††thanks: The work was supported in part by the National Natural Science
Foundation of China under Grants 61925303, 62173034, 62088101, U22B2058, the
China Scholarship Council under Grants 202206030127, and in part by the
Chongqing Natural Science Foundation under Grant 2021ZX4100027.
_(Corresponding author: Gang Wang.)_ ††thanks: This paper was not presented at
any IFAC meeting.

, , , ,

# Robust Control of Unknown Switched Linear
Systems from Noisy Data

Wenjie Liu<EMAIL_ADDRESS>Yifei Li<EMAIL_ADDRESS>Jian Sun
<EMAIL_ADDRESS>Gang Wang<EMAIL_ADDRESS>Jie Chen<EMAIL_ADDRESS>National Key Lab of Autonomous Intelligent Unmanned Systems and School of
Automation,
Beijing Institute of Technology, Beijing 100081, China  Beijing Institute of
Technology Chongqing Innovation Center, Chonqing 401120, China  Department of
Control Science and Engineering, Tongji University, Shanghai 201804, China

###### Abstract

This paper investigates the problem of data-driven stabilization for linear
discrete-time switched systems with unknown switching dynamics. In the absence
of noise, a data-based state feedback stabilizing controller can be obtained
by solving a semi-definite program (SDP) on-the-fly, which automatically
adapts to the changes of switching dynamics. However, when noise is present,
the persistency of excitation condition based on the closed-loop data may be
undermined, rendering the SDP infeasible. To address this issue, an auxiliary
function-based switching control law is proposed, which only requires
intermittent SDP solutions when its feasibility is guaranteed. By analyzing
the relationship between the controller and the system switching times, it is
shown that the proposed controller guarantees input-to-state practical
stability (ISpS) of the closed-loop switched linear system, provided that the
noise is bounded and the dynamics switches slowly enough. Two numerical
examples are presented to verify the effectiveness of the proposed controller.

###### keywords:

Data-driven control, online learning, switched system, noisy data, semi-
definite program.

##  1 Introduction

Since the early nineteenth century, classical control theory has yielded
abundant results in areas ranging from feedback control to optimal, adaptive,
robust, and nonlinear control. A critical intermediate step for synthesizing a
controller and associated stability analysis is acquiring the system model
using first-principles or the system identification method. However, with the
increasing complexity and inter-connectivity of engineered cyber-physical
systems, obtaining system models using first-principles methods has become
challenging, while system identification has become data-inefficient and/or
computationally expensive. In recent years, thanks to advances in data science
and big data technology, data-driven control has emerged as a promising
paradigm relative to classical model-based control. Data-driven control
methods circumvent the need for parametric system modeling Chua et al. (2018)
and mitigate the over-fitting of noise Krishnan and Pasqualetti (2021), which
have attracted significant attention.

A range of data-driven control methods have been proposed, including iterative
feedback tuning Hjalmarsson et al. (1998), adaptive control Åström and
Wittenmark (1989); Wu and Meng (2023), and reinforcement learning-based
control Sassano and Astolfi (2020). Further references can be found in Hou and
Wang (2013); De Persis and Tesi (2020). Recently, renewed interest has been
generated by the fundamental lemma developed in Willems et al. (2005), which
provides a sufficient condition for the existence of a data-based system
representation based on input-state data. This has led to a growing number of
publications, including data-enabled predictive control (DeePC) Coulson et al.
(2019) and various system analysis and controller design results that use data
directly; see De Persis and Tesi (2020) for a recent survey of these
developments. The DeePC framework Coulson et al. (2019) aims to design control
inputs by solving a convex optimization problem based on system trajectories
collected offline. Several extensions have been made, including establishing
theoretical stability guarantees in Berberich et al. (2021), enhancing
robustness in Dörfler et al. (2023), and improving resiliency against attacks
in Liu et al. (2023). Using input-state data, the work of De Persis and Tesi
(2020); van Waarde et al. (2020) provides simple data-based parametrizations
of linear state feedback systems. Moreover, it has been demonstrated in De
Persis and Tesi (2020) that many control problems can be formulated as data-
dependent linear matrix inequalities (LMIs), including linear quadratic
regulation (LQR) De Persis and Tesi (2021); Zhao et al. (2023), robust control
van Waarde et al. (2020); Li et al. (2023a), event- and self-triggered control
Li et al. (2023b); X. Wang et al. (2023); De Persis et al. (2022), complex and
network systems Baggio et al. (2021), time-delay systems Rueda-Escobedo et al.
(2022); Wang et al. (2023), nonlinear systems Guo et al. (2022); Hu et al.
(2023), and system identification Kang and You (2023).

All of the aforementioned results focused on systems that can be fully
characterized by a finite set of data, such as linear time-invariant (LTI)
systems or special nonlinear systems that can be expressed in a ‘linear-like’
form. In such cases, a stabilizing controller designed using offline data can
effectively stabilize the system during online implementation. However, when
the complexity of a system cannot be fully captured or approximated by a
finite set of data, the controller designed offline may no longer be effective
for the changing dynamics of the system. Switched systems, which are commonly
used to model real-world systems such as mobile robots Lee and Jiang (2008),
chemical processes Mhaskar et al. (2005), and power systems Cardim et al.
(2009); Wang et al. (2021), are one such example. Switched systems comprise a
set of subsystems that switch between each other according to some switching
signal. Since the switching signal can be arbitrary in general, even with just
two modes (i.e., subsystems), the sequence of activated modes present in the
data collected offline may be entirely different from that appearing in the
online operation. Therefore, designing a data-driven state feedback control
law for switched systems requires modifications to the offline solutions
discussed earlier, to account for the abrupt switching dynamics which cannot
be fully characterized by a finite set of offline data.

Several related contributions on data-based stabilization of switched linear
systems can be found in Rotulo et al. (2022); Eising et al. (2022); Bianchi et
al. (2022), with the works Eising et al. (2022) and Bianchi et al. (2022)
specifically dealing with noise-free and noisy data, respectively. However,
the assumption that sufficient input-state trajectories for each subsystem
must be collected may seem impractical and hard to validate in general. The
results of Rotulo et al. (2022) relaxed this assumption by proposing an online
perturbed data-based state feedback controller based on a semi-definite
program (SDP) that is updated and solved at each time step. Nevertheless, it
is important to note that the work Rotulo et al. (2022) assumes the offline
data are noiseless, which is often not the case in real-world applications.
Furthermore, its controller and stability results do not hold when noise is
present.

The aim of this paper is to extend the findings of Rotulo et al. (2022) by
incorporating noisy data. The challenge in implementing an online controller
lies in ensuring that the closed-loop data are persistently exciting, which is
crucial for recovering the system behavior and for parameterizing the state
feedback controller using solutions from the data-based SDP De Persis and Tesi
(2021). To address this challenge, this paper proposes a switched controller
that employs a switching control law based on variations of an auxiliary
function. A robust data-based SDP is formulated using noisy data collected in
real time, which is solved intermittently only when its feasibility is
guaranteed. By establishing the relationship between the controller’s
switching times and the system’s switching times, we show that the closed-loop
system is ISpS under mild assumptions on the noise and switching dynamics.

In summary, the paper offers the following contributions to data-based
stabilization of unknown linear switched systems.

  * c1)

A novel online data-based switched controller is designed, following an
auxiliary function based switching law that dictates whether a data-based SDP
is solved or not.

  * c2)

Conditions on the noise and control inputs are provided to guarantee the
feasibility of the robust data-based SDP.

  * c3)

The relationship between the system switching times and the controller
switching times is developed, and ISpS of the closed-loop system is
established.

_Notation:_ Denote the sets of real numbers, integers, and positive integers
by $\mathbb{R}$, $\mathbb{N}$, and $\mathbb{N}_{+}$ respectively. For a matrix
$M$, if it has full column rank, its left pseudo-inverse is denoted by
$M^{\dagger}$. Given a vector $x\in\mathbb{R}^{n_{x}}$, let $\|x\|$ denote its
Euclidean norm. Denote the spectral norm of a matrix $M$ by $\|M\|$. Given a
measurable time function $f:\mathbb{N}\rightarrow\mathbb{R}^{n}$ and a time
interval $[0,k)$ we denote the $\mathcal{L}_{\infty}$ norm of $f(\cdot)$ on
$[0,k)$ by $\|f_{k}\|_{\infty}:={\rm ess}\sup_{s\in[0,k)}\|f(s)\|$. For
matrices $A$, $B$, and $C$ with compatible dimensions, we abbreviate
$ABC(AB)^{\prime}$ as $AB\cdot C[\star]^{\prime}$. Let
$\underline{\lambda}_{P}$ [$\overline{\lambda}_{P}$] be the minimum (maximum)
singular value of matrix $P$. Let
$x_{[k_{1},k_{2}]}:=[x_{k_{1}}~{}x_{k_{1}+1}~{}\cdots~{}x_{k_{2}}]$ denote a
stacked window of signal $x$ in discrete time interval $[k_{1},k_{2}]$. The
Hankel matrix associated with sequence $\\{x(k)\\}_{k=0}^{N-1}$ is denoted by

$H_{L}(x_{[0,N-1]}):=\left[\begin{matrix}x_{0}&x_{1}&\ldots&x_{N-L}\\\
x_{1}&x_{2}&\ldots&x_{N-L+1}\\\ \vdots&\vdots&\ddots&\vdots\\\
x_{L-1}&x_{L}&\ldots&x_{N-1}\end{matrix}\right].$

The definition of $\mu$-persistent excitation as in (Coulson et al., 2022,
Definition 3.1) is given below.

######  Definition 1.1 ($\mu$-persistently exciting).

Let $\mu>0$. A signal $\\{x_{t}\\}_{t=0}^{N-1}\in\mathbb{R}^{n}$ with
$N\geq(n+1)L+1$ is $\mu$-persistently exciting of order $L$ if
$\underline{\lambda}_{H_{L}(x_{[0,N-1]})}\geq\mu$.

##  2 Preliminaries and Problem Formulation

In this section, we begin by reviewing the results in De Persis and Tesi
(2021), which dealt with the data-based stabilization of unknown linear time-
invariant (LTI) systems using noisy data. This plays an instrumental role in
deriving and explaining our results.

###  2.1 Data-driven control of LTI systems

Consider a discrete-time LTI system as follows

$x(k+1)=Ax(k)+Bu(k)+d(k),\quad k\in\mathbb{N}$ (1)

where $x(k)\in\mathbb{R}^{n_{x}}$ is the state, $u(k)\in\mathbb{R}^{n_{u}}$ is
the control input, and $d(k)\in\mathbb{R}^{n_{x}}$ is the noise or
disturbance. The system matrices $(A,B)$ are unknown and we do not have access
to the disturbance $d(k)$. Instead, we assume there are some input-state data
$(u_{[-T,-1]},x_{[-T,0]})$ obtained from e.g., offline experiments by exciting
the system using control inputs $u_{[-T,-1]}$ and collecting the corresponding
states $x_{[-T,0]}$. For consistency, negative indices are used to refer to
data collected offline. Define the data matrices as follows

$\displaystyle U_{-1}$ $\displaystyle=[u(-T)~{}u(-T+1)~{}\cdots~{}u(-1)],$
(2a) $\displaystyle X_{-1}$
$\displaystyle=[x(-T)~{}x(-T+1)~{}\cdots~{}x(-1)],$ (2b) $\displaystyle X_{0}$
$\displaystyle=[x(-T+1)~{}x(-T+2)~{}\cdots~{}x(0)],$ (2c) $\displaystyle
D_{-1}$ $\displaystyle=[d(-T)~{}d(-T+1)~{}\cdots~{}d(-1)],$ (2d)
$\displaystyle W_{-1}$
$\displaystyle=\left[U_{-1}^{\prime}~{}X_{-1}^{\prime}\right]^{\prime}.$ (2e)

A method for finding a matrix $K$ such that $A+BK$ is Schur stable using
disturbance-corrupted data in (2) was presented in (De Persis and Tesi, 2021,
Theorem 4).

######  Lemma 2.1.

Let $U_{-1}$, $X_{-1}$ and $X_{0}$ be data generated by system (5). If the
condition

${\rm rank}(W_{-1})=n_{x}+n_{u}$ (3)

holds, then there exists a constant $\delta>0$ such that for
$\|D_{-1}\|\leq\delta$ the following problem is feasible

$\displaystyle\min_{(\gamma,Q,P,L,V)}\gamma$ $\displaystyle{\rm subject~{}to}$
$\displaystyle\left\\{\begin{aligned}
&X_{0}QP^{-1}Q^{\prime}X_{0}^{\prime}-P+I\preceq 0\\\ &P\succeq I\\\
&L-U_{-1}QP^{-1}Q^{\prime}U_{-1}^{\prime}\succeq 0\\\
&V-QP^{-1}Q^{\prime}\succeq 0\\\ &X_{0}Q=P\\\ &{\rm tr}(P)+{\rm
tr}(L)+\alpha{\rm tr}(V)\preceq\gamma\end{aligned}\right.$ (4)

where $\alpha>0$ is arbitrary. Let
$(\bar{\gamma},\bar{Q},\bar{P},\bar{L},\bar{V})$ be an optimal solution of
(2.1). Then the matrix $K^{*}=U_{-1}\bar{Q}\bar{P}^{-1}$ is stabilizing.

In light of condition (3), it is notable that a stabilizing gain $K^{*}$ can
be parameterized using solely data. This implies that the state feedback
controller $u(k)=K^{*}x(k)$ can be implemented on system (1). According to
(Coulson et al., 2022, Theorem 3.1), for any bounded disturbance, i.e.,
$\|D_{-1}\|\leq\delta$ for some $\delta>0$, the condition (3) can be ensured
by utilizing a sufficiently exciting input sequence.

We present Lemma 2.2 below, which shows that a $\bar{w}$-persistently exciting
input sequence of order $n_{x}+1$ is sufficient to satisfy the condition (3).
We postpone the proof of this lemma to Appendix Appendix A.1.

######  Lemma 2.2.

Suppose the system (1) is controllable. For any $\delta\geq 0$, let the
disturbance obey $\|D_{-1}\|\leq\delta$. Then there exists a constant
$\bar{w}>0$ such that if the input sequence ${u(-T),\cdots,u(-1)}$ is
$\bar{w}$-persistently exciting of order $n_{x}+1$, then the condition (3) is
satisfied.

###  2.2 Switched system and problem formulation

Consider the following discrete-time linear switched system

$x(k+1)=A_{\sigma(k)}x(k)+B_{\sigma(k)}u(k)+d(k),$ (5)

where the switching signal $\sigma:\mathbb{N}\rightarrow\mathcal{M}$ is a
piecewise constant function of time taking values in the finite set
$\mathcal{M}:=\\{1,2,$ $\cdots,m\\}$, where $m>1$ is the total number of
modes. The matrices $(A_{\sigma(k)},B_{\sigma(k)})$ belong to a collection of
constant matrices $\mathcal{S}:=\\{(A_{i},B_{i}):i\in\mathcal{M}\\}$. Let
$k_{s_{j}}$ denote the time when the $j$-th switching occurs, i.e.,
$k_{s_{j}}=\min\\{k>k_{s_{j-1}}:\sigma(k)\neq\sigma(k_{s_{j-1}})\\}$ with
$j\in\mathbb{N}_{+}$. Without loss of generality, let $k_{s_{0}}=0$. Suppose
that the active mode selected by $k_{s_{j}}$ is indicated by $i$, so it holds
that $i=\sigma(k)$ for all $k\in[k_{s_{j}},k_{s_{j+1}}-1]$.

In this paper, we make the following assumptions.

######  Assumption 2.1 (Unknown system).

The pairs $(A_{i},B_{i})$ for all $i\in\mathcal{M}$, the switching signal
$\sigma$, and the switching instant $k_{s_{j}}$ with $j\in\mathbb{N}_{+}$ are
unknown.

######  Assumption 2.2 (Controllability).

For each $i\in\mathcal{M}$, the pair $(A_{i},B_{i})$ is controllable.

######  Assumption 2.3 (Bounded disturbance).

For all $k\in\mathbb{N}$, it holds that
$d(k)\in\mathbb{B}_{\bar{d}}:=\\{d|\|d\|\leq\bar{d}\\}$ for some known
constant $\bar{d}\geq 0$.

In addition, let us suppose that the controller side possesses a buffer of
size $T\in\mathbb{N}_{+}$. At each time instant $k\in\mathbb{N}$, the buffer
records the latest $T$ input-state samples, which are collected in $U_{k-1}$,
$X_{k-1}$, and $X_{k}$, as follows

$\displaystyle U_{k-1}$ $\displaystyle=[u(k-T)~{}u(k-T+1)~{}\cdots~{}u(k-1)],$
(6a) $\displaystyle X_{k-1}$
$\displaystyle=[x(k-T)~{}x(k-T+1)~{}\cdots~{}x(k-1)],$ (6b) $\displaystyle
X_{k}$ $\displaystyle=[x(k-T+1)~{}x(k-T+2)~{}\cdots~{}x(k)].$ (6c)

Let $D_{k-1}=[d(k-T)~{}d(k-T+1)~{}\cdots~{}d(k-1)]$ denote the disturbance
matrix corresponding to the most recent $T$ input-state samples in $U_{k-1}$,
$X_{k-1}$, and $X_{k}$. Notice that some $k\in[0,T-1]$, the indices of the
samples in (6) are negative, which refers to data obtained offline, as
described in Section 2.1. Specifically, there exists a positive constant
$\bar{w}>0$ such that the input-state data
$[U_{-1}^{\prime}X_{-1}^{\prime}]^{\prime}$ generated from the system (5) by
using a $\bar{w}$-persistently exciting input sequence ${u(-T),\cdots,u(-1)}$
of order $n_{x}+1$, has full row rank, according to Lemma 2.2. To simplify the
analysis, we assume that the samples in $X_{-1}$, $X_{0}$ are generated from
the same subsystem, which implies that the SDP (2.1) is feasible with
$X_{-1}$, $X_{0}$, and $U_{-1}$. At each $k\geq T$, the buffer’s window is
shifted one step forward, which means that the oldest sample (i.e., the first
column of the data matrices in (6)) is removed, and the new sample is added to
the buffer.

In order to stabilize the system (5) under the presence of disturbances and
unknown switching modes, we aim to design a control signal $u(k)$ for the
aforementioned setups. To accomplish this, we propose an online switched
controller composed of an exciting signal $\epsilon(k)$ and a dynamic state
feedback law $K(k)$, as given by

$u(k)\\!=\\!\begin{cases}\epsilon(k),\\!&{\rm if}~{}k\in\mathbb{I}_{E}\\\
K(k)x(k),\\!&{\rm else}\end{cases}$ (7)

Here, the exciting signal $\epsilon(k)$ is appropriately selected from the set
$\epsilon(k)\in\mathcal{B}_{\delta_{\epsilon}}:=\\{\epsilon\,|\,\|\epsilon(k)\|\leq\delta_{\epsilon}\\}$
to ensure the persistency of excitation of the noisy input-state data sequence
collected online. The set $\mathbb{I}_{E}$ is a collection of some event times
governed by a switching law, and $K(k)\in\mathbb{R}^{n_{u}\times n_{x}}$ is a
dynamic state feedback control gain. The design of $\epsilon(k)$, $K(k)$, and
$\mathbb{I}_{E}$ will be discussed later.

Moreover, to reflect the goal of stabilization under the unknown disturbance
$d(k)$ and the exciting signal $\epsilon(k)$, we invoke the input-to-state
practical stability (ISpS). The definition of ISpS treating $d(k)$ as an
unknown input is adapted from Definition 2.2 in Jiang et al. (1994).

######  Definition 2.1 (ISpS Jiang et al. (1994)).

System (5) in closed-loop with a control signal as in (7) is ISpS if, for any
$x(0)\in\mathbb{R}^{n_{x}}$ and measurable essentially bounded $d(k)$ on
$k\in[0,+\infty)$, its solution satisfies

$\|x(k)\|\leq\alpha(\|x(0)\|,k)+\beta(\|d_{k}\|_{\infty})+c_{0},\quad\forall
k\in\mathbb{N}$ (8)

where $\alpha$ is a $\mathcal{KL}$-function, $\beta$ is a
$\mathcal{K}$-function, and $c_{0}$ is some constant111 A function
$\beta:[0,\infty)\rightarrow[0,\infty)$ is said to be of class $\mathcal{K}$
if it is continuous, strictly increasing, and $\beta(0)=0$. A function
$\alpha:[0,\infty)\times[0,\infty)\rightarrow[0,\infty)$ is a
$\mathcal{KL}$-function if $\alpha(\cdot,k)$ is of class $\mathcal{K}$ for
each fixed $k\geq 0$ and $\alpha(s,k)$ decreases to $0$ as
$k\rightarrow\infty$ for any fixed $s\in\mathbb{N}$..

With the preliminaries above, the problem to be addressed is formally stated
as follows.

######  Problem 1.

For the switched system (5) under Assumptions 2.1–2.3, design a data-based
control input of the form in (7) to ensure ISpS of the closed-loop system.

##  3 Robust Data-driven Switched Control

To tackle Problem 1, we draw inspiration from the approach presented in Rotulo
et al. (2022). In this approach, the state feedback control gain $K(k)$ is
constructed based on an online version of the SDP (2.1), which allows for
automatic adaptation of the control input to the switching dynamics.
Specifically, we formulate a robust SDP at each $k\in\mathbb{N}$ using the
noisy data $U_{k-1}$, $X_{k-1}$, and $X_{k}$, as follows

$\displaystyle\min_{(\gamma,Q,P,L,V)}\gamma$ $\displaystyle{\rm subject~{}to}$
$\displaystyle\left\\{\begin{aligned}
&X_{k}QP^{-1}Q^{\prime}X_{k}^{\prime}-P+I\preceq 0\\\ &P\succeq I\\\
&L-U_{k-1}QP^{-1}Q^{\prime}U_{k-1}^{\prime}\succeq 0\\\
&V-QP^{-1}Q^{\prime}\succeq 0\\\ &X_{k-1}Q=P\\\ &{\rm tr}(P)+{\rm
tr}(L)+\alpha{\rm tr}(V)\preceq\gamma\end{aligned}\right.$ (9)

Here, $\alpha>0$ is chosen arbitrarily to balance performance and robustness,
as in Lemma 2.1. Suppose that SDP (3) is feasible at time $k$, and we denote
its optimal solution by $(\gamma^{*}(k),Q^{*}(k),P^{*}(k),L^{*}(k),V^{*}(k))$.
An important condition for the feasibility of SDP (3) at time $k$ is the rank
condition:

${\rm rank}(W_{k-1})=\left[\begin{matrix}U_{k-1}\\\
X_{k-1}\end{matrix}\right]=n_{u}+n_{x}$ (10)

where $n_{u}$ and $n_{x}$ are the dimensions of the input and state vectors,
respectively.

According to Lemma 2.2, this condition can be satisfied if the input sequence
${u(k-T),\cdots,u(k-1)}$ is $\bar{w}$-persistently exciting of order $n_{x}+1$
for some $\bar{w}>0$. Note from (7) that if $u(i)=\epsilon(i)$ for all
$i\in[k-T,k-1]$, the persistency of excitation of the input sequence can be
easily guaranteed. However, using $u(k)=\epsilon(k)$ places the system (5) in
open-loop, which can harm system stability. Moreover, the SDP (3) may not be
feasible when matrices $X_{k-1}$ and $X_{k}$ contain data generated from
different subsystems. Even if it is feasible, the resulting matrix $K(k)$
constructed from its optimal solution may not be stabilizing. This indicates
that there are times when solving SDP (3) is unnecessary. In other words,
using the open-loop control signal $\epsilon(k)$ sparingly may have little
influence on stability at times.

In pursuit of addressing Problem 1, we are faced with three fundamental
questions: i) How can we ensure the feasibility of SDP (3)? ii) When should we
solve SDP (3) and when can we use $\epsilon(k)$ instead? iii) What about the
ISpS of the closed-loop system? In the upcoming sections, we provide a
comprehensive explanation of the designated controller in (7) and answer each
of these questions.

Throughout this paper, we operate under the following two assumptions.

######  Assumption 3.1 (Data length).

The number of samples in (6) satisfies $T\geq 2N-1$, where
$N=(n_{x}+1)n_{u}+n_{x}$ is necessary for the persistency of excitation of
order $n_{x}+1$.

######  Assumption 3.2 (Dwell time).

The dwell time $\tau:=\min_{j\in\mathbb{N}}~{}k_{s_{j+1}}-k_{s_{j}}$ satisfies
$\tau>T$.

It is important to note that Assumption 3.1 has been utilized in Rotulo et al.
(2022) to ensure the feasibility of the SDP (3) in the disturbance-free case.
However, as the complexity of solving (3) scales with $T$, smaller values of
$T$ are often preferred for implementation purposes. Without loss of
generality, we set $T:=2N-1$ in the subsequent analysis.

Moreover, Assumptions 3.1 and 3.2 guarantee that the system switches slowly
enough such that the collected input-state data are generated by at most two
subsystems, and at least $N$ input-state data are generated from the same
subsystem. This simplifies the analysis to the question of ensuring the
feasibility of the SDP (3), which we address in the following subsections.

###  3.1 Robust data-driven controller

The matrix $K(k)$, the signal $\epsilon(k)$, and the switching laws in
$\mathbb{I}_{E}$ determine the activated times of signal $\epsilon(k)$ in (7),
as specified below. To decide when to solve SDP (3) and when to apply
$\epsilon(k)$, an auxiliary function $\mathcal{V}(x(k))=x^{\prime}(k)P(k)x(k)$
is designed in conjunction with the controller in (7). Specifically, for
$k=0$, as mentioned in Section 2.2, ${\rm rank}(W_{-1})=n_{x}+n_{u}$, and
samples in matrices $X_{-1}$, $X_{0}$ are generated from the same subsystem.
Therefore, SDP (3) is feasible, as per Lemma 2.1. Let $P^{*}(0)$ be an optimal
solution of SDP (3) at $k=0$. The initial condition is given by
$P(0)=P^{*}(0)$. For $k\in\mathbb{N}_{+}$, matrices $P(k)$ are the
$P$-solutions of SDP (3), which will be specified later. The confidence about
the feasibility of SDP (3) is measured by function $\mathcal{V}(x(k))$.

We begin by introducing some definitions before proceeding. Fix any small
constant $\delta_{V}>0$, and without loss of generality, assume that the
initial condition $x(0)\notin\ker(P(0))$ and $\mathcal{V}(x(0))>\delta_{V}$.
The set of times $k\in\mathbb{N}$ such that $\mathcal{V}(x(k))\leq\delta_{V}$
is denoted by
$\mathbb{I}_{\delta_{V}}:=\\{k\in\mathbb{N}\,|\,\mathcal{V}(x(k))\leq\delta_{V}\\}$.
For some $\lambda_{0}\in(0,1)$ and all $k\notin\mathbb{I}_{\delta_{V}}$, let
$\\{k_{j}\\}_{j\in\mathbb{N}}$ be the times $k_{j}$ when
$\mathcal{V}(x(k_{j}))>\lambda_{0}\mathcal{V}(x(k_{j}-1))$ and
$\mathcal{V}(x(k_{j}-1))\leq\lambda_{0}\mathcal{V}(x(k_{j}-2))$. Set
$k_{0}=0$. Similarly, let $\\{k^{j}\\}_{j\in\mathbb{N}}$ be the times $k^{j}$
when $\mathcal{V}(x(k^{j}-1))>\lambda_{0}\mathcal{V}(x(k^{j}-2))$ and
$\mathcal{V}(x(k^{j}))\leq\lambda_{0}\mathcal{V}(x(k^{j}-1))$. Assume that the
elements in $\\{k_{j}\\}_{j\in\mathbb{N}}$ and $\\{k^{j}\\}_{j\in\mathbb{N}}$
are ordered chronologically, adhering to
$0=k_{0}<k_{0}+N<k^{0}<k_{1}<k_{1}+N<k^{1}<\cdots$. This ordering will be
proved in the next subsection.

Expanding on the definition above, we provide a specification for the
controller (7) as follows

$u(k)\\!=\\!\begin{cases}\epsilon(k),\\!\\!&\\!\\!{\rm
if}~{}k\in[k_{j},k_{j}+N-1]\\\ K(k)x(k),\\!\\!&\\!\\!{\rm else}\end{cases}$
(11)

where $\epsilon(k)$ is chosen within the ball
$\mathcal{B}_{\delta_{\epsilon}}:=\\{\epsilon\,|\,\|\epsilon(k)\|\leq\delta_{\epsilon}\\}$,
and the controller gain $K(k)$ is set to

$K(k)=U_{k-1}Q(k)P(k)^{-1}.$ (12)

Furthermore, $P(k)$ and $Q(k)$ are determined by

$\displaystyle(P(k),Q(k))=$
$\displaystyle\qquad\begin{cases}(P^{*}(k),Q^{*}(k))~{}{\rm
from}~{}\eqref{eq:sdp_noise},\\!\\!\\!&{\rm if}~{}k\in[k_{j}+N,k^{j}]\\\
(P(k-1),Q(k-1)),\\!\\!\\!&{\rm else}.\end{cases}$ (13)

To implement the controller given by (11)–(3.1), we must solve the SDP given
by (3) for all $k\in[k_{j}+N,k^{j}]$ for each $j\in\mathbb{N}$. The
feasibility of this SDP is established in the next subsection.

###  3.2 Feasibility of SDP (3)

Based on the analysis presented in the previous section, the feasibility of
SDP (3) at time $k$ depends on the rank condition (10) and the samples in data
matrices $X_{k-1}$ and $X_{k}$. Note that for
$k\in[k_{s_{j}}+1,k_{s_{j}}+T-1]$, the data matrices contain a mixture of
samples from subsystems $\sigma(k_{s_{j}})$ and $\sigma(k_{s_{j-1}})$, as
indicated by Assumptions 3.1 and 3.2. Therefore, even if condition (10) is
satisfied, SDP (3) may be infeasible.

To address this issue, this subsection conducts the feasibility analysis of
SDP (3) in two steps. First, we show that if condition (10) is satisfied at
time $k$, then SDP (3) is always feasible, provided that $\bar{d}$ is small
enough. Moreover, if $k\in[k_{s_{j}}+T,k_{s_{j+1}}]$, then the resultant
matrix $K(k)$ in (12) is stabilizing. Second, we derive conditions on
$\epsilon(k)$ and $\bar{d}$ that guarantee (10) always holds at times
$k\in[k_{j}+N,k_{j}]$, thus confirming the feasibility of the controller
(11)–(3.1).

In light of Assumptions 3.1 and 3.2, the time interval
$[k_{s_{j}}+1,k_{s_{j+1}}]$ is partitioned into two sub-intervals:
$[k_{s_{j}}+1,k_{s_{j}}+T-1]$ and $[k_{s_{j}}+T,k_{s_{j+1}}]$. For
$k\in[k_{s_{j}}+1,k_{s_{j}}+T-1]$, the data matrices $X_{k-1}$ and $X_{k}$
comprise a mixture of states from subsystems $\sigma(k_{s_{j}})$ and
$\sigma(k_{s_{j-1}})$. In contrast, data matrices in the second sub-interval
collect only samples generated from subsystem $\sigma(k_{s_{j}})$. The
feasibility analysis of SDP (3) proceeds by examining these two sub-intervals.

Before stating the main results, the disturbance-free version of SDP (3),
which was used in Rotulo et al. (2022), is introduced below:

$\displaystyle\min_{(\gamma,Q,P,L)}\gamma$ $\displaystyle{\rm subject~{}to}$
$\displaystyle\left\\{\begin{aligned}
&(X_{k}\\!-\\!D_{k-1})QP^{-1}Q^{\prime}(X_{k}\\!-\\!D_{k-1})^{\prime}\\!-\\!P\\!+\\!I\preceq
0\\\ &P\succeq I\\\ &L-U_{k-1}QP^{-1}Q^{\prime}U_{k-1}^{\prime}\succeq 0\\\
&X_{k-1}Q=P\\\ &{\rm tr}(P)+{\rm tr}(L)\preceq\gamma.\end{aligned}\right.$
(14)

The optimal solutions of this SDP above will be utilized to construct feasible
solutions for SDP (3).

With this in mind, the following lemmas demonstrate the feasibility of SDP (3)
for $k\in[k_{s_{j}}+1,k_{s_{j}}+T-1]$ and $k\in[k_{s_{j}}+T,k_{s_{j+1}}]$,
respectively. Their proofs can be found in Appendices Appendix A.2 and
Appendix A.3.

######  Lemma 3.1.

Under Assumptions 2.1–3.2, let $X_{k}$, $X_{k-1}$, and $U_{k-1}$ be collected
from system (5). Let $i\in\mathcal{M}$ denote the subsystem selected by
$\sigma(k_{s_{j}})$, i.e., $\sigma(k_{s_{j}})=i$. Consider SDP (3) with any
$\alpha>0$. For every constant $\bar{w}\geq 0$, there exists a constant
$\delta_{d,1}\geq 0$ such that if the following conditions hold: i) the input
sequence ${u(k-T),\ldots,u(k-1)}$ is $\bar{w}$-persistently exciting of order
$n_{x}+1$, ii) $k\in[k_{s_{j}}+T,k_{s_{j+1}}]$, and iii)
$\bar{d}<\delta_{d,1}$, then condition (10) holds and SDP (3) is feasible. Let
$(\gamma^{*}(k),Q^{*}(k),P^{*}(k),L^{*}(k),V^{*}(k))$ be an optimal solution
of SDP (3). The matrix $K^{*}(k)=U_{k-1}Q_{i}^{*}(k)P_{i}^{*}(k)^{-1}$ is such
that $A_{i}+B_{i}K^{*}(k)$ is Schur stable.

######  Lemma 3.2.

Under Assumptions 2.1–3.2, let $X_{k}$, $X_{k-1}$, and $U_{k-1}$ be collected
from system (5). Consider SDP (3) with any $\alpha>0$. For constants $\bar{w}$
and $\delta_{d,1}$ specified in Lemma 3.1, if the following conditions hold:
i) the input sequence ${u(k-T),\ldots,u(k-1)}$ is $\bar{w}$-persistently
exciting of order $n_{x}+1$, ii) $k\in[k_{s_{j}}+1,k_{s_{j}}+T-1]$, and iii)
$\bar{d}<\delta_{d,1}$, then condition (10) holds and SDP (3) is feasible.

Lemmas 3.1 and 3.2 are based on the assumption that the input sequence
${u(k-T),\cdots,u(k-1)}$ is $\bar{w}$-persistently exciting of order $n_{x}+1$
at time instant $k$. The following two lemmas present conditions on
$\epsilon(k)$ and $\bar{d}$, which ensure that the persistency of excitation
holds for all $k\in[k_{j}+N,k^{j}]$ with $j\in\mathbb{N}$. The proofs are
provided in Appendices Appendix A.4 and Appendix A.5.

######  Lemma 3.3.

For any $j\in\mathbb{N}$ and $\delta_{\epsilon}>0$, there exists a constant
$\bar{w}>0$ such that the sequence
${\epsilon(k_{j}),\cdots,\epsilon(k_{j}+N-1)}$ can be designed to be
$\bar{w}$-persistently exciting of order $n_{x}+1$, with
$\epsilon(k)\in\mathbb{B}_{\delta_{\epsilon}}$ for all
$k\in[k_{j},k_{j}+N-1]$. If $\bar{d}<\delta_{d,1}$, where $\delta_{d,1}$ is
given in Lemma 3.1, then the SDP (3) is feasible for all
$k\in[k_{j}+N,k_{j}+T]$.

We can observe from (3.1) that the SDP (3) should be feasible for
$k\in[k_{j}+N,k^{j}]$. In light of Lemma 2.2, this can be ensured if
$k_{j}+T\geq k^{j}$. To show this, we first construct a set of Lyapunov
candidates. According to Assumption 2.2, for each subsystem $i\in\mathcal{M}$,
there exist constant $\beta_{i}>0$ and matrices $P_{i}$, $K_{i}$ such that
$\mathcal{A}_{i}^{\prime}P_{i}\mathcal{A}_{i}-P_{i}=-\beta_{i}I$ holds with
$\mathcal{A}_{i}=A_{i}+B_{i}K_{i}$. Consider Lyapunov candidate functions
$\\{\mathcal{W}_{i}(x(k))=x(k)^{\prime}P_{i}x(k)\,|\,i\in\mathcal{M}\\}$, each
obeying

$\underline{\lambda}_{P}\|x(k)\|^{2}\leq\mathcal{W}_{i}(x(k))\leq\bar{\lambda}_{P}\|x(k)\|^{2}$
(15)

where
$\underline{\lambda}_{P}=\min_{i\in\mathcal{M}}\underline{\lambda}_{P_{i}}$
and $\bar{\lambda}_{P}=\max_{i\in\mathcal{M}}\bar{\lambda}_{P_{i}}$ denote the
minimum and maximum eigenvalues of $P_{i}$. Utilizing the Lyapunov functions
above, the following lemma reveals the relationships between $k_{j}+T$,
$k^{j}$, and the switching time $k_{s_{j}}$.

######  Lemma 3.4.

There exist constants $\delta_{d,2}$ and
$0<\check{\lambda}_{0}<\hat{\lambda}_{0}\leq 1$ such that for all
$\bar{d}<\delta_{d,2}$ and
$\lambda_{0}\in[\check{\lambda}_{0},\hat{\lambda}_{0})$, the following facts
hold for all $j\in\mathbb{N}$: i) $k_{j}=k_{s_{j}}$, and ii)
$k^{j}=k_{s_{j}}+T$. Moreover, let $i\in\mathcal{M}$ denote the subsystem
selected by $\sigma(k_{s_{j}})$. For all $k\in[k_{s_{j}}+T,k_{s_{j}+1}-1]$, it
holds that $\mathcal{W}_{i}(x(k+1))\leq\lambda_{0}\mathcal{W}_{i}(x(k))$.

The above lemmas provide solutions for questions i) to iii). For simplicity,
we consider the worst case scenario where any matrix $K_{i}$ that stabilizes
subsystem $(A_{i},B_{i})$ cannot stabilize subsystems $(A_{j},B_{j})$ for all
$j\neq i\in\mathcal{M}$. We will derive conditions for the ISpS.

###  3.3 Stability analysis

This section investigates the stability of switched system (5) under the
controller strategy (11)–(3.1). Noticing from Lemma 3.2 that $K(k)$ is not
stabilizing for $k\in[k_{s_{j}},k_{s_{j}}+T-1]$, our analysis is carried out
in two steps. First, we establish the uniform boundedness of matrix $K(k)$ in
(12) for all $k\notin[k_{j},k_{j}+N-1]$, where $j\in\mathbb{N}$. Using the
fact that $\epsilon(k)\in\mathbb{B}_{\delta_{\epsilon}}$ holds for all
$k\in[k_{j},k_{j}+N-1]$, it can be deduced that the state remains bounded for
all $k\in[k_{s_{j}},k_{s_{j}}+T-1]$. Building on these findings and recalling
Lemma 3.4, we prove the ISpS provided that the disturbance is upper bounded
and the dynamics switches slowly enough.

We use Lemmas 3.1–3.4 to derive an upper bound on $\|K(k)\|$ as follows.

######  Theorem 3.1.

Consider the switched system (5) subject to Assumptions 2.1–3.2. If
$\bar{d}<\delta_{d,2}$, and
$\lambda_{0}\in[\check{\lambda}_{0},\hat{\lambda}_{0})$, where $\delta_{d,2}$,
$\check{\lambda}_{0}$, and $\hat{\lambda}_{0}$ are as defined in Lemma 3.4,
then there exists a constant $\delta_{K}>0$ such that matrix $K(k)$ in (12)
satisfies $\|K(k)\|\leq\delta_{K}$ for all $k\notin[k_{j},k_{j}+N-1]$, where
$j\in\mathbb{N}$.

Suppose that the SDPs (3) and (3.2) are feasible at time $k$. The proof of
Lemmas 3.1 and 3.2 has shown that there exists constant $\eta_{2}\geq 1$ such
that a feasible solution of SDP (3) can be constructed as
$\eta_{2}(\bar{\gamma}(k),\bar{Q}(k),\bar{P}(k),\bar{L}(k),\bar{Q}(k)\bar{P}(k)^{-1}\bar{Q}(k)^{\prime})$,
for $\bar{d}<\delta_{d,2}$ with $\delta_{d,2}$ defined in Lemma 3.4. Here,
$(\bar{\gamma}(k),\bar{Q}(k),\bar{P}(k),\bar{L}(k))$ is any optimal solution
of SDP (3.2). Let $(\gamma^{*}(k),Q^{*}(k),P^{*}(k),L^{*}(k),V^{*}(k))$ denote
any optimal solution of SDP (3). Then, it follows that

${\rm tr}(P^{*}(k))+{\rm tr}(L^{*}(k))+\alpha{\rm
tr}(V^{*}(k))\leq\eta_{2}\bar{\gamma}(k)$

where $\bar{\gamma}(k)=\alpha{\rm
tr}(\bar{Q}(k)\bar{P}(k)^{-1}\bar{Q}(k)^{\prime})+{\rm tr}(\bar{P}(k))+{\rm
tr}(\bar{L}(k))$. From the second constraint of SDP (3), it can be observed
that $P^{*}(k)\succeq I$. Hence, ${\rm tr}(L^{*}(k))={\rm
tr}(K^{*}(k)P^{*}(k)K^{*}(k)^{\prime})\geq{\rm
tr}(K^{*}(k)K^{*}(k)^{\prime})\geq\|K^{*}(k)\|^{2}$. Thus, it follows that

$\|K^{*}(k)\|\leq\sqrt{\eta_{2}\bar{\gamma}(k)-n_{x}}.$ (16)

Here, since $k$ can be infinitely large, the inequality (16) can result in an
infinite number of upper bounds on $\|K^{*}(k)\|$. In the following, we
demonstrate that by taking the modes of subsystems into account, there exist a
finite number of $\bar{\gamma}(k)$. As a consequence, $\|K^{*}(k)\|$ can be
upper bounded by using the maximum one.

Let us consider any $j\in\mathbb{N}$ and $k=k^{j}=k_{s_{j}}+T$. The samples in
matrices $X_{k-1}$ and $X_{k}$ are generated from the same subsystem denoted
by $i$, where $\sigma(k_{s_{j}})=i\in\mathcal{M}$. We denote the unique LQR
control gain of system $(A_{i},B_{i})$ by $\bar{K}_{i}$, the controllability
Gramian matrix by $\bar{P}_{i}$, and $\bar{\gamma}_{i}:={\rm
tr}(\bar{P}_{i})+{\rm tr}(\bar{K}_{i}\bar{P}_{i}\bar{K}_{i}^{\prime})$ the
associated cost. According to (Rotulo et al., 2022, Lemma 4), the SDP (3.2) is
feasible and the matrix $\bar{K}(k)=U_{k-1}\bar{Q}(k)\bar{P}(k)^{-1}$, where
$(\bar{Q}(k),\bar{P}(k))$ is any optimal solution of SDP (3.2), is such that
$\bar{K}(k)=\bar{K}_{i}$, $\bar{P}(k)=\bar{P}_{i}$, and
$\bar{\gamma}(k)=\bar{\gamma}_{i}$. Moreover, observing from (12), (3.1), and
Lemma 3.4 that $K(k)=K(k-1)$ holds for all $k\in[k_{s_{j}}+T+1,k_{s_{j+1}}]$,
we obtain from (16) that for
$k\in\cup_{j\in\mathbb{N}}[k_{s_{j}}+T,k_{s_{j+1}}]$,

$\|K(k)\|\leq\max_{i\in\mathcal{M}}\sqrt{\eta_{2}\bar{\gamma}_{i}-n_{x}}.$
(17)

Lemma 3.4 confirms that $k_{j}=k_{s_{j}}$. According to (12), for any
$j\in\mathbb{N}$ and $k\in[k_{s_{j}}+N,k_{s_{j}}+T-1]$, it has been shown in
(Rotulo et al., 2022, Lemma 5) that if there are at least $N$ data generated
from subsystem $\sigma(k_{s_{j}}-1)=z$, then
$(\bar{\gamma}_{z},\bar{Q}_{z},\bar{P}_{z},\bar{K}_{z}\bar{P}_{z}\bar{K}_{z}^{\prime})$
is a feasible solution of SDP (3.2) with

$\bar{Q}_{z}=W_{k-1}^{\dagger}\left[\begin{matrix}\bar{K}_{z}\\\
I\end{matrix}\right]\bar{P}_{z}.$

Alternatively, if there are at least $N$ samples from subsystem
$\sigma(k_{s_{j}})=i$, then
$(\bar{\gamma}_{i},\bar{Q}_{i},\bar{P}_{i},\bar{K}_{i}\bar{P}_{i}\bar{K}_{i}^{\prime})$
is a feasible solution of SDP (3.2). Based on Lemma 3.2, it can be concluded
that inequality (17) is valid for all $k$ within the intervals
$\cup_{j\in\mathbb{N}}[k_{s_{j}}+N,k_{s_{j}}+T-1]$.

Therefore, for all $k$ outside the range of $[k_{j},k_{j}+N-1]$, it can be
established that $\|K(k)\|$ is less than or equal to $\delta_{K}$. Here,
$\delta_{K}$ can be defined as
$\delta_{K}\triangleq\max_{i\in\mathcal{M}}\sqrt{\eta_{2}\bar{\gamma}_{i}-n_{x}}$.
This inequality completes the proof.

The following stability result for system (5) is established.

######  Theorem 3.2.

Consider the switched system (5) with controller (11)–(3.1). Let Assumptions
2.1–3.2 hold. For all $k\in\mathbb{N}$, if $\bar{d}<\delta_{d,2}$, and
$\lambda_{0}\in[\check{\lambda}_{0},\hat{\lambda}_{0})$ with $\delta_{d,2}$,
$\check{\lambda}_{0}$, and $\hat{\lambda}_{0}$ as in Lemma 3.4, then there
exist constants $\bar{\delta}_{\epsilon}>\underline{\delta}_{\epsilon}>0$,
$\bar{\delta}_{x}>\underline{\delta}_{x}>0$, and $\bar{\tau}>0$ such that, for
all
$\delta_{\epsilon}\in(\underline{\delta}_{\epsilon},\bar{\delta}_{\epsilon})$,
$\delta_{x}\in(\underline{\delta}_{x},\bar{\delta}_{x})$, and
$\tau\geq\bar{\tau}$, the closed-loop system achieves ISpS.

From (11)–(3.1), we observe that the controller gain matrix $K(k)$ stops
updating when function $\mathcal{V}(x(k))$ is small, i.e.,
$\mathcal{V}(x(k))\leq{\delta}_{V}$. Since
$\|P(k)\|\leq\max_{i\in\mathcal{M}}\eta_{2}\bar{\gamma}_{i}$ as shown in the
proof of Lemma 3.4, this condition is equivalent to $\|x(k)\|\leq\delta_{x}$
for some small $\delta_{x}>0$. Consider the time instant $k_{in}$ when the
state enters the range $\mathbb{B}_{\delta_{x}}$ for the first time, where
$\mathbb{B}_{\delta_{x}}$ denotes a ball centered at the origin with radius
$\delta_{x}$. Formally,
$k_{in}:=\min\\{k\in\mathbb{N}\,|\,x(k)\in\mathbb{B}_{\delta_{x}}\\}$.

The proof is divided into two steps with respect to $k_{in}$. First, we
demonstrate that the state converges for all $k\in[0,k_{in})$. Then, we show
that for all $k\geq k_{in}$, the state either converges to zero, or escapes
from $\mathbb{B}_{\delta_{x}}$ and returns within finite time.

_Step 1: State convergence before entering $\mathbb{B}_{\delta_{x}}$._

Assuming that $\forall k\leq T$, $\mathcal{V}(x(k))>\delta_{V}$, and
$x(k)\notin\mathbb{B}_{\delta_{x}}$, we consider an arbitrary time interval
$[k_{s_{j}},k_{s_{j}+1}-1]\subseteq[0,k_{in})$. Let $i=\sigma(k_{s_{j}})$
denote the activated subsystem for $k\in[k_{s_{j}},k_{s_{j}+1}-1]$. We now
show that the system achieves stability for all
$k\in[k_{s_{j}}+T,k_{s_{j+1}}-1]$. According to Lemmas 3.1 and 3.4, it holds
for $\lambda_{0}\in[\check{\lambda}_{0},\hat{\lambda}_{0})$ as in Lemma 3.4,
that $\mathcal{W}_{i}(x(k+1))\leq\lambda_{0}\mathcal{W}_{i}(x(k))$, and
recursively

${\mathcal{W}_{i}(x(k))\leq\lambda_{0}^{k-k_{s_{j}}-T}\mathcal{W}_{i}(x(k_{s_{j}}+T))}$

then

$\|x(k)\|\leq\phi_{0}\tilde{\lambda}_{0}^{k-k_{s_{j}}-T}\|x(k_{s_{j}}+T)\|$
(18)

where
$\phi_{0}:=[(\max_{i\in\mathcal{M}}\eta_{2}\bar{\gamma}_{i})/{\underline{\lambda}_{P}}]^{1/2}$
and $\tilde{\lambda}_{0}:=\sqrt{\lambda_{0}}$.

Based on (11) and Theorem 3.1, the growth of states between two consecutive
times can be upper bounded as follows

$\displaystyle\|x(k+1)\|$
$\displaystyle\leq\max\\!\Big{\\{}\|A_{\sigma(k)}+B_{\sigma(k)}K(k)\|\|x(k)\|+\bar{d},$
$\displaystyle\quad\|A_{\sigma(k)}\|\|x(k)\|+\|B_{\sigma(k)}\|\delta_{\epsilon}+\bar{d}\Big{\\}}$
(19) $\displaystyle\leq C\|x(k)\|+\bar{d}$ (20)

where $C:=\max\\{C_{0},C_{1},1\\}$ with
$C_{0}:=\max_{i\in\mathcal{M}}(\|A_{i}\|+\|B_{i}\|\delta_{K})$ and
$C_{1}:=\max_{i\in\mathcal{M}}(\|A_{i}\|+\|B_{i}\|\delta_{\epsilon}/\delta_{x})$.

Noting inequality (20) holds for all $k\in[k_{s_{j}}+1,k_{s_{j}}+T]$.
Combining this fact with (18), we obtain the following inequality for all
$t\in[1,T]$

$\displaystyle\|x(k_{s_{j}}+t)\|$ $\displaystyle\leq
C^{t}\phi_{0}^{s}\tilde{\lambda}_{0}^{k_{s_{j}}-k_{0}}\Big{(}\frac{C}{\lambda_{0}}\Big{)}^{sT}\|x(k_{0})\|+\bar{d}\sum_{i=0}^{t-1}C^{i}$
$\displaystyle~{}~{}~{}+\frac{C^{t+T}}{C-1}\bar{d}\sum_{i=1}^{j}\phi_{0}^{i}\tilde{\lambda}_{0}^{k_{s_{j}}-k_{s_{j}-i}}\Big{(}\frac{C}{\tilde{\lambda}_{0}}\Big{)}^{iT}.$
(21)

For $t\in[T+1,k_{s_{j+1}}-k_{s_{j}}]$, we have that

$\displaystyle\|x(k_{s_{j}}+t)\|\leq\tilde{\lambda}_{0}^{t}\phi_{0}^{s+1}\tilde{\lambda}_{0}^{k_{s_{j}}-k_{0}}\Big{(}\frac{C}{\tilde{\lambda}_{0}}\Big{)}^{(s+1)T}\|x(k_{0})\|$
$\displaystyle\qquad\qquad~{}+\frac{\lambda_{0}^{t}C^{T}}{C-1}\bar{d}\sum_{i=0}^{j}\phi_{0}^{i+1}\tilde{\lambda}_{0}^{k_{s_{j}}-k_{s_{j}-i}}\Big{(}\frac{C}{\tilde{\lambda}_{0}}\Big{)}^{(i+1)T}.$
(22)

Choose constant $\mu$ such that $0<\tilde{\lambda}_{0}<\mu<1$. Let
$\tau>\bar{\tau}$ with

$\bar{\tau}:=\frac{\ln({C}/{\tilde{\lambda}_{0}})}{\ln(\phi_{0}({\mu}/{\tilde{\lambda}_{0}})^{T})}.$
(23)

Hence, for all $t\in[1,T]$, it follows that

$\displaystyle
C^{t}\phi_{0}^{s}\tilde{\lambda}_{0}^{k_{s_{j}}-k_{s_{0}}}\Big{(}\\!\frac{C}{\lambda_{0}}\Big{)}^{jT}\\!\\!$
$\displaystyle=$
$\displaystyle~{}\mu^{t\\!+\\!k_{s_{j}}\\!-k_{0}}\\!\Big{(}\frac{C}{\mu}\Big{)}^{t}\\!\Big{(}\frac{\tilde{\lambda}_{0}}{\mu}\Big{)}^{k_{s_{j}}-k_{s_{0}}}\\!\phi_{0}^{j}\Big{(}\frac{C}{\tilde{\lambda}_{0}}\Big{)}^{jT}$
$\displaystyle\\!\\!\leq$
$\displaystyle~{}\mu^{t+k_{s_{j}}-k_{s_{0}}}\Big{(}\frac{C}{\mu}\Big{)}^{T}\Big{(}\frac{\tilde{\lambda}_{0}}{\mu}\Big{)}^{j\tau}\phi_{0}^{j}\Big{(}\frac{C}{\tilde{\lambda}_{0}}\Big{)}^{jT}$
$\displaystyle\leq$
$\displaystyle~{}\phi_{0}\Big{(}\frac{C}{\mu}\Big{)}^{T}\mu^{t+k_{s_{j}}-k_{s_{0}}}.$

Similarly, for all $t\in[T+1,k_{s_{j+1}}-k_{s_{j}}]$, it holds that

$\displaystyle\tilde{\lambda}_{0}^{t}\phi_{0}^{j+1}\tilde{\lambda}_{0}^{k_{s_{j}}\\!\\!-k_{s_{0}}}\\!\Big{(}\frac{C}{\tilde{\lambda}_{0}}\Big{)}^{\\!(j+1)T}\\!\\!\\!\\!$
$\displaystyle\leq$
$\displaystyle~{}\mu^{k_{s_{j}}\\!+t-k_{s_{0}}}\Big{(}\frac{\tilde{\lambda}_{0}}{\mu}\Big{)}^{\\!j\tau}\\!\\!\\!\phi_{0}^{j+1}\Big{(}\frac{C}{\tilde{\lambda}_{0}}\Big{)}^{\\!(j+1)T}$
$\displaystyle\leq$
$\displaystyle~{}\phi_{0}\Big{(}\frac{C}{\mu}\Big{)}^{T}\mu^{k_{s_{j}}+t-k_{s_{0}}}.$

Therefore, it can be deduced from (21) and (22) that

$\displaystyle\|x(k_{s_{j}}+t))\|\leq\phi_{0}\Big{(}\frac{C}{\mu}\Big{)}^{T}\mu^{t+k_{s_{j}}-k_{s_{0}}}\|x(k_{s_{0}})\|+\varphi(\bar{d})$

where $\varphi(\cdot)\in\mathcal{K}$, which implies that for sufficiently
small $\bar{d}$, the state converges. Hence, for $\delta_{x}>0$, there exists
a time $0<k_{in}<\infty$ such that $x(k_{in})\in\mathbb{B}_{\delta_{x}}$.

_Step 2: State convergence after first entering $\mathbb{B}_{\delta_{x}}$_.

Suppose that for some switching time $k_{s^{\prime}_{j}}\in\mathbb{N}$ and
$k_{in}\in[k_{s_{j}^{\prime}}+T,k_{s_{j}^{\prime}+1}]$ for some
$k_{s_{j}^{\prime}}$, the mode of the switched system remains unchanged, i.e.,
$j=\sigma(k)=\sigma(k_{in})$ for all $k\geq k_{in}$. Noticing that the
disturbance satisfies $\bar{d}<\delta_{d,2}$, the Lyapunov function decreases
with at least rate $\lambda_{0}$. Therefore,
$\mathcal{W}_{j}(x(k))\leq\lambda_{0}\mathcal{W}_{i}(x(k-1))$ for $k\geq
k_{in}$. Recursively, the state satisfies
$\|x(k)\|\leq\phi_{0}\tilde{\lambda}_{0}^{(k-k_{in})/2}\delta_{x}$, indicating
that the state converges to the origin as $k\rightarrow\infty$.

Next, we show that if the state escapes from the range
$\mathbb{B}_{\delta_{x}}$, it will come back within finite times. For
$k_{in}\in[k_{s_{j}^{\prime}}+1,k_{s_{j}^{\prime}}+T-1]$, and
$\sigma(k_{s_{j}^{\prime}+1})\neq\sigma(k_{s_{j}^{\prime}})$, from (20), the
state satisfies $\|x(k_{in}+1)\|=C\|x(k_{in})\|+\bar{d}$ with $C\geq 1$. This
implies that the state may diverge. Therefore, there exists a time
$k_{out}\in(k_{in},\infty)$ such that
$x(k_{out})\notin\mathbb{B}_{{\delta}_{x}}$, and the state feedback controller
gain $K(k_{out})$ continues to be updated. It follows from Step 1 that the
state gets back to this range within finite times.

In conclusion, the state satisfies

$|x(k_{s_{j}}+t)|\leq\phi_{0}\Big{(}\frac{C}{\mu}\Big{)}^{T}\mu^{t+k_{s_{j}}-k_{s_{0}}}\|x(k_{s_{0}})\|+\varphi(\bar{d})+\delta_{x},$

which completes the proof according to Definition 2.1.

It is worth remarking that the disturbance-free data-driven controller
proposed in Rotulo et al. (2022) cannot be directly extended to handle cases
with disturbances, as we show below.

######  Remark 3.1 (Relative to Rotulo et al. (2022)).

In the absence of disturbances, the controller in Rotulo et al. (2022)
utilizes a perturbed feedback control law given by
$u(k)=K(k)x(k)+\bar{\epsilon}(k)\|x(k)\|$, where $K(k)x(k)$ is added to an
auxiliary signal $\bar{\epsilon}(k)\|x(k)\|$ with $\bar{\epsilon}(k)$ selected
within the ball
$\mathbb{B}_{\delta_{\bar{\epsilon}}}:=\\{\bar{\epsilon}\in\mathbb{R}^{n_{u}}\,|\,\|\epsilon\|\leq\delta_{\bar{\epsilon}}\\}$.
It has been proven in (De Persis and Tesi, 2021, Lemma 3) that the rank
condition (10) can be guaranteed at every time $k$ by the auxiliary signal
$\bar{\epsilon}(k)$, ensuring the feasibility of SDP (3.2) for all
$k\in\mathbb{N}$. Building on this fact, the disturbance-free system can be
stabilized with the perturbed controller by taking
$K(k)=U_{k-1}\bar{Q}(k)\bar{P}(k)^{-1}$, where $(\bar{P}(k),\bar{Q}(k))$ are
optimal solutions of SDP (3.2) at each time $k$. Intuitively, one might be
tempted to directly apply this controller when disturbances are present. In
other words, if $\bar{\epsilon}(k)$ can guarantee the rank condition (10) for
any $k$ with $d(k)\neq 0$, then we can adopt the controller and update $K(k)$
by solving SDP (3). However, as we illustrate through an example, the
condition (10) can be violated when nonzero disturbance $d(k)$ is involved.

Consider the case where $(A_{\sigma(k)},B_{\sigma(k)})=(1,1/2)$ for all
$k\in[0,4]$ in (5) and $x(0)=1$. Under the assumption that
$T=2((n_{x}+1)n_{u}+n_{x})-1=5$ (cf. Assumption 3.1), let us take
$\bar{\epsilon}(k)=-1/2$ and $K(k)=-1$ for all $k\in[0,4]$ for simplicity. In
the absence of disturbances, i.e., $d(0)=d(1)=d(2)=d(3)=0$, we can obtain the
input-state sequence $({x}(0),u(0))=(1,-3/2)$, $({x}(1),u(1))=(1/4,-3/4)$,
$({x}(2),u(2))=(-1/8,1/16)$, $({x}(3),u(3))=(-3/32,3/64)$,
$({x}(4),u(4))=(-9/128,9/256)$. As a result, the matrix $W_{5}$ defined by

$W_{5}=\left[\begin{matrix}U_{5}\\\
X_{5}\end{matrix}\right]=\left[\begin{matrix}-{3}/{2}\\!&-{3}/{4}\\!&{1}/{16}\\!&-{3}/{64}\\!&-{9}/{256}\\\
1\\!&{1}/{4}\\!&-{1}/{8}\\!&-{3}/{32}\\!&-{9}/{128}\end{matrix}\right]$

has full row rank, and hence condition (10) is satisfied.

However, when the disturbance sequence is nonzero, with e.g., $d(0)=3/4$,
$d(1)=(1/8)$, $d(2)=5/48$, and $d(3)=1/12$, while retaining the same initial
condition $x(0)=1$ and variables $\epsilon(k)=-1/2$, $K(k)=-1$ for all
$k\in[0,4]$, the input-state sequence becomes $({x}(0),u(0))=(1,-3/2)$,
$({x}(1),u(1))=(1/2,-3/4)$, $({x}(2),u(2))=(1/4,-3/8)$,
$({x}(3),u(3))=(1/6,-1/4)$, and $({x}(4),u(4))=(1/8,-3/16)$. In this case, the
matrix $W_{5}$ given by

$W_{5}=\left[\begin{matrix}U_{5}\\\
X_{5}\end{matrix}\right]=\left[\begin{matrix}-{3}/{2}\\!&-{3}/{4}\\!&-{3}/{8}\\!&-{1}/{4}\\!&-{3}/{16}\\\
1\\!&{1}/{2}\\!&{1}/{4}\\!&{1}/{6}\\!&{1}/{8}\end{matrix}\right]$

is such that ${\rm rank}(W_{5})=1$, violating condition (10). Therefore, SDP
(3) is not feasible.

##  4 Numerical Examples

In this section, we examine two numerical examples to validate the
effectiveness of the proposed controller, both of which have been previously
considered in Rotulo et al. (2022) under the disturbance-free case.

###  4.1 Flight control system

The first example considers the stabilization problem of the linearized
longitudinal dynamics of an F-18 aircraft operating at Mach $0.3$ and altitude
$26$ kft, and Mach $0.7$ and altitude $14$ kft, respectively. This problem can
be described using two linear subsystems $(A_{1},B_{1})$ and $(A_{2},B_{2})$.
Both subsystems are controllable and have the following matrices:

$\displaystyle A_{1}$ $\displaystyle=\left[\begin{matrix}0.977&0.097\\\
0.002&0.981\end{matrix}\right],~{}~{}~{}B_{1}=\left[\begin{matrix}-0.013&-0.004\\\
-0.171&-0.051\end{matrix}\right],$ $\displaystyle A_{2}$
$\displaystyle=\left[\begin{matrix}0.852&0.088\\\
-0.753&0.87\end{matrix}\right],~{}B_{2}=\left[\begin{matrix}-0.106&-0.021\\\
-1.8143&-0.358\end{matrix}\right].$

We set $T=15$, $\bar{d}=0.03$, $\delta_{V}=0.05$, $\alpha=1$, and
$\lambda_{0}=0.945$. We generate an arbitrary switching signal $\sigma$ with
dwell time $\tau\geq 15$. We first collect an offline input-state trajectory
by applying a sequence of inputs $u$ uniformly distributed in $[-0.3,0.3]$ to
the subsystem $(A_{1},B_{1})$ only. Additionally, we prepare an
$\mu$-persistently exciting input sequence $\\{u(k)\\}_{k=0}^{N-1}$ of order
$3$ with $\mu=0.01$. We then run the system online using the proposed control
strategy (11)–(3.1).

The top panel of Fig. 1 depicts the convergence of the state trajectory over a
simulation horizon of $200$ time instants. The offline data-collection phase
is shown in the interval $t\in[0,15]$. The bottom panel of Fig. 1 shows the
evolution of the Lyapunov function and the smallest singular value of the
matrix $W_{k-1}$. The gray shades represent the phase when the auxiliary
function $\mathcal{V}(x(k))$ converges with rate $\lambda_{0}$, indicating the
convergence of the Lyapunov function $\mathcal{W}(x(k))$. The orange shades
indicate the phase when $\mathcal{V}(x(k))\leq\delta_{V}$. We observe that at
$k=76$, function $\mathcal{V}(x(k))$ first enters the range
$\mathbb{B}_{\delta_{V}}$ (state $x(k)$ entering the range
$\mathbb{B}_{\delta_{x}}$). However, the system switches at $k=80$, causing
the state to escape from this range. Nonetheless, the proposed control
strategy (11)–(3.1) eventually causes the state to converge to a small range.

Figure 1: State-input trajectory of flight control systems: disturbance-free
data with the controller in Rotulo et al. (2022) (top two panels) and noisy
data with the proposed controller (11) (bottom two panels).
Figure 2: State-input trajectory of aircraft engine systems: disturbance-free
data with the controller in Rotulo et al. (2022) (top two panels) and noisy
data with the proposed controller (11) (bottom two panels).

###  4.2 Aircraft engine systems

In the second example, we consider our approach for stabilizing fault tolerant
systems as switched systems. Specifically, we apply our proposed online
controller to an F-$404$ aircraft engine system subject to system and actuator
faults, originally considered in Liu et al. (2017). The system is composed of
three states, namely, the sideslip angle, the roll rate, and the yaw rate,
with two control inputs representing the engine thrust and the flight path
angle. We consider a discretized linearized version of this system with a
sampling period of $0.1$s, and the system matrices are given by

$\displaystyle A=\left[\begin{matrix}0.867&0&0.202\\\ 0.015&0.961&-0.032\\\
0.026&0&0.803\\\ \end{matrix}\right],~{}B=\left[\begin{matrix}0.011&0\\\
0.014&-0.039\\\ 0.009&0\\\ \end{matrix}\right].$

Similarly to the previous section, we collect an input-state trajectory of
length $T=21$ by simulating the system offline using a persistently exciting
input sequence uniformly generated in $[-3.5,3.5]$. Fig. 2 shows the offline
phase represented by the interval $t\in[0,21]$. We then run the system online,
where external disturbances and unknown faults such as wind gusts or
structural vibrations can undermine system stability, characterized by changes
in the system matrices $A$ and $B$. Specifically, system faults are captured
by the changes in the system matrices as $\tilde{A}=A+\beta(k)D$ with

$\displaystyle D=\left[\begin{matrix}0.075&0&0\\\ 0.5&1&0\\\
0&0&-0.75\end{matrix}\right],~{}\beta(k)=\begin{cases}0.1&k\in[0,27]\\\
0.05&k\in[27,52]\\\ -0.5&k\in[52,95]\\\ 0&{\rm else}\end{cases}$

while failures of the engine generating thrust and the motor moving the path
angle are modeled by the changes in the input matrix $B$ as
$\tilde{B}=B\alpha(k)$ with

$\displaystyle\alpha(k)=\left[\begin{matrix}1&0\\\
0&0\end{matrix}\right],\forall
k\in[27,52],~{}~{}~{}\alpha(k)=\left[\begin{matrix}0&0\\\
0&1\end{matrix}\right],\forall k\geq 52.$

During online operation, we track the state trajectory (top panel), evolution
of the function $\mathcal{V}(x(k))$, and the smallest singular value of
$W_{k-1}$ (bottom two panels) as shown in Fig. 2. We set $\bar{d}=0.015$,
$\delta_{u}=0.01$, $\delta_{x}=10^{-4}$, and $\alpha=0.2$.

##  5 Acknowledgments

The authors would like to thank Prof. Claudio De Persis for his kind help and
valuable suggestions on Remark 3.1 and the paper’s structure.

##  6 Conclusions

In this paper, we presented a data-driven switched controller for stabilizing
unknown linear switched systems, utilizing noisy input-state data. Our
approach employs an auxiliary function-based switching law, where the state
feedback gain is updated by solving a robust data-based SDP online. We
provided conditions such that the feasibility of the robust SDP is guaranteed,
and established the ISpS under the assumptions that the disturbance is bounded
and the system switches slowly enough. Two numerical examples were employed to
demonstrate the practical benefits of our proposed controller.

###  Appendix A.1 Proof of Theorem 2.2

Partition matrix $W_{-1}$ in (2e) into $W_{-1}:=W_{-1,\bar{x}}+W_{-1,d}$ with

$W_{-1,\bar{x}}:=\left[\begin{matrix}U_{-1}\\\
\bar{X}_{-1}\end{matrix}\right],\quad W_{-1,d}:=\left[\begin{matrix}0\\\
\tilde{D}_{-1}\end{matrix}\right]$ (24)

where $\bar{X}_{-1}=[\bar{x}(-T)~{}\cdots~{}\bar{x}(-1)]$ with
$\bar{x}(-T)=x(-T)$ and subsequent ones computed by the disturbance-free
recursion $\bar{x}(i+1)=A\bar{x}(i)+Bu(i)$ for all $i\in[-T,-2]$, and, where
$\tilde{D}_{-1,p}:=\sum_{i=0}^{p-1}\Pi_{j=1}^{i}Ad(T+p-i-1)$ is the $p$-th
column of matrix $\tilde{D}_{-1}$. Since $\|D_{-1}\|\leq\delta$, it can be
deduced from (1) that
$\|\tilde{D}_{-1}\|\leq\sqrt{n_{x}}\|\Omega_{0}\|\|D_{-1}\|\leq\sqrt{n_{x}}\|\Omega_{0}\|\delta$
with

$\displaystyle\Omega_{0}:=\left[\begin{matrix}0&0&0&\cdots&0\\\
I&0&0&\cdots&0\\\ A&I&0&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
A^{T-2}&A^{T-3}&A^{T-4}&\cdots&0\end{matrix}\right].$

On the other hand, it follows from (Coulson et al., 2022, Theorem 3.1) that
for $\bar{w}$-persistently exciting input sequence $u(-T),\cdots,u(-1)$, the
smallest singular value of matrix $W_{-1,\bar{x}}$ satisfies
$\underline{\lambda}_{W_{-1,\bar{x}}}\geq\bar{w}\rho/\sqrt{n_{x}+1}$ where
$\rho>0$ is an internal parameter of system $(A,B)$; see (Coulson et al.,
2022, Lemma 3.1) for its detailed definition. If constant $\bar{w}$ satisfies

$\bar{w}>\sqrt{n_{x}(n_{x}+1)}\|\Omega_{0}\|\delta/\rho$ (25)

then it holds that
$\underline{\lambda}_{W_{-1,\bar{x}}}>\sqrt{n_{x}}\|\Omega_{0}\|\delta\geq\|\tilde{D}_{-1}\|=\|W_{-1,d}\|$.
Recall the fact that for any matrices $Y$, $Z$ of the same size, inequality
$\underline{\lambda}_{Y+Z}\geq\underline{\lambda}_{Y}-\|Z\|$ holds true.
Hence, one has from (24) that
$\underline{\lambda}_{W_{-1}}\geq\underline{\lambda}_{W_{-1,\bar{x}}}-\|W_{-1,d}\|>0$
indicating that condition (3) holds.

###  Appendix A.2 Proof of Theorem 3.1

Partition matrix $W_{k-1}$ following the same step as in the proof of Lemma
2.2, i.e., $W_{k-1}:=W_{k-1,\bar{x}}+W_{k-1,d}$ with

$W_{k-1,\bar{x}}:=\left[\begin{matrix}U_{k-1}\\\
\bar{X}_{k-1}\end{matrix}\right],\quad W_{k-1,d}:=\left[\begin{matrix}0\\\
\tilde{D}_{k-1}\end{matrix}\right]$

where $\bar{X}_{k-1}=[\bar{x}(k-T)~{}\cdots~{}\bar{x}(k-1)]$ with
$\bar{x}(k-T)=x(k-T)$ and subsequent ones computed by the disturbance-free
recursion $\bar{x}(i+1)=A_{\sigma(i)}\bar{x}(i)+B_{\sigma(i)}u(i)$ for all
$i\in[k-T,k-2]$, and, where
$\tilde{D}_{k-1,p}:=\sum_{i=0}^{p-1}\Pi_{j=1}^{i}A_{\sigma(k-T+j)}d(k-T+p-i-1)$
is the $p$-th column of matrix $\tilde{D}_{k-1}$.

Since the input sequence ${u(k-T),\cdots,u(k-1)}$ is $\bar{w}$-persistently
exciting of order $n_{x}+1$, it follows from the proof of Lemma 2.2 that
$\underline{\lambda}_{W_{k-1,\bar{x}}}\geq\hat{w}:=\bar{w}\rho/\sqrt{n_{x}+1}$
for some constant $\rho>0$. In addition, note that
$\|W_{k-1,d}\|\leq\sqrt{n_{x}}\|\Omega_{k-1}\|\|D_{k-1}\|$ where
$\Omega_{k-1}$ is defined in (26), presented at the top of the next page.
Since the system switches in a finite set, there exists a constant
$\bar{\Omega}$ such that $\|\Omega_{k}\|\leq\bar{\Omega}$ for all
$k\in\mathbb{N}$. Moreover, it can be deduced from $\|D_{k-1}\|\leq
T\bar{d}^{2}$ that $\|W_{k-1,d}\|\leq\sqrt{Tn_{x}}\bar{\Omega}\bar{d}$ for all
$k\in\mathbb{N}$. Therefore, if

$\bar{d}<{\hat{w}}/(2\sqrt{Tn_{x}}\bar{\Omega})$ (27)

then $\underline{\lambda}_{W_{k-1,\bar{x}}}>\|W_{k-1,{d}}\|$ and
$\underline{\lambda}_{W_{k-1}}\geq\underline{\lambda}_{W_{k-1,\bar{x}}}-\|W_{k-1,{d}}\|>0$
implying that condition (10) holds. Hence, SDPs (3) and (3.2) are feasible,
where the feasibility of SDP (3.2) is guaranteed by (Rotulo et al., 2022,
Lemma 4). Let $(\bar{\gamma}(k),\bar{Q}(k),\bar{P}(k),\bar{L}(k))$ denote an
optimal solution of SDP (3.2). It follows from (De Persis and Tesi, 2021,
Lemma 7) that for some given constant $\eta_{2}\geq 1$, if
$-\bar{\Psi}(k)\leq(1-{1}/{\eta_{2}})I$ with
$\bar{\Psi}(k)=D_{k-1}\bar{M}(k)D_{k-1}^{\prime}-X_{k}\bar{M}(k)D_{k-1}^{\prime}-D_{k-1}\bar{M}(k)X_{k}^{\prime}$
and $\bar{M}(k)=\bar{Q}(k)\bar{P}(k)^{-1}\bar{Q}(k)^{\prime}$, then SDP (3) is
feasible, and any optimal solution
$(\gamma^{*}(k),Q^{*}(k),P^{*}(k),L^{*}(k),V^{*}(k))$ constructs
$K^{*}(k)=U_{k-1}Q_{i}^{*}(k)P_{i}^{*}(k)^{-1}$ resulting to a Schur stable
matrix $A_{i}+B_{i}K^{*}(k)$.

$\Omega_{k-1}=\left[\begin{matrix}0&0&0&\cdots&0\\\ I&0&0&\cdots&0\\\
A_{\sigma(k-T+1)}&I&0&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
\Pi_{j=1}^{T-2}A_{\sigma(k-T+j)}&\Pi_{j=0}^{T-3}A_{\sigma(k-T+j)}&\Pi_{j=0}^{T-4}A_{\sigma(k-T+j)}&\cdots&0\end{matrix}\right]$
(26)

In the following, a bound on the disturbance is derived ensuring that
$-\bar{\Psi}(k)\leq(1-{1}/{\eta_{2}})I$ always holds true. Define
$\bar{\Phi}(k)=[\bar{K}(k)^{\prime}~{}I]^{\prime}$, where
$\bar{K}=U_{k-1}\bar{Q}(k)\bar{P}(k)^{-1}$. Combining
$\bar{M}(k)=\bar{Q}(k)\bar{P}(k)^{-1}\bar{Q}(k)^{\prime}$ with
$\bar{P}(k)^{-1}\succeq 0$, one gets that $\bar{M}(k)\succeq 0$, and hence a
sufficient condition for $-\bar{\Psi}(k)\leq(1-{1}/{\eta_{2}})I$ is

$X_{k}\bar{M}(k)D_{k-1}^{\prime}+D_{k-1}\bar{M}(k)X_{k}^{\prime}\leq(1-{1}/{\eta_{2}})I.$

Taking the $2$-norm for both sides of the above inequality, if

$2\|X_{k}\|\|\bar{M}(k)\|\|D_{k-1}^{\prime}\|\leq 1-{1}/{\eta_{2}}$ (28)

then $-\bar{\Psi}(k)\leq(1-{1}/{\eta_{2}})I$.

Combining $X_{k-1}\bar{Q}(k)=\bar{P}(k)$ (the second constraint in (3.2)) with
$\bar{K}(k)=U_{k-1}\bar{Q}(k)\bar{P}(k)^{-1}$, also considering that $W_{k-1}$
has full row rank, matrix $\bar{Q}(k)$ can be expressed as
$\bar{Q}(k)=W_{k-1}^{{\dagger}}\bar{\Phi}(k)\bar{P}(k)$. Hence,
$\bar{M}(k)=W_{k-1}^{\dagger}\bar{\Phi}(k)\bar{P}(k)\bar{\Phi}^{\prime}(W_{k-1}^{\dagger})^{\prime}$,
and
$\|\bar{M}(k)\|\leq\|\bar{\Phi}(k)\|^{2}\|\bar{P}(k)\|\|W_{k-1}^{\dagger}\|^{2}$.
Noticing from (De Persis and Tesi, 2020, Theorem 4) that $\bar{K}(k)$ is the
unique solution of
$(A_{i}+B_{i}\bar{K}(k))\bar{P}(k)(A_{i}+B_{i}\bar{K}(k))^{\prime}-\bar{P}(k)+I=0$,
there exists a constant $\phi$ such that
$\|\bar{\Phi}(k)\|^{2}\|\bar{P}(k)\|\leq\phi$ holds for all $i\in\mathcal{M}$.
In addition, since
$\underline{\lambda}_{W_{k-1,\bar{x}}}=\|W_{k-1,\bar{x}}^{\dagger}\|^{-1}$ and
$\|W_{k-1}^{\dagger}\|^{-1}\geq\|W_{k-1,\bar{x}}^{\dagger}\|^{-1}-\|W_{k-1,d}\|$,
under condition (27), inequality $\|W_{k-1}^{\dagger}\|\leq
2\|W_{k-1,\bar{x}}^{\dagger}\|$ holds. Furthermore, since columns in $X_{k}$
are generated by subsystem $(A_{i},B_{i})$, it follows that
$X_{k}=[B_{i}~{}A_{i}]W_{k-1}+D_{k-1}$. Based on the fact that
$\|W_{k-1}\|\leq\|W_{k-1,\bar{x}}\|+\sqrt{Tn_{x}}\bar{\Omega}\bar{d}$,
inequality (28) becomes

$\displaystyle
8\sqrt{T}\bar{d}\phi\|W_{k-1,\bar{x}}^{\dagger}\|^{2}(\|\Xi_{i}\|(\|W_{k-1,\bar{x}}\|+\sqrt{Tn_{x}}\bar{\Omega}\bar{d}))$
$\displaystyle+8\sqrt{T}\bar{d}\phi\|W_{k-1,\bar{x}}^{\dagger}\|^{2}\sqrt{T}\bar{d}\leq
1-{1}/{\eta_{2}}$

where $\Xi_{i}:=[A_{i}~{}B_{i}]$. Since the switched system has finite modes,
there exists a constant $\bar{\Xi}$ such that $\|\Xi_{i}\|\leq\bar{\Xi}$ for
all $i\in\mathcal{M}$. Considering (27), a sufficient condition for (28) is

$\displaystyle\bar{d}<\delta_{d,1}:=\hat{w}\min\bigg{\\{}$
$\displaystyle\frac{1}{2\sqrt{Tn_{x}}\bar{\Omega}},\frac{1-1/\eta_{2}}{24\sqrt{T}\phi\bar{\Xi}c(W_{k-1,\bar{x}})},$
$\displaystyle\frac{1-1/\eta_{2}}{\sqrt{24T\sqrt{n_{x}}\phi\bar{\Xi}\bar{\Omega}}},\frac{1-1/\eta_{2}}{\sqrt{24T\phi}}\bigg{\\}}$
(29)

where $c(W_{k-1,\bar{x}}):=\|W_{k-1,\bar{x}}\|\|W_{k-1,\bar{x}}^{\dagger}\|$
is the condition number of matrix $W_{k-1,\bar{x}}$. Since
$\underline{\lambda}_{W_{k-1,\bar{x}}}\geq\hat{w}$, the condition number obeys
$c(W_{k-1,\bar{x}})=\|W_{k-1,\bar{x}}\|/\underline{\lambda}_{W_{k-1,\bar{x}}}\geq
1$. This indicates that $\delta_{d,1}$ is independent of disturbance $d(k)$
and time $k$. This bound is similar to the bound in (De Persis and Tesi, 2021,
(37)). The only difference here is that $\|W_{k-1}^{\dagger}\|^{-1}$ is
replaced by $\bar{w}$. This is because that the selected input sequence is
$\bar{w}$-persistently exciting of order $n_{x}+1$, and hence provides a
positive constant lower bound on the smallest singular value of matrix
$W_{k-1}$. In conclusion, for sufficiently small disturbance
$\|d(k)\|\leq\bar{d}$ with $\bar{d}<\delta_{d,1}$, a candidate solution of (3)
can be constructed by
$\eta_{2}(\bar{\gamma}(k),\bar{Q}(k),\bar{P}(k),\bar{L}(k),\bar{Q}(k)\bar{P}(k)^{-1}\bar{Q}(k)^{\prime})$.
Moreover, for any optimal solution of SDP (3), i.e.,
$(\gamma^{*}(k),Q^{*}(k),P^{*}(k),L^{*}(k),V^{*}(k))$, matrix $K^{*}(k)$
satisfying $K^{*}(k)=U_{k-1}Q^{*}(k)P^{*}(k)^{-1}$ is such that
$A_{i}+B_{i}K^{*}(k)$ is Schur stable, which completes the proof.

###  Appendix A.3 Proof of Lemma 3.2

Similar from the proof of Lemma 3.2 in Appendix A.3, condition (10) holds due
to the fact that input sequence ${u(k-T),\cdots,u(k-1)}$ is
$\bar{w}$-persistently exciting of order $n_{x}+1$. According to Assumption
(3.1) and (3.2), matrix $X_{k-1}$ contains at least $N$ data from the same
subsystem, it follows from (Rotulo et al., 2022, Lemma 5) that , SDP (3.2) is
feasible. In addition, it has been shown in the proof of Lemma 3.1 that under
condition (Appendix A.2), a candidate solution of (3) can be constructed by
$\eta_{2}(\bar{\gamma}(k),\bar{Q}(k),\bar{P}(k),\bar{L}(k),\bar{Q}(k)\bar{P}(k)^{-1}\bar{Q}(k)^{\prime})$,
where $(\bar{\gamma}(k),\bar{Q}(k),\bar{P}(k),\bar{L}(k))$ is any optimal
solution of SDP (3.2). This completes the proof.

###  Appendix A.4 Proof of Lemma 3.3

For a given $\delta_{\epsilon}$ and
$\epsilon(k)\in\mathbb{B}_{\delta_{\epsilon}}$, there exists a $\bar{w}>0$
such that sequence $\\{\epsilon(k_{j}),\cdots,\epsilon(k_{j}+N-1)\\}$ is
$\bar{w}$-persistently exciting for order $n_{x}+1$. This indicates that
${u(k_{j}),\cdots,u(k_{j}+N-1)}$ is $\bar{w}$-persistently exciting for order
$n_{x}+1$. One can further deduce that ${u(k-T),\cdots,u(k-1)}$ is
$\bar{w}$-persistently exciting for order $n_{x}+1$ for all
$k\in[k_{j}+N,k^{j}]$ with $j\in\mathbb{N}$. To be specific, let matrix
$Y\in\mathbb{R}^{m\times n}$ with $n\geq m$ and ${\rm rank}(Y)=m$. Define
matrix $Z=[Y,b_{j}]\in\mathbb{R}^{m\times(n+1)}$ with
$b_{j}\in\mathbb{R}^{m}$. Since $ZZ^{\prime}=YY^{\prime}+b_{j}b_{j}^{\prime}$,
the smallest singular value of matrices $Y$ and $Z$ satisfies
$\underline{\lambda}_{Z}\geq\underline{\lambda}_{Y}$. Since the smallest
singular value of matrix $H_{n_{x}+1}(u_{[k_{j},k_{j}+N-1]})$ is lower bounded
by $\bar{w}$, it can be deduced that the smallest singular value of matrix
$H_{n_{x}+1}(u_{[k_{j},k_{j}+2N-1]})$ is larger than or equal to $\bar{w}$.
This implies that ${\rm rank}(W_{k-1})=n_{x}+n_{u}$ holds for all
$k\in[k_{j}+N+1,k_{j}+2N-1]$. According to Assumptions 3.1 and the
computational complexity concerns, $T=2N-1$. Based on the results in Lemmas
3.1 and 3.2, we conclude that SDP (3) is feasible for all
$k\in[k_{j}+N,k_{j}+T]$.

###  Appendix A.5 Proof of Lemma 3.4

Let $i\in\mathcal{M}$ denote the subsystem selected by $\sigma(k_{s_{j}})$,
i.e., $i=\sigma(k),k\in[k_{s_{j}},k_{s_{j+1}}]$. According to Lemma 3.1, for
$k\in[k_{s_{j}}+T,k_{s_{j+1}}-1]$, the difference between the Lyapunov
function at two consecutive time instants satisfies

$\displaystyle\Delta\mathcal{W}_{i}(x(k+1))$
$\displaystyle=\mathcal{W}_{i}(x(k+1))-\mathcal{W}_{i}(x(k))$
$\displaystyle=x(k)^{\prime}\\!\mathcal{A}_{i}^{\prime}\\!\cdot\\!P_{i}(\star)-x(k)^{\prime}P_{i}(\star)\\!+\\!d(k)^{\prime}P_{i}d(k)$
$\displaystyle~{}~{}~{}+\\!2x(k)^{\prime}\mathcal{A}_{i}^{\prime}P_{i}d(k)$
$\displaystyle\leq\mathcal{W}^{1}(x(k))\\!-\\!(\beta_{i}/2)\|x(k)\|^{2}\\!+\\!\bar{\lambda}_{P}\bar{d}^{2}$
$\displaystyle~{}~{}~{}+2\bar{\lambda}_{P}\|\mathcal{A}_{i}\|\|x(k)\|\bar{d}$

where
$\mathcal{W}^{1}(x(k))=x(k)^{\prime}\\!\mathcal{A}_{i}^{\prime}\\!\cdot\\!P_{i}(\star)-x(k)^{\prime}P_{i}x(k)+(\beta_{i}/2)\|x(k)\|^{2}$.
Assuming that $k\notin\mathbb{I}_{\delta_{V}}$, there exists a constant
$\delta_{x}$ such that $\|x(k)\|\geq\delta_{x}$, which implies that

$\displaystyle\Delta\mathcal{W}_{i}(x(k+1))$
$\displaystyle\leq\mathcal{W}^{1}(x(k))$
$\displaystyle~{}~{}~{}+\\!\Big{(}\\!\underbrace{-\frac{\beta_{i}}{2}\\!+\frac{\underline{\lambda}_{P}}{\delta_{x}^{2}}\bar{d}^{2}+2\frac{\underline{\lambda}_{P}}{\delta_{x}}\|\mathcal{A}_{i}\|\bar{d}}_{\Delta_{\mathcal{W}_{\bar{d}}}}\Big{)}\\!\|x(k)\|^{2}.$

Let $\bar{\beta}:=\max_{i\in\mathcal{M}}\\{\beta_{i}\\}$ and
$\bar{\delta}_{\mathcal{A}}:=\max_{i\in\mathcal{M}}\\{\mathcal{A}_{i}\\}$. If
$\bar{d}<\delta_{d,2}$ with

$\delta_{d,2}\\!:=\min\Bigg{\\{}\\!\delta_{d,1},~{}\delta_{x}\frac{-2\bar{\lambda}_{P}\bar{\delta}_{\mathcal{A}}\\!+\\!\sqrt{4\bar{\lambda}_{P}^{2}\bar{\delta}_{\mathcal{A}}^{2}\\!+\\!2\bar{\lambda}_{P}\bar{\beta}}}{2\bar{\lambda}_{P}}\Bigg{\\}}$
(31)

and $\delta_{d,1}$ defined in (Appendix A.2), then
$\Delta_{\mathcal{W}_{\bar{d}}}\leq 0$ and
$\Delta\mathcal{W}_{i}(x(k+1))\leq\mathcal{W}^{1}(x(k))$. Leveraging (15), the
following inequality holds for all $k\in[k_{s_{j}}+T,k_{s_{j+1}}]$

$\mathcal{W}_{i}(x(k+1))\leq\big{(}1-\bar{\beta}/(2\bar{\lambda}_{P})\big{)}\mathcal{W}_{i}(x(k)).$
(32)

In fact, due to disturbance $d(k)$, optimal solutions of SDP (3) for a
subsystem $i\in\mathcal{M}$ activated at different times, i.e.,
$\sigma(k_{s_{1}})=\sigma(k_{s_{2}})=\cdots=i$ with $k_{s_{1}}\neq
k_{s_{2}}\neq\cdots$, are generally different. Suppose that condition (10)
holds at $k=k_{s_{j}},j\in\mathbb{N}$. According to Lemma 3.1, for a given
$\eta_{2}\geq 1$ and $\bar{d}<\delta_{d,2}$ with $\delta_{d,2}$ in (31),
$\eta_{2}(\bar{\gamma}(k),\bar{Q}(k),\bar{P}(k),\bar{L}(k),\bar{Q}(k)\bar{P}(k)^{-1}\bar{Q}(k)^{\prime})$
is a candidate solution of SDP (3). Let
$(\gamma^{*}(k),Q^{*}(k),P^{*}(k),L^{*}(k),\\\ V^{*}(k))$ be an optimal
solution of SDP (3), and $K^{*}(k)=U_{k-1}Q^{*}(k)P^{*}(k)^{-1}$. It follows
from the last constraint of SDP (3) that
$\bar{\lambda}_{P^{*}(k_{s_{j}})}\leq{\rm
tr}(P^{*}(k_{s_{j}}))\leq\eta_{2}\gamma^{*}(k_{s_{j}})$. Since
$(\bar{\gamma}(k),\bar{Q}(k),\bar{P}(k),\bar{L}(k))$ is the unique LQR
solution of SDP (3.2) (Rotulo et al., 2022, Lemma 4), it holds that
$\bar{\gamma}(k_{s_{j}})=\bar{\gamma}_{i}$ for all $j\in\mathbb{N}$. Let

$\check{\lambda}_{0}:=1-\frac{\bar{\beta}}{2\max_{i\in\mathcal{M}}\eta_{2}\bar{\gamma}_{i}}.$
(33)

Hence, for any $i\in\mathcal{M}$ and $j\in\mathbb{N}$, matrix
$K^{*}(k_{s_{j}})$ is such that $A_{i}+B_{i}K^{*}(k_{s_{j}})$ is Schur stable,
the Lyapunov function converges following
$\mathcal{W}_{i}(x(k+1))\leq\check{\lambda}_{0}\mathcal{W}_{i}(x(k))$ for all
$k\in[k_{s_{j}}+T,k_{s_{j+1}}]$. This further implies that for some
$\lambda_{0}\in[\check{\lambda}_{0},1)$, it holds that
$\mathcal{V}(x(k+1))\leq\check{\lambda}_{0}\mathcal{V}(x(k))$.

On the other hand, for $k=k^{j}$, it follows from (11)–(3.1) that
$\mathcal{V}(x(k^{j}+1))\leq\lambda_{0}\mathcal{V}(x(k^{j}))$,
$P(k^{j}+1)=P(k^{j})$, and $K(k^{j}+1)=K(k^{j})$. The difference between the
auxiliary function at two consecutive time instants satisfies

$\displaystyle\Delta\mathcal{V}(x(k^{j}+1))$
$\displaystyle=\mathcal{V}(x(k^{j}+1))-\mathcal{V}(x(k^{j}))$
$\displaystyle=x(k^{j})^{\prime}\\!\mathcal{A}(k^{j}\\!+\\!1)^{\prime}\\!\cdot\\!P(k^{j})(\star)$
$\displaystyle~{}~{}~{}-x(k^{j})^{\prime}P(k^{j})(\star)\\!+\\!d(k^{j})^{\prime}P(k^{j})d(k^{j})$
$\displaystyle~{}~{}~{}+\\!2x(k^{j})^{\prime}\mathcal{A}(k^{j}\\!+\\!1)^{\prime}P(k^{j})d(k^{j})$
$\displaystyle=\mathcal{V}_{1}(x(k^{j}))\\!-\\!(\bar{\beta}/2)\|x(k^{j})\|^{2}\\!+\\!\bar{\lambda}_{P(k^{j})}\bar{d}^{2}$
$\displaystyle~{}~{}~{}+2\bar{\lambda}_{P(k^{j})}\|\mathcal{A}(k^{j}\\!+\\!1)\|\|x(k^{j})\|\bar{d}$
$\displaystyle\leq\mathcal{V}_{1}(x(k^{j}))+\Big{(}\underbrace{-\frac{\bar{\beta}}{2}+\frac{\underline{\lambda}_{P(k^{j})}}{\delta_{x}^{2}}\bar{d}^{2}}$
$\displaystyle~{}~{}~{}\underbrace{+2\frac{\underline{\lambda}_{P(k^{j})}}{\delta_{x}}\|\mathcal{A}(k^{j}\\!+\\!1)\|\bar{d}}_{\Delta_{\mathcal{V}_{\bar{d}}}}\Big{)}\\!\|x(k^{j})\|^{2}$

where
$\mathcal{A}(k^{j}\\!+\\!1)\\!:=\\!A_{\sigma(k^{j}\\!+\\!1)}+B_{\sigma(k^{j}\\!+\\!1)}K(k^{j})$,
$\mathcal{V}_{1}(x(k^{j}\\!+\\!1))=x(k^{j})^{\prime}\\!\mathcal{A}(k^{j}\\!+\\!1)^{\prime}\\!\cdot\\!P(k^{j})(\star)-x(k^{j})^{\prime}P(k^{j})x(k^{j})+\bar{\beta}/2\|x(k^{j})\|^{2}$,
and the last inequality holds since $\mathcal{V}(x(k^{j}))\geq\delta_{V}$.
Since $\bar{d}<\delta_{d,2}$, it follows that
$\Delta_{\mathcal{V}_{\bar{d}}}\leq 0$ and
$\Delta\mathcal{V}(x(k^{j}+1))\leq\mathcal{V}_{1}(x(k^{j}))$. Since
$\lambda_{0}\in[\check{\lambda}_{0},1)$ with $\check{\lambda}_{0}$ in (33),
one has that

$\displaystyle\Delta\mathcal{V}(x(k^{j}+1))$
$\displaystyle=\mathcal{V}(x(k^{j}+1))-\mathcal{V}(x(k^{j}))$
$\displaystyle\leq(\lambda_{0}-1)\mathcal{V}(x(k^{j}))\leq-(\bar{\beta}/2)\|x(k^{j})\|^{2}$

hence
$x(k^{j})^{\prime}\\!\mathcal{A}(k^{j}\\!+\\!1)^{\prime}\\!\cdot\\!P(k^{j})(\star)-x(k^{j})^{\prime}P(k^{j})x(k^{j})+{\bar{\beta}}/{2}\|x(k^{j})\|^{2}\leq-{\bar{\beta}}/{2}\|x(k^{j})\|^{2}$
and consequently
$\mathcal{A}(k^{j}\\!+\\!1)^{\prime}P(k^{j})\mathcal{A}(k^{j}\\!+\\!1)-P(k^{j})\leq-\bar{\beta}I$.
This implies that $K(k^{j})$ derived from (3) stabilizes subsystem
$(A_{\sigma(k^{j}+1)},B_{\sigma(k^{j}+1)})$. Hence, it follows from Lemma 3.1
that $k^{j}\in[k_{s_{j}}+T,k_{s_{j+1}}]$.

Moreover, there exists some constant $\hat{\lambda}_{0}$ obeying
$\check{\lambda}_{0}<\hat{\lambda}_{0}\leq 1$ such that condition
$\mathcal{V}(x(k^{j}-1))>\lambda_{0}\mathcal{V}(x(k^{j}-2))$ holds when
$K(k^{j}-2)$ cannot stabilize $(A_{\sigma(k^{j}-1)},B_{\sigma(k^{j}-1)})$ or
the Lyapunov function converges with a rate smaller than
$\check{\lambda}_{0}$. Since the disturbance satisfies (31), based on Lemma
3.2, this only occurs when matrices $X_{k-1}$ and $X_{k}$ contain samples
generated from two subsystems. Hence, $k^{j}-1\in[k_{s_{j}},k_{s_{j}}+T-1]$,
and consequently $k^{j}=k_{s_{j}}+T$ holds for all $j\in\mathbb{N}$.

Similarly, for time $k_{j}$, condition
$\mathcal{V}(x(k_{j}))>\lambda_{0}\mathcal{V}(x(x_{j}-1))$ means that
$k_{j}\in[k_{s_{j}},k_{s}+T-1]$, and condition
$\mathcal{V}(x(k_{j}-1))\leq\lambda_{0}\mathcal{V}(x(x_{j}-2))$ implies that
$k_{j}-1\in[k_{s-1}+T,k_{s}]$. Hence, $k_{j}=k_{s_{j}}$ for all
$j\in\mathbb{N}$. This further implies that $k_{j}$ and $k^{j}$ in sequences
$\\{k_{j}\\}_{j\in\mathbb{N}}$ and $\\{k^{j}\\}_{i\in\mathbb{N}}$ are ordered
such that $0=k_{0}<k_{0}+N<k^{0}<k_{1}<k_{1}+N<k^{1}<\cdots$.

Noticing that system (5) may switch its mode at $k_{s_{j}}$ while condition
$\mathcal{V}(x(k_{s_{j}}))\leq\lambda_{0}\mathcal{V}(x(k_{s_{j}}-1))$ still
holds. According to (11)–(3.1), SDP (3) is not solved, and both matrices
$K(k_{s_{j}})$ and $P(k_{s_{j}})$ remain unchanged. In this case, the system
does not aware that a switching happens, and the current controller can
guarantee that the convergence of Lyapunov function satisfies (32). Hence,
such switches will not affect system stability.

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