in $\mathcal{C}_{1}$ defined in (51).

###### Theorem 3

Given $T\in\mathbb{Z}_{\geq 1}$, consider system $\mathcal{H}$. Suppose there
exist $T^{\star}:\mathcal{S}_{2}\to\mathbb{Z}_{\geq 0}$ and
$\mu_{1}:\mathcal{S}_{2}\to\mathbb{R}_{\geq 0}$ such that the following holds
for any solution $q$ and $(k,j)\in\text{dom}\,q$.

  1. (i)

When $k\geq T^{\star}(q_{2}(0,0))$, $q(k,j)\in\mathcal{C}_{1}$.

  2. (ii)

When $k\leq T^{\star}(q_{2}(0,0))$,
$\pi(q,k,j)\frac{\lambda_{\max}(S(0,0))}{\lambda_{\min}(S(k,j))}\leq\mu_{1}(q_{2}(0,0))$.

Then $\mathcal{A}$ is stable. In addition,

  * •

if $\mu_{1}$ is constant, then $\mathcal{A}$ is uniformly stable,

  * •

if $\sigma(a_{1}(T^{\star}(q_{2}(0,0)),j_{T^{\star}}(q_{2}(0,0))))\in[0,1)$
where $j_{T^{\star}}(q_{2}(0,0))=\max\\{j\in\mathbb{Z}_{\geq
0}\,:\,(T^{\star}(q_{2}(0,0)),j)\in\text{dom}\,q\\}$, then $\mathcal{A}$ is
GAS,

  * •

if $\mu_{1}$ and $T^{\star}$ are constant and there exists $\mu_{2}\in(0,1)$
such that $\sigma(a_{1}(T^{\star},j_{T}^{\star}))\leq\mu_{2}$ for any
solution, then $\mathcal{A}$ is UGES. $\Box$

Proof: Let $q$ be a solution to $\mathcal{H}$. We denote for the sake of
convenience $(T^{\star}(q_{2}(0,0)),j_{T^{\star}}(q_{2}(0,0))$ by
$(T^{\star},j_{T^{\star}})$. Let $(k,j)\in\text{dom}\,q$ with $k\geq
T^{\star}$. We observe that by definition of $\pi$ in (54),

$\begin{array}[]{rlllll}\pi(q,k,j)&\leq\pi(q,T^{\star},j_{T^{\star}})\pi(\widetilde{q},k-T^{\star},j-j_{T^{\star}}),\end{array}$
(57)

with $\widetilde{q}(k,j)=q(k+T^{\star},j+j_{T^{\star}})$, which is a solution
to $\mathcal{H}$ initialized at $q(T^{\star},j_{T^{\star}})$. Hence, by item
(ii) of Theorem 3,

$\begin{array}[]{rlllll}\pi(q,k,j)\frac{\lambda_{\max}(S(0,0))}{\lambda_{\min}(S(k,j))}&\leq\mu_{1}(q_{2}(0,0))\pi(\widetilde{q},k-T^{\star},j-j_{T^{\star}}).\end{array}$
(58)

On the other hand, by item (i) of Theorem 3 and the definition of $\pi$ in
(54), we derive that
$\pi(\widetilde{q},k-T^{\star},j-j_{T^{\star}})=\sigma(a_{1}(T^{\star},j_{T^{\star}}))^{k-T^{\star}}$
and thus that

$\begin{array}[]{rlllll}\pi(q,k,j)\frac{\lambda_{\max}(S(0,0))}{\lambda_{\min}(S(k,j))}&\leq\mu_{1}(q_{2}(0,0))\sigma(a_{1}(T^{\star},j_{T^{\star}}))^{k-T^{\star}}.\end{array}$
(59)

As $\sigma(a_{1}(T^{\star},j_{T^{\star}}))\in[0,1]$ in view of (45),

$\begin{array}[]{rlllll}\pi(q,k,j)\frac{\lambda_{\max}(S(0,0))}{\lambda_{\min}(S(k,j))}&\leq\mu_{1}(q_{2}(0,0)).\end{array}$
(60)

We deduce from item (ii) of Theorem 3 and (60) that item (i) of Theorem 2 is
verified with $\mu(s_{1},s_{2})=\mu_{1}(s_{2})$. Consequently, $\mathcal{A}$
is stable.

When $\mu_{1}$ is constant, we have by item (i-a) by Theorem 2 that
$\mathcal{A}$ is uniformly stable.

When $\sigma(a_{1}(T^{\star},j_{T^{\star}}))\in[0,1)$, by (59),
$\pi(q,k,j)\lambda_{\min}(S(k,j))^{-1}\to 0$ as $k\to\infty$ when $q$ is
maximal; recall that it is $k$-complete by Proposition 3. As a consequence,
$\mathcal{A}$ is GAS.

Finally, when $\mu_{1}$ and $T^{\star}$ are constant, and
$\sigma(a_{1}(T^{\star},j_{T^{\star}}))\leq\mu_{2}\in(0,1)$ for any solution,
(59) becomes for any $(k,j)\in\text{dom}\,q$ with $k\geq T^{\star}$

$\begin{array}[]{rlllll}\pi(q,k,j)\frac{\lambda_{\max}(S(0,0))}{\lambda_{\min}(S(k,j))}\leq\mu_{1}\mu_{2}^{k-T^{\star}}\\!=\\!\mu_{1}\mu_{2}^{-j_{T^{\star}}-T^{\star}}\mu_{2}^{k+j_{T^{\star}}}\end{array}$
(61)

where we write $\mu_{1}(q_{2}(0,0))=\mu_{1}$ with some slight abuse of
notation. Noting that $j=j_{T^{\star}}$ when $k\geq T^{\star}$ by item (i) of
Theorem 3, we derive

$\begin{array}[]{rlllll}\pi(q,k,j)\frac{\lambda_{\max}(S(0,0))}{\lambda_{\min}(S(k,j))}&\leq\mu_{1}\mu_{2}^{-T^{\star}-j_{T^{\star}}}\mu_{2}^{k+j}.\end{array}$
(62)

On the other hand, from item (ii) of Theorem 3, for any
$(k,j)\in\text{dom}\,q$ with $k\leq T^{\star}$,

$\begin{array}[]{rlllll}\pi(q,k,j)\frac{\lambda_{\max}(S(0,0))}{\lambda_{\min}(S(k,j))}&\leq\mu_{1}\leq\mu_{1}\mu_{2}^{-T^{\star}-j_{T^{\star}}}\mu_{2}^{k+j}.\end{array}$
(63)

We deduce from (62) and (63) that item (iii) of Theorem 2 holds with
$c_{1}=\mu_{1}\mu_{2}^{-T^{\star}-j_{T^{\star}}}$ and $c_{2}=\ln(\mu_{2})$. As
a result, $\mathcal{A}$ is UGES. $\blacksquare$

Item (i) of Theorem 3 means that all solutions to $\mathcal{H}$ eventually lie
for all future hybrid times (recall that hybrid times are pairs $(k,j)$, see
Section 3.2) in set $\mathcal{C}_{1}$. Item (ii) of Theorem 3 implies that we
can upper-bound the norm of $q_{1}$ from physical time $0$ to physical time
$T^{\star}(q_{2}(0,0))$ (at which $q$ enters forever in $\mathcal{C}_{1}$) by
the product of a function of the initial value of $q_{2}$ with the initial
value of $|q_{1}|$. Then, stricter conditions are derived to ensure stronger
stability properties. The next corollary essentially provides conditions on
$A$, $B$ and $K$ such that the requirements of Theorem 3 hold.

###### Corollary 1

Given $T\in\mathbb{Z}_{\geq 1}$, consider system $\mathcal{H}$ with
$\sigma(a_{1})\in[a_{1},1)$ for any $a_{1}\in[0,1)$ and $\sigma(1)=1$. Suppose
there exist $T^{\star}:\mathcal{S}_{2}\to\mathbb{Z}_{\geq T}$ and
$\overline{m}\in\mathbb{R}_{\geq 0}$, such that the following holds for any
solution $q$ and any $(k,j)\in\text{dom}\,q$.

  1. (i)

When $k\geq T^{\star}$ and $j\geq j_{T}+1$,

$[B(\kappa(k,j))\,\,\,A(\kappa(k,j))]^{\top}\in\mathcal{E}(\mathrm{d}(T^{\star},j_{T^{\star}}),F(T^{\star},j_{T^{\star}})),$

where we omit the dependence of $T^{\star}$ on $q_{2}(0,0)$ and
$j_{T^{\star}}=\max\\{j^{\prime}\in\mathbb{Z}_{\geq
0}\,:\,(T^{\star},j^{\prime})\in\text{dom}\,q\\}$.

  2. (ii)

$\max\\{\|A(\kappa(k,j))\|,\|B(\kappa(k,j))\|,\|K(k,j)\|\\}\leq\overline{m}$.

Then $\mathcal{A}$ is GAS. In addition,

  * •

if $T^{\star}$ is constant, then $\mathcal{A}$ is uniformly stable,

  * •

if $a_{1}(T^{\star},j_{T^{\star}})\in[0,1)$, then $\mathcal{A}$ is GAS,

  * •

if $T^{\star}$ is constant and there exists $c\in(0,1)$ independent of $q$
such that

$a_{1}(T^{\star},j_{T^{\star}})\leq c,$ (64)

then $\mathcal{A}$ is UGES. $\Box$

Proof: Let $q$ be a solution to $\mathcal{H}$ and $(k,j)\in\text{dom}\,q$ with
$k\geq T^{\star}$ and $j\geq j_{T}+1$. Since $T^{\star}\geq T$ in item (i) of
Corollary 1, by definition of $L$ in (39),
$\mathcal{P}_{T}(\mathrm{d}(T^{\star},j_{T^{\star}}))$ holds. As a
consequence, for any $k>T^{\star}$,
$V(x(k,j_{T^{\star}}),S(k,j_{T^{\star}}))\leq
a_{1}(T^{\star},j_{T^{\star}})V(\hat{x}(k,j_{T^{\star}}),S(k,j_{T^{\star}}))$.
Since
$a_{1}(T^{\star},j_{T^{\star}})\leq\sigma(a_{1}(T^{\star},j_{T^{\star}}))$ by
definition of $\sigma$ in Corollary 1, the above inequality implies that
$q(k,j_{T^{\star}})\in\mathcal{C}_{1}$ for any $k>T^{\star}$. As a
consequence, item (i) of Theorem 3 holds with $T^{\star}+1$. Item (ii) of
Corollary 1 implies that
$|x(k,j)|\leq(\overline{m}+\overline{m}^{2})^{T^{\star}}$ for any
$(k,j)\in\text{dom}\,q$ with $k\leq T^{\star}$. We can then follow similar
lines as in the proof of Theorem 3 to derive that $\mathcal{A}$ is stable as,
although the term $(\overline{m}+\overline{m}^{2})^{T^{\star}}$ does not
necessarily upper-bounds
$\pi(q,k,j)\frac{\lambda_{\max}(S(0,0))}{\lambda_{\min}(S(k,j))}$, it does so
for $|x(k,j)|$ which is what all we need in view of the proof of Theorem 1.
When $T^{\star}$ is constant, $|x(k,j)|$ is upper-bounded by a constant for
$k\leq T^{\star}$, and we derive that $\mathcal{A}$ is uniformly stable like
in Theorem 3. When $a_{1}(T^{\star},j_{T^{\star}})\in[0,1)$ for any solution
$q$, $\sigma(a_{1}(T^{\star},j_{T^{\star}}))\in[0,1)$ and we derive from
Theorem 3 that $\mathcal{A}$ is GAS. Finally, when $T^{\star}$ is constant and
(64) holds, we conclude that $\mathcal{A}$ is UGES by the last item of Theorem
3. $\blacksquare$

####  7.4.2 Solutions frequently enough in $\mathcal{C}_{1}$

This second result ensure the satisfaction of the requirements of Theorem 2 by
formalizing the intuition that, when every solution to system $\mathcal{H}$
lie in $\mathcal{C}_{1}$ frequently enough, then desirable stability
properties hold.

###### Theorem 4

Given $T\in\mathbb{Z}_{\geq 1}$, consider system $\mathcal{H}$ and suppose
there exist $\lambda_{c}:\mathcal{S}_{2}\to[0,1]$,
$\lambda_{d}:\mathcal{S}_{2}\to\mathbb{R}_{\geq 0}$ with
$\lambda_{c}\leq\lambda_{d}$, $m_{1},m_{2}:\mathcal{S}_{2}\to\mathbb{R}_{\geq
0}$ and $\underline{s},\overline{s}\in\mathbb{R}_{>0}$ such that the following
holds for any solution $q$ and any $(k,j)\in\text{dom}\,q$.

  1. (i)

$\underline{s}I_{n_{x}}\preceq S(k,j)\preceq\overline{s}I_{n_{x}}$.

  2. (ii)

$\sigma(a_{1}(k,j))\leq\lambda_{c}(q_{2}(0,0))$.

  3. (iii)

$\max\\{\theta(q_{2}(k,j)),\nu_{d}(q_{2}(k,j))\\}\leq\lambda_{d}(q_{2}(0,0))$
with $\nu_{d}$ and $\theta$ as in Proposition 5.

  4. (iv)

If $q(k,j)\notin\mathcal{C}_{1}$ and $\tau(k,j)=1$,
$L(\widehat{X}(k,j),X(k,j),U(k,j))\neq\emptyset$.

  5. (v)

$(k-2j)\ln(\lambda_{c}(q_{2}(0,0)))+(3j+1)\ln(\lambda_{d}(q_{2}(0,0)))\leq
m_{1}(q_{2}(0,0))-m_{2}(q_{2}(0,0))(k+j)$.

Then $\mathcal{A}$ is stable. In addition,

  * •

if $m_{1}$ is constant, then $\mathcal{A}$ is uniformly stable,

  * •

if $m_{2}(q_{2}(0,0))<1$, then $\mathcal{A}$ is GAS,

  * •

if $m_{1}$, $m_{2}$ are constant and $m_{2}<1$, then $\mathcal{A}$ is UGES.
$\Box$

Proof: Let $q$ be a solution to $\mathcal{H}$ and $(k,j)\in\text{dom}\,q$. By
items (ii)-(iv) of Theorem 4 and (54),
$\pi(q,k,j)\leq\lambda_{c}(q_{2}(0,0))^{k-2j}\lambda_{d}(q_{2}(0,0))^{3j+1}$.
By item (v) of Theorem 4, we derive
$\pi(q,k,j)\leq\displaystyle\exp(m_{1}(q_{2}(0,0))-m_{2}(q_{2}(0,0))(k+j))$.
By invoking item (i) of Theorem 4 and as $m_{2}(q_{2}(0,0))\geq 0$, we derive
that item (i) of Theorem 2 holds with
$\mu(s,z)=\frac{\overline{s}}{\underline{s}}\exp(m_{1}(z))$ for any
$(s,z)\in\mathbb{R}_{\geq 0}\times\mathcal{S}_{2}$. Hence $\mathcal{A}$ is
stable by Theorem 2. The last three items of Theorem 4 follow by application
of Theorem 2. $\blacksquare$

Item (i) of Theorem 4 states that $S$ is uniformly lower and upper-bounded by
positive definite matrices for any solution $q$ to system $\mathcal{H}$, in
which case it means that
$\mathcal{S}_{S}\subset\\{S\in\mathbb{S}^{n_{x}}_{\succ
0}\,:\,\underline{s}I_{n_{x}}\preceq S\preceq\overline{s}I_{n_{x}}\\}$; a
similar assumption is made in [7]. Item (ii) upper-bounds the decay rate of
$\mathcal{U}$ in view of Proposition 5 using $\lambda_{c}(q_{2}(0,0))$. This
property is for instance verified with $\lambda_{c}\equiv c$ when
$\sigma(a_{1})\leq c\in(0,1]$ along any solution to $\mathcal{H}$, which can
be imposed by design. Item (iii), on the other hand, upper-bounds the growth
rate of $\mathcal{U}$ using $\lambda_{d}$, which is typically equal or larger
to $1$. This property holds when $\theta$ admits a constant upper-bound, say
$c^{\prime}$, with
$\lambda_{d}\equiv\max\\{c^{\prime},\overline{s}/\underline{s}\\}$ as
$\nu_{d}(q_{2})\leq\overline{s}/\underline{s}$ by item (i) of Theorem 4.
Recall that we can relate $\theta$ to the parameters of Property 1, see Lemma
6. Item (iv) means that, whenever the solution is not in $\mathcal{C}_{1}$ and
an episode has not just occurred, it is possible to update the controller gain
$K$; this is the case in the numerical examples of Section 8. Finally, item
(v) is inspired by [15, Proposition 3.29], and is a condition that the decay
rate of $\mathcal{U}$ on $\mathcal{C}_{1}$, which is captured by
$\lambda_{c}$, compensates the growth of $\mathcal{U}$ when solutions leaves
$\mathcal{C}_{1}$ described by $\lambda_{d}$. This condition is thus satisfied
when solutions remain sufficiently frequently in $\mathcal{C}_{1}$.

##  8 Numerical illustration

###  8.1 Algorithmic implementation of $L$

We discuss here the implementation details on the design of the map $L$ in
(42). We propose solving an SDP with variables $\varsigma,Y,H$ subject to LMI
constraints (41) and with the objective of maximizing det($H$) to promote an
increase of the matrices set to which the controller is robust by design. From
the solution to this SDP we get the controller gain $K$ and the matrix $S$
defining the quadratic Lyapunov-like function $V(x,S)$ as in Proposition 2. To
uniquely determine the remaining output of $L$, we fix $\epsilon_{F}\in(0,1)$
and obtain

$\displaystyle\hat{\varsigma}$
$\displaystyle=\tfrac{\varsigma}{\varsigma+1},\quad
F=(1-\epsilon_{F})\hat{\varsigma},\quad a_{1}=1-a,$ (65) $\displaystyle a_{2}$
$\displaystyle=1+\varsigma^{-1}a=\max\left\\{a^{\prime}\in\mathbb{R}_{\geq
0}\,:\,a^{\prime}S^{-1}\preceq\epsilon_{F}H\right\\}.$

The parameter $\epsilon_{F}$ influences the closed-loop properties we can
conclude from this control design. Precisely, choosing smaller $\epsilon_{F}$
enlarges the set $\mathcal{E}(\mathrm{d},F)$ at the cost of a lower decay
rate, and viceversa. In all the analyses we selected $\epsilon_{F}=0.1$, and
adaptively chose $\sigma(a_{1})=1-0.1(1-a_{1})$ consistently with Corollary 1
and Theorem 4.

In the following, we study the problem of controlling LTV systems obtained
from time-varying perturbations applied to a baseline LTI system defined by

$A_{0}=\begin{bmatrix}1.1&0.1\\\ 0.1&0.2\\\ \end{bmatrix},\quad
B_{0}=\begin{bmatrix}0.5&1\\\ 0.1&0.2\\\ \end{bmatrix},$ (66)

which has one unstable mode. We consider a control horizon $[0,100]$ and,
slightly departing from the theoretical framework, we impose a first episode
of fixed length $[0,T],T=n_{x}+n_{u}$, where we excite the system with i.i.d
uniformly distributed input with values in666This could also have been done
before time $(0,0)$, see Remark 1. $[-1,1]$. We denote hereafter by $K_{0}$
the control gain obtained at the end of this initial interval by evaluating
map $L$. The SDPs were solved using MOSEK [26] and the code used to generate
the results is available at the linked repository777 https://gitlab.com/col-
tasas/data-adaptive-et-c.

###  8.2 Switching plant

In the first example, we consider an LTV plant that periodically switches
between two LTI systems

$\begin{array}[]{rlll}A(k)&=A_{0}\;\forall k\in\mathbb{Z}_{\geq
0},\vspace{0.1in}\\\ B(k)&=\begin{cases}\begin{bmatrix}0.5&1\\\ 0.1&0.2\\\
\end{bmatrix},&k\in[1+p(z-1),pz],z=2n+1,\vspace{0.1in}\\\
\begin{bmatrix}0.5&-1\\\ 0.1&-0.2\\\
\end{bmatrix},&k\in[1+p(z-1),pz],z=2n,\end{cases}\end{array}$ (67)

where $n\in\mathbb{Z}_{\geq 0}$ and $p=12$. That is, the state matrix is
constant whereas the second column of the input matrix periodically changes
sign with period $p$. The switch can simulate for example a faulty behaviour
of on an actuator, and is inspired by the theoretical scenario analyzed in
Theorem 4 where the system remains sufficiently frequently in the region with
certified Lyapunov function decrease. We compare the proposed adaptive event-
triggered scheme with the strategy of never updating the initial controller
$K_{0}$ (fixed controller) and with a time-triggered strategy that triggers a
new episode according to a preset period $n_{p}$ and starting at a preset
time, similarly to what was proposed in [7]. Specifically, we consider the
oracle case where the time-triggered strategy has knowledge of the exact
period (i.e., $n_{p}=p$) and moreover triggers at $k=T+pn,n\in\mathbb{Z}_{\geq
0}$, i.e., exactly $T$ times after the system has switched and hence the data
matrices (16) contain only data relative to the current system to be
controlled. We study also _imperfect_ time-triggered scenarios where:
$n_{p}=p$ but the episode is triggered $T+4$ times after the system has
switched (offset start); $n_{p}=p-4$ without offset; and $n_{p}=p+4$ without
offset. The asterisk marker denotes, in all plots, the end of the data
collection phase at $k=T$.

Figure 1: Closed-loop state response of (67) with different controllers
(fixed, adaptive periodic time-triggered, and adaptive event-triggered).

The results in Figure 1 show that the event-triggered scheme (solid black
curve) outperforms all the other schemes at regulating the system around the
origin. From the reported triggering instants (cross markers) it can be seen
that the scheme only triggers after the plant first switch has occurred (cross
markers denote the plant switches) and does this only using observed data. On
the other hand, the time-triggered strategy shows low robustness to inexact
choice of period and first triggering instant. Indeed, while the performance
of the oracle case (solid blue curve) is only slightly worse than the even-
triggered case, the other three cases all display undesired oscillations and,
in some cases, non-convergent behaviours. Note that in this case the non-
adaptive solution using $K_{0}$ (solid red curve) leads to a converging
response after some initial transient.

We then also consider the case where the switch affects the second column of
the input matrix not only by changing its sign but also by scaling it up, that
is

$\begin{array}[]{rlll}B(k)&=\begin{cases}\begin{bmatrix}0.5&1\\\ 0.1&0.2\\\
\end{bmatrix},&k\in[1+p(z-1),pz],z=2n+1,\vspace{0.1in}\\\
\begin{bmatrix}0.5&-\ell 1\\\ 0.1&-\ell 0.2\\\
\end{bmatrix},&k\in[1+p(z-1),pz],z=2n,\\\ \end{cases}\end{array}$ (68)

where $\ell=2.5$. The time-triggered solutions show similar trends, thus we
compare in Figure 2 the non-adaptive case based on $K_{0}$ with the event-
triggered solution. The former now clearly shows diverging behaviour, whereas
the response of the adaptive even-triggered controlled system is qualitatively
unchanged with respect to the case with $\ell=1$ (which is reported for
reference in Figure 2 with a black dashed line).

Figure 2: Closed-loop state response of the system with both sign switch and
scaling change of the control matrix second column (68). Comparison between
fixed and adaptive time-triggered controller.

Overall, these results show that the proposed event-triggered solution,
without any prior knowledge of the time-variations, robustly addresses a
challenging regulation problem where other more common approaches would fail
or lack robustness.

###  8.3 Sinusoidal variations

We consider now the following perturbations to (66)

$\begin{array}[]{rlll}A(k)&=\begin{bmatrix}1.1&0.1\\\ 0.1&0.2\\\
\end{bmatrix}\left(I_{2}+\delta_{a}\text{diag}(\cos{\frac{2\pi}{p}k},-\cos{\frac{2\pi}{p}k})\right),\vspace{0.1in}\\\
B(k)&=B_{0}\;\forall k\in\mathbb{Z}_{\geq 0}\end{array}$ (69)

with $\delta_{a}=0.8$. The input matrix is now constant and the state matrix
undergoes a structured sinusoidal perturbation of period $p$ which determines
a change up to $80\%$ for the diagonal values. Figure 3 shows the closed-loop
the response of (69) with the event-triggered scheme for different values of
$p$.

Figure 3: Closed-loop state response of system (69) for 3 values of $p$
controlled with the adaptive event-triggered controller.

The results show that in all cases the proposed controllers successfully
regulate the system to the origin with a limited number of episodes (each
triggering instant is denoted with a differently shaped marker on the
corresponding curve). The simulation was carried out until $k=100$ but the
$x$-axis is stopped before as all curves reached the origin and no further
episode is triggered. The fixed controller $K_{0}$ led to unstable responses
for all the considered values of $p$.

##  9 Conclusions

This work addresses the problem of data-based stabilization of linear time-
varying systems by on-line adaptation of the feedback gain. We propose a
hybrid systems framework, which casts the adaptation as a jump in the dynamics
triggered by events depending on some relevant closed-loop properties. The
control design uses robust LMI conditions based on the most recently collected
data to guarantee a Lyapunov-like property for all the systems sufficiently
close to those that generated the data. The formulation allows the problem of
establishing how and when adaptation should take place to be formally
addressed, and a closed-loop analysis of the resulting nonlinear feedback loop
to be performed using nonlinear control techniques. We establish sufficient
conditions under which the interconnection achieves different notions of
stability, and we provide examples where this can be established under
conditions related to the time-variations of the plant. Besides studying
different triggering conditions, open research questions include the effect of
both process and measurement noise, the addition of exploratory signals to
guarantee richness of the collected data and the extension to nonlinear
plants.

## Appendix: Proof of Lemma 4

The proof is organized as follows. Under the full raw rank assumption on $Z$,
we first prove the “only if” part of item (i), which we then exploit to show
that item (ii) holds. We finally invoke item (ii) to establish the “if” part
of item (i). We finally relax the assumption on the rank of $Z$ and prove
items (iii) and (iv).

_Item (i) (“only if”)_. Let $R:=Z^{\top}M^{-1}Z$. Since $Z$ has full row rank
by assumption, $M$ is positive definite and therefore $R\succeq 0$. On the
other hand,

$\begin{array}[]{rlllll}\Delta&=XZ^{\top}M^{-1}ZX^{\top}-XX^{\top}+F\\\
&=XRX^{\top}-XX^{\top}+F\\\ &=X(R-I_{T})X^{\top}+F.\end{array}$ (70)

We first show that

$\Delta=\widetilde{D}(\mathrm{d},M_{A},M_{B})(R-I_{T})\widetilde{D}(\mathrm{d},M_{A},M_{B})^{\top}+F,\,\forall
M_{A},M_{B}.$ (71)

For this, we fix $M_{A}\in\mathbb{R}^{n_{x}\times n_{x}}$ and
$M_{B}\in\mathbb{R}^{n_{u}\times n_{x}}$ and observe that (28), (32) and (34)
yield

$\begin{array}[]{rlllll}X&=M_{A}\widehat{X}(q)+M_{B}U-\widetilde{D}(\mathrm{d},M_{A},M_{B})\\\
&=[M_{B}\,\,M_{A}]Z-\widetilde{D}(\mathrm{d},M_{A},M_{B}).\end{array}$ (72)

Hence

$\begin{array}[]{ll}(R-I_{T})X^{\top}=\\\
-(R-I_{T})\widetilde{D}(\mathrm{d},M_{A},M_{B})^{\top}+(R-I_{T})Z^{\top}[M_{B}\,M_{A}]^{\top}.\end{array}$
(73)

By definition of $R$ and $M$, we note that

$\begin{array}[]{rlllll}\begin{aligned}
&(R-I_{T})Z^{\top}[M_{B}\,M_{A}]^{\top}=\Big{(}Z^{\top}M^{-1}Z-I_{T}\Big{)}Z^{\top}[M_{B}\,M_{A}]^{\top}\\\
&=Z^{\top}\Big{(}M^{-1}ZZ^{\top}-I_{T}\Big{)}[M_{B}\,M_{A}]^{\top}\\\
&=Z^{\top}\Big{(}M^{-1}M-I_{T}\Big{)}[M_{B}\,M_{A}]^{\top}=0.\end{aligned}\end{array}$
(74)

Consequently, in view of (73), we deduce
$(R-I_{T})X^{\top}=-(R-I_{T})\widetilde{D}(\mathrm{d},M_{A},M_{B})^{\top}$,
and thus, as $R$ is symmetric,

$X(R-I_{T})=-\widetilde{D}(\mathrm{d},M_{A},M_{B})(R-I_{T}).$ (75)

Applying these last two equations to (70) yields (71) as $M_{A}$ and $M_{B}$
have been arbitrarily selected.

We take $\left[M_{B}\,\,M_{A}\right]^{\top}\in\mathcal{E}(\mathrm{d},F)$,
where the existence of such a pair is justified by the assumed non-emptiness
of $\mathcal{E}(\mathrm{d},F)$. By (71),

$\begin{array}[]{rlllll}\Delta&=\widetilde{D}(\mathrm{d},M_{A},M_{B})(R-I_{T})\widetilde{D}(\mathrm{d},M_{A},M_{B})^{\top}+F\\\
&\succeq-\widetilde{D}(M_{A},M_{A})\widetilde{D}(\mathrm{d},M_{A},M_{B})^{\top}+F\succeq
0,\end{array}$ (76)

where we first use that $R\succeq 0$ and then the fact that
$\widetilde{D}(\mathrm{d},M_{A},M_{B})\widetilde{D}(\mathrm{d},M_{A},M_{B})^{\top}\preceq
F$ as $[M_{B}\,M_{A}]^{\top}\in\mathcal{E}(\mathrm{d},F)$. We have proved the
desired result.

_Item (ii)_. Let $M_{A}$ and $M_{B}$ be of compatible dimensions, we have by
(32) and (34)

$\begin{array}[]{rllll}\widetilde{D}(\mathrm{d},M_{A},M_{B})&=&\left[M_{B}\,\,M_{A}\right]Z-X.\end{array}$
(77)

Therefore

$\begin{array}[]{rlllll}&\widetilde{D}(\mathrm{d},M_{A},M_{B})\widetilde{D}(\mathrm{d},M_{A},M_{B})^{\top}\preceq
F\\\
\Leftrightarrow&\left(\left[M_{B}\,\,M_{A}\right]Z-X\right)\left(Z^{\top}\left[M_{B}\,\,M_{A}\right]^{\top}-X^{\top}\right)\preceq
F\\\
\Leftrightarrow&\left[M_{B}\,\,M_{A}\right]ZZ^{\top}\left[M_{B}\,\,M_{A}\right]^{\top}-\left[M_{B}\,\,M_{A}\right]ZX^{\top}\\\
&\hfill-XZ^{\top}\left[M_{B}\,\,M_{A}\right]^{\top}+XX^{\top}\preceq F.\\\
\end{array}$ (78)

Define now $N:=-ZX^{\top}$ and $C:=XX^{\top}-F$. Hence
$\widetilde{D}(\mathrm{d},M_{A},M_{B})\widetilde{D}(\mathrm{d},M_{A},M_{B})^{\top}\preceq
F$ if and only if
$\left[M_{B}\,\,M_{A}\right]^{\top}\in\widetilde{\mathcal{E}}(\mathrm{d},F)$
where

$\displaystyle\widetilde{\mathcal{E}}(\mathrm{d},F)$
$\displaystyle:=\Big{\\{}\widetilde{Z}\,:\,\widetilde{Z}^{\top}M\widetilde{Z}+\widetilde{Z}^{\top}N+N^{\top}\widetilde{Z}+C\preceq
0\Big{\\}}.$ (79)

This shows $\mathcal{E}(\mathrm{d},F)=\widetilde{\mathcal{E}}(\mathrm{d},F)$.
Note that showing this equality does not require any assumption on the rank of
$Z$. When $Z$ has full row rank, then $M\succ 0$ and
$N^{\top}M^{-1}N-C=\Delta\succeq 0$ from item (i) because
$\left[M_{B}\,\,M_{A}\right]^{\top}\in\mathcal{E}(\mathrm{d},F)$. Therefore
$\widetilde{\mathcal{E}}(\mathrm{d},F)$ is a matrix ellipsoidal. The desired
result is then obtained by noticing that the ellipsoidal set
$\widehat{\mathcal{E}}(\mathrm{d},F)$ is equivalent to
$\widetilde{\mathcal{E}}(\mathrm{d},F)$ in (37) by taking $Z_{c}=-M^{-1}N$ and
$\Delta=N^{\top}M^{-1}N-C$.

_Item (i) (“if”)_. If $\Delta\in\mathbb{S}^{n_{x}}_{\succeq 0}$, then set
$\widehat{\mathcal{E}}(\mathrm{d},F)$ is non-empty as
$Z_{c}\in\widehat{\mathcal{E}}(\mathrm{d},F)$. By item (ii), the equivalence
(36) holds and thus set $\mathcal{E}(\mathrm{d},F)$ is also non-empty.

_Item (iii)_. Define $\widetilde{V}^{\top}\in\mathbb{R}^{m\times n_{x}+n_{u}}$
and $\widetilde{W}^{\top}\in\mathbb{R}^{(n_{x}+n_{u}-m)\times n_{x}+n_{u}}$ as
in the statement of the lemma. Using the range-null space decomposition, we
can write any matrix $\widetilde{Z}\in\mathbb{R}^{(n_{x}+n_{u})\times n_{x}}$
as

$\widetilde{Z}^{\top}=\widetilde{P}\widetilde{W}^{\top}+\widetilde{Q}\widetilde{V}^{\top}=\left[\widetilde{P}\,\,\widetilde{Q}\right]\begin{bmatrix}\widetilde{W}^{\top}\\\
\widetilde{V}^{\top}\end{bmatrix}$

for appropriate matrices $\widetilde{P},\widetilde{Q}$. Recall moreover that
the equivalence
$\mathcal{E}(\mathrm{d},F)=\widetilde{\mathcal{E}}(\mathrm{d},F)$ in item (ii)
does not require any assumption on the rank of $Z$. For these reasons, we can
write

$\displaystyle\widetilde{\mathcal{E}}(\mathrm{d},F):=\Big{\\{}\widetilde{Z}\,:\,\widetilde{Z}^{\top}M\widetilde{Z}+\widetilde{Z}^{\top}N+N^{\top}\widetilde{Z}+C\preceq
0\Big{\\}},$ (80)
$\displaystyle=\Big{\\{}[\widetilde{P}\,\,\widetilde{Q}]\in\mathbb{R}^{(n_{x}+n_{u})\times
n_{x}}\,:\,[\widetilde{P}\;\;\widetilde{Q}]\begin{bmatrix}\widetilde{W}^{\top}\\\
\widetilde{V}^{\top}\end{bmatrix}M(\star)+$
$\displaystyle\quad\quad+[\widetilde{P}\;\;\widetilde{Q}]\begin{bmatrix}\widetilde{W}^{\top}\\\
\widetilde{V}^{\top}\end{bmatrix}N+N^{\top}(\star)+C\preceq 0\Big{\\}},$
$\displaystyle=\Big{\\{}[\widetilde{P}\,\,\widetilde{Q}]\,:\,\widetilde{Q}\widetilde{M}\widetilde{Q}^{\top}+\widetilde{Q}\widetilde{N}+\widetilde{N}^{\top}\widetilde{Q}^{\top}+C\preceq
0\Big{\\}},$

where we have defined $\widetilde{M}:=\widetilde{V}^{\top}M\widetilde{V}$ and
$\widetilde{N}:=\widetilde{V}^{\top}N$. Note that $\widetilde{M}$ has full row
rank by choice of $\widetilde{V}$, and the conditions defining (80) do not
depend on $\widetilde{P}$. We can thus use the same arguments of item (ii) to
show that $\widetilde{\mathcal{E}}(\mathrm{d},F)$ is equivalent to

$\displaystyle\widehat{\mathcal{E}}(\mathrm{d},F):=\big{\\{}[\widetilde{P}\,\,\widetilde{Q}]\,:\,(\widetilde{Q}^{\top}-\widetilde{Z}_{c})^{\top}\widetilde{M}(\widetilde{Q}^{\top}-\widetilde{Z}_{c})\preceq\widetilde{\Delta}\big{\\}}$
(81)

by taking $\widetilde{Z}_{c}=-\widetilde{M}^{-1}\widetilde{N}$ and
$\widetilde{\Delta}=\widetilde{N}^{\top}\widetilde{M}^{-1}\widetilde{N}-C$,
where the latter expression coincides with (38). This shows
$\mathcal{E}(\mathrm{d},F)=\widetilde{\mathcal{E}}(\mathrm{d},F)=\widehat{\mathcal{E}}(\mathrm{d},F)$
and thus (81) characterizes $\mathcal{E}(\mathrm{d},F)$ when $Z$ is not full
row rank. Moreover, following the same reasoning of item (ii), this set is
non-empty if and only if $\widetilde{\Delta}\succeq 0$. Note that this set is
either empty or unbounded, because if
$[\widetilde{P}\,\,\widetilde{Q}]\in\widetilde{\mathcal{E}}(\mathrm{d},F)$ for
$\widetilde{P},\widetilde{Q}$, then clearly
$[\widehat{P}\,\,\widetilde{Q}]\in\widetilde{\mathcal{E}}(\mathrm{d},F)$ for
all $\widehat{P}$ of compatible dimensions.

## Appendix: Proof of Proposition 2

We rely on the next intermediate lemma.

###### Lemma 7

Under the conditions of Proposition 2, for any $\varepsilon\in\mathbb{R}_{\geq
0}$,
$\left[M_{B}\,\,M_{A}\right]^{\top}\in\mathcal{E}(\mathrm{d},F+\varepsilon
S^{-1})$ implies
$S-(M_{A}+M_{B}K)^{\top}\Big{(}\star\Big{)}^{-1}(M_{A}+M_{B}K)\succ 0$, with
$S=(\widehat{X}Y)^{-1}$, $\star=S^{-1}-H+(1+\varsigma^{-1})(F+\varepsilon
S^{-1})$. $\Box$

Proof: The proof follows the main steps of [3, proof of Theorem 5] adapted to
the LTV context of this work. Let $\mathrm{d}\in\mathcal{S}_{\mathrm{d}}$ be
such that (40) holds. We restrict the controller $K$ we search over to be of
the form

$\begin{array}[]{rllll}\left[\begin{matrix}K\\\
I_{n_{x}}\end{matrix}\right]&=&\left[\begin{matrix}U\\\
\widehat{X}\end{matrix}\right]G,\end{array}$ (82)

where $G\in\mathbb{R}^{T\times n_{x}}$. Let $M_{A}\in\mathbb{R}^{n_{x}\times
n_{x}}$ and $M_{B}\in\mathbb{R}^{n_{u}\times n_{x}}$, we have from (82)

$\begin{array}[]{rllll}M_{A}+M_{B}K&=&\big{[}\begin{matrix}M_{B}&M_{A}\end{matrix}\big{]}\left[\begin{matrix}K\\\
I_{n_{x}}\end{matrix}\right]\vspace{0.2cm}\\\
&=&\big{[}\begin{matrix}M_{B}&M_{A}\end{matrix}\big{]}\left[\begin{matrix}U\\\
\widehat{X}\end{matrix}\right]G\\\ &=&(M_{A}\widehat{X}+M_{B}U)G.\end{array}$
(83)

We deduce from (32) and (40) that

$\begin{array}[]{rllll}M_{A}+M_{B}K=(X+\widetilde{D}(q,M_{A},M_{B}))G.\end{array}$
(84)

To ease the notation, we omit in the reminder the arguments of $\widetilde{D}$
unless necessary. The first main step of the proof consists in showing the
existence of $P\in\mathbb{S}^{n_{x}}_{\succ 0}$ and
$\widetilde{W}\in\mathbb{S}^{n_{x}}$ such that

$\begin{array}[]{rlllll}(X+\widetilde{D})GPG^{\top}(X+\widetilde{D})^{\top}-P&\prec&-\widetilde{W}.\end{array}$
(85)

To this end, we define $Y:=GP$. Then, from the second block row of (82) it
follows that

$P=\widehat{X}GP=\widehat{X}Y.$ (86)

Using the second LMI in (41), this shows immediately that $\widehat{X}Y=P\succ
0$. Indeed, by Schur complement of the second LMI in (41),
$\widehat{X}Y-Y^{\top}Y\succ 0$, and thus $\widehat{X}Y\succ Y^{\top}Y\succeq
0$.

Using the invertibility of $P$, we can write

$GPG^{\top}=Y(\widehat{X}Y)^{-1}Y^{\top}.$ (87)

Leveraging Young’s inequality we know that for any real matrices of compatible
dimensions $M_{1},M_{2},N$ with $N\succ 0$ and any
$\epsilon\in\mathbb{R}_{>0}$,
$M_{1}NM_{2}^{\top}+M_{2}NM_{1}^{\top}\preceq\epsilon
M_{1}NM_{1}^{\top}+\epsilon^{-1}M_{2}NM_{2}^{\top}$. We derive, by taking
$M_{1}=XY(\widehat{X}Y)^{-1}$, $N=\widehat{X}Y$,
$M_{2}=\widetilde{D}Y(\widehat{X}Y)^{-1}$ and $\epsilon=\varsigma$, and (86)
and (87)

$\begin{array}[]{llllll}(X+\widetilde{D})GPG^{\top}(X+\widetilde{D})^{\top}-P\\\
\quad=(X+\widetilde{D})Y(\widehat{X}Y)^{-1}Y^{\top}(X+\widetilde{D})^{\top}-\widehat{X}Y\\\
\quad\preceq(1+\varsigma)XY(\widehat{X}Y)^{-1}(XY)^{\top}\\\
\quad\quad+(1+\varsigma^{-1})\widetilde{D}Y(\widehat{X}Y)^{-1}(\widetilde{D}Y)^{\top}-\widehat{X}Y.\end{array}$
(88)

By Schur complement of the second LMI in (41),
$I_{T}-Y(\widehat{X}Y)^{-1}Y^{\top}\succ 0$, equivalently

$I_{T}\succ Y(\widehat{X}Y)^{-1}Y^{\top}.$ (89)

As a consequence,

$\begin{array}[]{llllll}(X+\widetilde{D})GPG^{\top}(X+\widetilde{D})^{\top}-P\\\
\quad\quad\quad\quad\preceq(1+\varsigma)XY(\widehat{X}Y)^{-1}(XY)^{\top}\\\
\quad\quad\quad\quad\quad+(1+\varsigma^{-1})\widetilde{D}\widetilde{D}^{\top}-\widehat{X}Y.\end{array}$
(90)

Let $F\in\mathbb{S}_{\succ 0}^{n_{x}}$ be such that
$F\prec\frac{\varsigma}{1+\varsigma}H$ and $\varepsilon\in\mathbb{R}_{\geq
0}$. Take
$\big{[}M_{B}\,\,M_{A}\big{]}^{\top}\in\mathcal{E}(\mathrm{d},F+\varepsilon
P)$, it holds that
$\widetilde{D}(M_{A},M_{B})\widetilde{D}(q,M_{A},M_{B})^{\top}\preceq
F+\varepsilon P$ and thus

$\displaystyle(X+\widetilde{D})GPG^{\top}(X+\widetilde{D})^{\top}-P\preceq(1+\varsigma)XY(\widehat{X}Y)^{-1}(XY)^{\top}$
(91) $\displaystyle+(1+\varsigma^{-1})(F+\varepsilon P)-\widehat{X}Y=:\Theta.$

By Schur complement of the first LMI in (41), $\widehat{X}Y-\varsigma
XX^{\top}-H-XY(\widehat{X}Y)^{-1}(XY)^{\top}\succ 0$, which we rewrite as, in
view of the definition of $\Theta$ in (91),
$(1+\varsigma)XY(\widehat{X}Y)^{-1}(XY)^{\top}+(1+\varsigma^{-1})(F+\varepsilon
P)-\Theta-\varsigma XX^{\top}-H-XY(\widehat{X}Y)^{-1}(XY)^{\top}\succ 0$.
Equivalently,

$\begin{array}[]{rlllll}\Theta&\prec&\varsigma
XY(\widehat{X}Y)^{-1}(XY)^{\top}+(1+\varsigma^{-1})(F+\varepsilon P)\\\
&&-\varsigma XX^{\top}-H.\end{array}$ (92)

From (89) we obtain

$\begin{array}[]{rlllll}\Theta&\prec&(1+\varsigma^{-1})(F+\varepsilon
P)-H.\end{array}$ (93)

Define $W=H-(1+\varsigma^{-1})F$ and observe that $W\succ 0$ as
$F\prec\frac{\varsigma}{1+\varsigma}H$. We can then write

$\begin{array}[]{rlllll}\Theta&\prec&-W+(1+\varsigma^{-1})\varepsilon
P=:-\widetilde{W}.\end{array}$ (94)

Piecing together (91) and (94) yields (85).

Now, rewrite (85) as
$(X+\widetilde{D})GPG^{\top}(X+\widetilde{D})^{\top}-(P-\widetilde{W})\prec 0$
and do a Schur complement to obtain

$\left[\begin{matrix}P-\widetilde{W}&(X+\widetilde{D})G\\\
\star&P^{-1}\end{matrix}\right]\succ 0.$ (95)

Considering the other Schur complement of the matrix above, we get

$P^{-1}-((X+\widetilde{D})G)^{\top}(P-\widetilde{W})^{-1}(X+\widetilde{D})G\succ
0,$ (96)

note that $P-\widetilde{W}\in\mathbb{S}^{n_{x}}_{\succ 0}$ by definition of
$\widetilde{W}$ and because from the first LMI in (41) we have $P\succ H$, as
principal submatrices of positive definite matrices are positive definite.

Let $S=P^{-1}$, $\varepsilon\in\mathbb{R}_{\geq 0}$ and
$\big{[}M_{B}\,\,M_{A}\big{]}^{\top}\in\mathcal{E}(\mathrm{d},F+\varepsilon
S^{-1})$, by (96) and the definition of $\widetilde{W}$,
$S-(M_{A}+M_{B}K)^{\top}(S^{-1}-W+(1+\varsigma^{-1})\varepsilon
S^{-1})^{-1}(M_{A}+M_{B}K)\succ 0$. We have obtained the desired result.
$\blacksquare$

We are now ready to prove Proposition 2. Let $\varepsilon\in\mathbb{R}_{\geq
0}$,
$\left[M_{B}\,\,M_{A}\right]^{\top}\in\mathcal{E}(\mathrm{d},F+\varepsilon
S^{-1})$ with $S$ as in Lemma 7, and $x\in\mathbb{R}^{n_{x}}$. By Lemma 7,
$x^{\top}(M_{A}+M_{B}K)^{\top}(S^{-1}-\widetilde{W})^{-1}(M_{A}+M_{B}K)x-x^{\top}Sx\leq
0$ with $\widetilde{W}=H-(1+\varsigma^{-1})(F+\varepsilon S^{-1})$ as in the
proof of Lemma 7, which we rewrite as

$\begin{array}[]{llllll}x^{\top}(M_{A}+M_{B}K)^{\top}S(M_{A}+M_{B}K)x-x^{\top}Sx\leq\\\
-x^{\top}(M_{A}+M_{B}K)^{\top}\big{[}\big{(}S^{-1}-\widetilde{W}\big{)}^{-1}-S\big{]}(M_{A}+M_{B}K)x.\end{array}$
(97)

By taking $V(x,S)=x^{\top}Sx$ and
$Q:=\big{(}S^{-1}-\widetilde{W}\big{)}^{-1}-S$, we obtain

$\displaystyle V((M_{A}+M_{B}K)x,S)-V(x,S)\leq$ (98)
$\displaystyle\quad-x^{\top}(M_{A}+M_{B}K)^{\top}Q(M_{A}+M_{B}K)x.$

We show next that there exists $\lambda(\varepsilon)\in(-1,+\infty)$ such that
$Q\succeq\lambda(\varepsilon)S$. We note that
$a=\max\big{\\{}a^{\prime}\in\mathbb{R}_{\geq 0}\,:\,a^{\prime}S^{-1}\preceq
W\big{\\}}$ belongs to $(0,1)$ as $S,W\in\mathbb{S}^{n_{x}}_{\succ 0}$ and
$P\succ H$ (from (41)) implies $S^{-1}-W\succ 0$. Take

$\lambda(\varepsilon)=\frac{a-(1+\varsigma^{-1})\varepsilon}{1-a+(1+\varsigma^{-1})\varepsilon},$
(99)

where the denominator is positive because $\varepsilon\in\mathbb{R}_{\geq 0}$,
$a\in(0,1)$ and $\varsigma>0$. Moreover, it holds

$\displaystyle\lambda(\varepsilon)+1$
$\displaystyle=\frac{1}{1-a+(1+\varsigma^{-1})\varepsilon}>0.$ (100)

This proves that $\lambda(\varepsilon)\in(-1,+\infty)$. To show
$Q\succeq\lambda(\varepsilon)S$, observe that

$\displaystyle Q\succeq\lambda(\varepsilon)S$
$\displaystyle\Leftrightarrow\left(S^{-1}-W+(1+\varsigma^{-1})\varepsilon
S^{-1}\right)^{-1}\\!-\\!S\succeq\lambda(\varepsilon)S$ (101)
$\displaystyle\Leftrightarrow
S^{-1}\\!-\\!W\\!+\\!(1+\varsigma^{-1})\varepsilon
S^{-1}\\!\\!\preceq\tfrac{1}{1+\lambda(\varepsilon)}S^{-1}$
$\displaystyle\Leftrightarrow\big{(}1-\tfrac{1}{1+\lambda(\varepsilon)}+(1+\varsigma^{-1})\varepsilon\big{)}S^{-1}\preceq
W$
$\displaystyle\Leftrightarrow\big{(}\tfrac{\lambda(\varepsilon)}{1+\lambda(\varepsilon)}+(1+\varsigma^{-1})\varepsilon\big{)}S^{-1}\preceq
W.$

By (99) and (100),
$\tfrac{\lambda(\varepsilon)}{1+\lambda(\varepsilon)}+(1+\varsigma^{-1})\varepsilon=a$
therefore $Q\succeq\lambda(\varepsilon)S\Leftrightarrow aS^{-1}\preceq W$. The
last inequality holds by definition of $a$. We deduce that
$Q\succeq\lambda(\varepsilon)S$. We use this property together with (98) to
derive that for all
$\left[M_{B}\,\,M_{A}\right]^{\top}\in\mathcal{E}(\mathrm{d},F+\varepsilon
S^{-1})$
$V((M_{A}+M_{B}K)x,S)-V(x,S)\leq-\lambda(\varepsilon)V((M_{A}+M_{B}K)x,S)$ and
thus, by (100)
$V((M_{A}+M_{B}K)x,S)\leq\frac{1}{1+\lambda(\varepsilon)}V(x,S)=(1-a+(1+\varsigma^{-1})\varepsilon)V(x,S)$.
The desired result holds with $a_{1}=1-a\in(0,1)$ and
$a_{2}=1+\varsigma^{-1}$.

## References

## References

  * [1] R. Sanfelice and A. Teel, “Dynamical properties of hybrid systems simulators,” Automatica, vol. 46, no. 2, pp. 239–248, 2010.
  * [2] W. Heemels, K. Johansson, and P. Tabuada, “An introduction to event-triggered and self-triggered control,” in IEEE Conference on Decision and Control, pp. 3270–3285, 2012.
  * [3] C. De Persis and P. Tesi, “Formulas for data-driven control: Stabilization, optimality, and robustness,” IEEE Trans. on Automatic Control, vol. 65, no. 3, pp. 909–924, 2020.
  * [4] H. van Waarde, M. Camlibel, and M. Mesbahi, “From noisy data to feedback controllers: Nonconservative design via a matrix S-lemma,” IEEE Trans. on Automatic Control, vol. 67, no. 1, pp. 162–175, 2020.
  * [5] J. Berberich, C. Scherer, and F. Allgöwer, “Combining prior knowledge and data for robust controller design,” IEEE Trans. on Automatic Control, vol. 68, no. 8, pp. 4618–4633, 2023.
  * [6] M. Rotulo, C. De Persis, and P. Tesi, “Online learning of data-driven controllers for unknown switched linear systems,” Automatica, vol. 145, p. 110519, 2022.
  * [7] S. Liu, K. Chen, and J. Eising, “Online data-driven adaptive control for unknown linear time-varying systems,” in IEEE Conference on Decision and Control, pp. 8775–8780, 2023.
  * [8] B. Nortmann and T. Mylvaganam, “Direct data-driven control of linear time-varying systems,” IEEE Trans. on Automatic Control, vol. 68, no. 8, pp. 4888–4895, 2023.
  * [9] M. Mazo, A. Anta, and P. Tabuada, “An ISS self-triggered implementation of linear controllers,” Automatica, vol. 46, no. 8, pp. 1310–1314, 2010\.
  * [10] X. Wang and M. Lemmon, “On event design in event-triggered feedback systems,” Automatica, vol. 47, no. 10, pp. 2319–2322, 2011.
  * [11] A. Maass, W. Wang, D. Nešić, R. Postoyan, and W. Heemels, “Event-triggered control through the eyes of a hybrid small-gain theorem,” IEEE Trans. on Automatic Control, vol. 68, no. 10, pp. 5906–5921, 2023\.
  * [12] C. De Persis, R. Postoyan, and P. Tesi, “Event-triggered control from data,” IEEE Trans. on Automatic Control, vol. available on-line, 2023.
  * [13] W.-L. Qi, K.-Z. Liu, R. Wang, and X.-M. Sun, “Data-driven $\mathcal{L}_{2}$-stability analysis for dynamic event-triggered networked control systems: A hybrid system approach,” IEEE Trans. on Industrial Electronics, vol. 70, no. 6, pp. 6151–6158, 2022.
  * [14] X. Wang, J. Berberich, J. Sun, G. Wang, F. Allgöwer, and J. Chen, “Model-based and data-driven control of event-and self-triggered discrete-time linear systems,” IEEE Trans. on Cyb., vol. 53, no. 9, pp. 6066 – 6079, 2023.
  * [15] R. Goebel, R. Sanfelice, and A. Teel, Hybrid Dynamical Systems.  Princeton University Press, Princeton, U.S.A., 2012.
  * [16] H. van Waarde, J. Eising, H. Trentelman, and M. Camlibel, “Data informativity: a new perspective on data-driven analysis and control,” IEEE Trans. on Automatic Control, vol. 65, no. 11, pp. 4753–4768, 2020.
  * [17] J. C. Willems, P. Rapisarda, I. Markovsky, and B. D. Moor, “A note on persistency of excitation,” Syst. & Contr. Lett., vol. 54, no. 4, pp. 325–329, 2005.
  * [18] A. Girard, “Dynamic triggering mechanisms for event-triggered control,” IEEE Trans. on Automatic Control, vol. 60, no. 7, pp. 1992–1997, 2015.
  * [19] S. Benahmed, R. Postoyan, M. Granzotto, L. Buşoniu, J. Daafouz, and D. Nešić, “Stability analysis of optimal control problems with time-dependent costs,” Automatica, vol. 157, p. 111272, 2023.
  * [20] A. Bisoffi, C. De Persis, and P. Tesi, “Trade-offs in learning controllers from noisy data,” Systems & Control Letters, vol. 154, 2021.
  * [21] A. Bisoffi, C. De Persis, and P. Tesi, “Data-driven control via Petersen’s lemma,” Automatica, vol. 145, p. 110537, 2022.
  * [22] D. Borgers, V. Dolk, G. Dullerud, A. Teel, and W. Heemels, “Time-regularized and periodic event-triggered control for linear systems,” in Control Subject to Computational and Communication Constraints, pp. 121–149, Springer, 2018.
  * [23] M. Abdelrahim, R. Postoyan, J. Daafouz, and D. Nešić, “Robust event-triggered output feedback controllers for nonlinear systems,” Automatica, vol. 75, pp. 96–108, 2017.
  * [24] P. Tallapragada and N. Chopra, “Event-triggered dynamic output feedback control for LTI systems,” in IEEE Conference on Decision and Control, Maui, U.S.A., pp. 6597–6602, 2012.
  * [25] W. Wang, D. Nešić, R. Postoyan, I. Shames, and W. Heemels, “State-feedback event-holding control for nonlinear systems,” in IEEE Conf. on Decision and Control, Nice, France, pp. 1650–1655, 2019.
  * [26] M. ApS, The MOSEK optimization toolbox for MATLAB manual. Version 9.0., 2019\.

[] Andrea Iannelli (Member, IEEE) is an Assistant Professor in the Institute
for Systems Theory and Automatic Control at the University of Stuttgart. He
completed his B.Sc. and M.Sc. degrees in Aerospace Engineering at the
University of Pisa and received his PhD from the University of Bristol. He was
also a postdoctoral researcher in the Automatic Control Laboratory at ETH
Zurich. His main research interests are at the intersection of control theory,
optimization, and learning, with a particular focus on robust and adaptive
control, uncertainty quantification, and sequential decision-making. [] Romain
Postoyan (Senior Member, IEEE) received the “Ingénieur” degree in Electrical
and Control Engineering from ENSEEIHT (France) in 2005. He obtained the M.Sc.
by Research in Control Theory & Application from Coventry University (United
Kingdom) in 2006 and the Ph.D. in Control Engineering from Université Paris-
Sud (France) in 2009. In 2010, he was a research assistant at the University
of Melbourne (Australia). Since 2011, he is a CNRS researcher at CRAN (Nancy,
France).