††thanks: The work was supported in part by the National Key R&D Program of
China under Grant 2021YFB1714800, the National Natural Science Foundation of
China under Grants 62088101, 62173034, 61925303, and the Chongqing Natural
Science Foundation under Grant 2021ZX4100027. ††thanks: This paper was not
presented at any IFAC meeting.

, , , ,

# Self-triggered Resilient Stabilization of Linear Systems with Quantized
Output

Wenjie Liu<EMAIL_ADDRESS>Masashi Wakaiki<EMAIL_ADDRESS>Jian
Sun<EMAIL_ADDRESS>Gang Wang<EMAIL_ADDRESS>Jie Chen
<EMAIL_ADDRESS>Key Lab of Intelligent Control and Decision of Complex
Systems, Beijing Institute of Technology, Beijing 100081, China  Beijing
Institute of Technology Chongqing Innovation Center, Chonqing 401120, China
Graduate School of System Informatics, Kobe University, Hyogo 657-8501, Japan
Department of Control Science and Engineering, Tongji University, Shanghai
201804, China

###### Abstract

This paper studies the problem of stabilizing a self-triggered control system
with quantized output. Employing a standard observer-based state feedback
control law, a self-triggering mechanism that dictates the next sampling time
based on quantized output is co-developed with an output encoding scheme. If,
in addition, the transmission protocols at the controller-to-actuator (C-A)
and sensor-to-controller (S-C) channels can be adapted, the self-triggered
control architecture can be considerably simplified, leveraging a delicate
observer-based deadbeat controller to eliminate the need for running the
controller in parallel at the encoder side. To account for denial-of-service
(DoS) in the S-C channel, the proposed output encoding and self-triggered
control schemes are further made resilient. It is shown that a linear time-
invariant system can be exponentially stabilized if some conditions on the
average DoS duration time are met. There is a trade-off between the maximum
inter-sampling time and the resilience against DoS attacks. Finally, a
numerical example is presented to demonstrate the practical merits of the
proposed self-triggered control schemes and associated theory.

###### keywords:

Self-triggered control, quantized output, Denial-of-Service attack, encoding,
deadbeat control.

##  1 Introduction

With the development of communication and networking technologies, networked
control systems (NCSs), in which data are transmitted over wired or wireless
networks, have been widely integrated in modern engineering systems (e.g.,
Pasqualetti et al. (2012); Chen et al. (2022)). In the study of NCSs, a basic
problem relates to determine how frequently different devices (e.g., sensors,
controllers, and actuators) of a control system shall be executed so as to
balance between the communication cost and the overall system performance.
Transmission frequency and limited bandwidth are two major factors affecting
the communication cost, both of which have been considerably studied in past
decades.

To reduce transmission frequency, research on aperiodic sampling techniques
has lately aroused great interest. Among many, event-triggered control and
self-triggered control schemes are two practically appealing solutions Heemels
et al. (2012). In the former, the state or output of a plant is periodically
or continuously sampled to examine the event-triggering condition, but the
sampled value is transmitted only when the condition is met, e.g., Åarzén
(1999); Heemels (2013); Wakaiki and Sano (2020). In self-triggered control
systems however, measurements are taken and transmitted only at sampling times
which are determined based on previous sampled measurements, e.g., Anta and
Tabuada (2010); Gommans and Heemels (2015); Matsume et al. (2021); Wakaiki
(2022). Hence, both sensors and communication channels are activated only at
sampling times. In general, self-triggered control outperforms event-triggered
control in terms of prolonging the sensor lifetime, and reducing the
communication load at the sensor-to-controller (S-C) channel.

On the other hand, data transmitting through a channel with limited bandwidth
are quantized before sent out. It has been shown that coarse quantization can
deteriorate the system performance, and may even lead to instability Bullo and
Liberzon (2006). To secure system stability, the quantizer should be carefully
designed. Proposed in Brockett and Liberzon (2000), the so-called “zooming-in”
and “zooming-out” method offers an elegant way to design dynamic quantizers
for linear time-invariant (LTI) systems. Its generalizations can be found to
deal with, e.g., nonlinear systems in Liberzon and Hespanha (2005), switched
systems in Wakaiki and Yamamoto (2017), and systems under DoS attacks in
Wakaiki et al. (2020); Liu et al. (2022a).

Although it is natural to consider self-triggering mechanism and constrained
bandwidth simultaneously in the real scenario, design a self-triggering
mechanism using only coarse measurements at sampling times is challenging. For
this reason, it has not been fully addressed in the literature, and only few
works have jointly studied these two factors; see e.g., De Persis and Frasca
(2013); Zhou et al. (2018); Wakaiki (2022); Ikeda et al. (2016); Liu et al.
(2023). In De Persis and Frasca (2013), consensus protocols considering both
self-triggered and quantized communication were proposed for linear multi-
agent systems. Although limited bandwidth was considered in Ikeda et al.
(2016); Zhou et al. (2018), both works designed the self-triggering mechanisms
using non-quantized states, which has addressed most of the difficulties
encountered in self-triggered control. Most recently, the work Wakaiki (2022)
proposed a self-triggering mechanism based on quantized states.

The goal of this paper is to generalize the results of Wakaiki (2022) to
systems where only quantized output rather than quantized state are available.
Specifically, we consider that the controller-to-actuator (C-A) channel is
ideal with adequate bandwidth, whereas the S-C channel has limited
communication resources. The S-C channel hosts an encoder, a decoder, and a
self-triggering module that are responsible for when/how the output shall be
sampled, quantized, and transmitted. In this context, a self-triggering
mechanism and an output encoding scheme are co-designed for a linear system
with a standard observer-based controller. In addition, if a higher
transmission rate can be afforded at the C-A channel, an observer-based
deadbeat control law is designed such that the self-triggering mechanism can
be considerably simplified and the computational overhead at the encoder can
be reduced. The key lies in synchronizing the design of the deadbeat
controller gain and the transmission period of the C-A channel, in the use of
the system controllability index. It is worth remarking that, under the
proposed deadbeat controller, the self-triggering mechanism for linear systems
with quantized outputs is as simple as that with quantized states in Wakaiki
(2022). To account for DoS attacks, a resilient self-triggered control scheme
with quantized output is put forth, for which exponential system stability is
established under certain conditions on the DoS duration time. We also reveal
an intriguing trade-off between the maximum inter-sampling time and the system
resilience in the presence of DoS attacks. As the self-triggering parameter
decreases to yield the minimum inter-sampling time, the DoS condition in this
paper coincides with that for the time-triggered sampling in Wakaiki et al.
(2020).

In succinct form, the main contributions of this work are summarized as
follows.

  * c1)

To reduce the communication cost, an output encoding scheme and a self-
triggering mechanism are co-developed for LTI systems by employing a standard
observer-based control law;

  * c2)

When a higher communication rate can be afforded at the C-A channel, a
delicate observer-based deadbeat control scheme significantly simplifies the
self-triggering design, which is as simple as that for systems with quantized
state feedback control in the literature; and,

  * c3)

Finally, the proposed self-triggered control and the encoding scheme are
further made resilient against DoS attacks, and a trade-off between the
maximum inter-sampling time and the system resilience is unraveled.

Denote the set of real numbers by $\mathbb{R}$. Given $\alpha\in\mathbb{R}$,
let $\mathbb{R}_{>\alpha}$ ($\mathbb{R}_{\geq\alpha}$) denote the set of real
numbers greater than (greater than or equal to) $\alpha$. Let $\mathbb{N}$
denote the set of natural numbers and define
$\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$. For a vector
$v=[v_{1},v_{2},\cdots\\!,v_{n}]^{T}\in\mathbb{R}^{n}$, denote its maximum
norm by $\|v\|_{\infty}:=\max\\{|v_{1}|,\cdots\\!,|v_{n}|\\}$ and the
corresponding induced norm of a matrix $M\in\mathbb{R}^{m\times n}$ by
$\|M\|_{\infty}:=\sup\\{\|Mv\|_{\infty}:v\in\mathbb{R}^{n},\|v\|_{\infty}=1\\}$.

##  2 Preliminaries and Problem Formulation

In this paper, we study the networked control architecture in Fig. 1, where a
plant is to be stabilized based on quantized output transmitted over a
bandwidth-limited network. The plant is described by the following linear
discrete-time dynamics

$\displaystyle x_{s+1}$ $\displaystyle=Ax_{s}+Bu_{s}$ (1a) $\displaystyle
y_{s}$ $\displaystyle=Cx_{s}$ (1b)

where $x_{s}\in\mathbb{R}^{n_{x}},u_{s}\in\mathbb{R}^{n_{u}}$, and
$y_{s}\in\mathbb{R}^{n_{y}}$ denote the state, control input, and output,
respectively. Here, the plant is connected to a sensor that samples the output
$y_{s}$ at time instants
$\\{s_{\ell}\\}_{\ell\in\mathbb{N}_{0}}\subseteq\\{s\\}_{s\in\mathbb{N}_{0}}$
dictated by a self-triggering mechanism which we design later. The sampled
output $y_{s_{\ell}}$ then passes through an encoder and gets quantized before
sent to the self-triggering module and the controller. Using the quantized
output, the self-triggering module then calculates the next sampling time and
sends it back to the encoder side.

Figure 1: Closed-loop system with self-triggered control.

We consider in this paper that the C-A channel is ideal with sufficient
bandwidth and low communication cost. In other words, neither quantized nor
self-triggered communication is required for the C-A channel. We adopt a
standard observer-based state feedback controller that receives quantized
output measurements from the network, and generates as well as transmits
control inputs to a zero-order holder (ZOH) before entering the plant.

Before moving on, we make the following standard assumptions on the system
(1).

######  Assumption 2.1 (_Controllability and observability_).

The pair $(A,B)$ is controllable, and the pair $(C,A)$ is observable.

######  Assumption 2.2 (_Initial state bound_).

An upper bound on the initial state $\|x_{0}\|_{\infty}\leq E_{in}$ is known.

######  Remark 2.1.

In fact, there are a number of ways for obtaining upper bounds on the initial
state, including, e.g., the zooming-out method in Liberzon (2003).

For clarity, this paper investigates the following closed-loop stability, as
also used in Wakaiki (2022).

######  Definition 2.1.

The discrete-time system (1) is exponentially stable under Assumptions 2.1—2.2
if there exist constants $\Omega\geq 1$ and $\omega\in(0,1)$ independent of
$E_{in}$, such that $\|x_{s}\|_{\infty}\leq\Omega E_{in}\omega^{s}$ holds for
all $s\in\mathbb{N}_{0}$

In addition, we also call for the following norm introduced in Wakaiki (2022)
for stability analysis of event-triggered control systems.

######  Lemma 2.1 (Wakaiki (2022),Lem. 3.2).

If there exists a matrix $G\\!\in\\!\mathbb{R}^{n_{x}\\!\times n_{x}}$,
constants $\Gamma>1$, and $\gamma>0$ such that
$\|G^{s}\|_{\infty}\leq\Gamma\gamma^{s}$ holds for all $s\in\mathbb{N}_{0}$,
then the function $\|\cdot\|_{G}:\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}$
defined by
$x\mapsto\|x\|_{G}:=\sup_{s\in\mathbb{N}_{0}}\|\gamma^{-s}G^{s}x\|_{\infty}$
is a norm on $\mathbb{R}^{n_{x}}$. Moreover, for every
$x\in\mathbb{R}^{n_{x}}$ and $s\in\mathbb{N}_{0}$, it holds that
$\|x\|_{\infty}\leq\|x\|_{G}\leq\Gamma\|x\|_{\infty},~{}{\rm
and}~{}\|G^{s}x\|_{G}\leq\gamma^{s}\|x\|_{G}$.

##  3 Self-triggered Control: DoS-free Case

For stabilization of linear systems with output available only, it is common
to adopt an observer-based feedback control law, namely using an observer to
estimate the state and then using the estimated state to construct a state
feedback controller. In addition, to compensate for the coarse transmission
caused by the limited bandwidth, a dynamic quantizer is equipped such that the
real output is quantized before transmitted. In this section, we begin by
developing a self-triggered observer-based state feedback controller for
linear systems with quantized output. When a faster transmission rate can be
afforded at the C-A channel, e.g., $b\in\mathbb{N}_{0}$ time steps at the S-C
channel equal one time step at the S-C channel, an observer-based deadbeat
controller is proposed to simplify the self-triggering mechanism while further
saving the communication resources.

###  3.1 Standard Observer-based Control

The commonly used standard observer-based controller (see e.g., O’Reilly
(1983)) is described as follows

$\displaystyle\hat{x}_{s+1}$
$\displaystyle=A\hat{x}_{s}+Bu_{s}+L[\mathcal{Q}(y_{s_{\ell}})-\hat{y}_{s}],$
(2a) $\displaystyle\hat{y}_{s}$ $\displaystyle=C\hat{x}_{s}$ (2b)
$\displaystyle u_{s}$ $\displaystyle=K\hat{x}_{s}$ (2c)

with the initial condition $\hat{x}_{0}=0$, where $\mathcal{Q}(y_{s_{\ell}})$
denotes the quantized output received from the network at previous self-
triggered time $s_{\ell}$ (which is the most recent measurement at the
controller side). The gain matrices $L$ and $K$ are determined such that
$A-LC$ and $A+BK$ are Schur stable. Equations (2a) and (2b) constitute the
standard Luenberger observer, and (2c) follows a feedback control law based on
the estimated state $\hat{x}_{s}$. The rest of this subsection is devoted to
designing a self-triggering mechanism as well as associated encoding scheme
for a dynamic quantizer.

Let $e^{x}_{s}:=x_{s}-\hat{x}_{s}$ denote the error between the estimated
state and the actual state. Suppose we can construct a sequence
$\\{E^{x}_{s_{\ell}}\geq 0\\}_{\ell\in\mathbb{N}_{0}}$ satisfying

$\|e^{x}_{s_{\ell}}\|_{\infty}\leq
E^{x}_{s_{\ell}},\quad\forall\ell\in\mathbb{N}_{0}.$ (3)

It follows from the definition of the induced $\infty$-norm that

$\|y_{s_{\ell}}-\hat{y}_{s_{\ell}}\|_{\infty}=\|Ce^{x}_{s_{\ell}}\|_{\infty}\leq\|C\|_{\infty}E^{x}_{s_{\ell}}\triangleq
E_{s_{\ell}}.$ (4)

Consider the quantizer has e.g., $N$ quantization levels. We take the
quantization range to be $E_{s_{\ell}}$, and the quantization center to be
$\hat{y}_{s_{\ell}}$ that can be provided by running in parallel (2) at the
encoder side. The dynamic output quantizer is presented as follows.

At each self-triggering time $s_{\ell}$, the encoder partitions the hypercube
$\\{y\in\mathbb{R}^{n_{y}}:\|y_{s_{\ell}}-\hat{y}_{s_{\ell}}\|_{\infty}\leq
E_{s_{\ell}}\\}$ into $N^{n_{y}}$ equal-sized boxes, and uses a value in
$\\{1,2,\cdots\\!,$ $N^{n_{y}}\\}$ to one-to-one index each of those boxes.
Let $\mathcal{Q}_{s_{\ell}}^{\rm ind}$ denote the partitioned box(es)
containing $y_{s_{\ell}}$, which is sent to the decoder. If $y_{s_{\ell}}$ is
on the boundary of multiple boxes, then any of the corresponding indices of
these boxes can be used. The decoder receives the index
$\mathcal{Q}_{s_{\ell}}^{\rm ind}$, and can easily recover the quantized
output $\mathcal{Q}(y_{s_{\ell}})$ thanks to the one-to-one correspondence.
Therefore, the quantization error of $y_{s_{\ell}}$ can be bounded by

$\|y_{s_{\ell}}-\mathcal{Q}(y_{s_{\ell}})\|_{\infty}\leq\frac{\|C\|_{\infty}}{N}E^{x}_{s_{\ell}}=\frac{E_{s_{\ell}}}{N}.$
(5)

####  3.1.1 Self-triggering scheme

In this part, we introduce a self-triggering mechanism that determines the
sampling times $\\{s_{\ell}\\}_{\ell\in\mathbb{N}_{0}}$ based on the quantized
output $\mathcal{Q}(y_{s_{\ell}})$.

Similarly to the event-triggered state feedback control in Heemels (2013), a
general self-triggering mechanism computes the inter-sampling time
$\tau_{\ell}:=s_{\ell+1}-s_{\ell}$ by constructing a function measuring some
sort of deviation between the current output $y_{s}$ and the most recently
sampled output $y_{s_{\ell}}$. However, due to the limited datarate, only the
quantized output $\mathcal{Q}(y_{s_{\ell}})$ is available. Therefore, we
consider the following self-triggering mechanism

$\bigg{\\{}\begin{aligned}
&s_{\ell+1}\\!:=\\!s_{\ell}+\min\\{\tau_{\max},\tau_{\ell}\\},~{}~{}\ell\in\mathbb{N}_{0},~{}~{}s_{0}:=0\\\
&\tau_{\ell}\\!:=\\!\min\\!\big{\\{}\tau\in\mathbb{N}:g(\mathcal{Q}(y_{s_{\ell}}),E_{s_{\ell}},\hat{x}_{s_{\ell}},\tau)\\!>\\!\sigma
E_{s_{\ell}}\big{\\}}\end{aligned}$ (6)

where $\sigma>0$ is a threshold, and $\tau_{\max}$ is an upper bound on the
inter-sampling times $s_{\ell+1}-s_{\ell}$ for all $\ell\in\mathbb{N}_{0}$.
The function $g$ measures the ‘informativeness’ of the current output $y_{s}$
relative to the previous quantized output $\mathcal{Q}(y_{s_{\ell}})$.
Obviously, if the value of function $g$ is large enough, a new triggering time
occur, and $y_{s}$ will be quantized and sent to the controller. The next
lemma provides a way to construct such a triggering function.

######  Lemma 3.1.

Let Assumptions 2.1—2.2 hold. Consider system (1) adopting the controller (2)
and a quantizer such that condition (4) is satisfied with a sequence
$\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ for all $\ell\in\mathbb{N}_{0}$.
If the self-triggering function $g$ in (6) is given by

$\displaystyle
g(q,E,\hat{x},\tau):=\Big{(}\frac{\|C(A^{\tau}-I)\|_{\infty}}{\|C\|_{\infty}}+\frac{1}{N}\Big{)}E$
(7)
$\displaystyle+\Big{\|}C\Big{(}\\!(A^{\tau}-I)\\!+\\!\sum_{i=0}^{\tau-1}A^{i}BK(A\\!+\\!BK\\!-\\!LC)^{\tau-i-1}\Big{)}\hat{x}$
$\displaystyle+C\sum_{i=0}^{\tau-1}A^{i}BK\sum_{j=0}^{\tau-j-2}(A\\!+\\!BK\\!-\\!LC)^{i}Lq\Big{\|}_{\infty},\tau\in\mathbb{N}$

then the output error $\|\mathcal{Q}(y_{s_{\ell}})-y_{s}\|_{\infty}$ satisfies

$\|\mathcal{Q}(y_{s_{\ell}})-y_{s}\|_{\infty}\leq\sigma
E_{{s_{\ell}}},\quad\forall s\in[s_{\ell},s_{\ell+1}).$ (8)

We start the proof by deriving an upper bound on the output quantization error
$\|\mathcal{Q}(y_{s_{\ell}})-y_{s}\|_{\infty}$. It follows from (2) that
$\hat{x}_{s_{\ell}+1}=(A+BK-LC)\hat{x}_{s_{\ell}}+L\mathcal{Q}(y_{s_{\ell}})$.
Let an integer $p$ satisfy $p\in[1,s_{\ell+1}-s_{\ell})$. It can be
recursively deduced that

$\displaystyle\hat{x}_{s_{\ell}+p}$ $\displaystyle=(A+BK-
LC)^{p}\hat{x}_{s_{\ell}}$ $\displaystyle\quad+\sum_{i=0}^{p-1}(A+BK-
LC)^{i}L\mathcal{Q}(y_{s_{\ell}}).$ (9)

From (1), one gets that

$\displaystyle
x_{s_{\ell}+p}=A^{p}x_{s_{\ell}}+\sum_{i=0}^{p-1}A^{i}BK\hat{x}_{s_{\ell}+p-i-1}.$
(10)

Substituting (3.1.1) into (10), we arrive at

$\displaystyle x_{s_{\ell}+p}$
$\displaystyle=A^{p}x_{s_{\ell}}\\!+\\!\sum_{i=0}^{p-1}A^{i}BK\big{[}(A\\!+\\!BK\\!-\\!LC)^{p-i-1}\hat{x}_{s_{\ell}}$
$\displaystyle\quad+\sum_{j=0}^{p-i-2}(A\\!+\\!BK\\!-\\!LC)^{j}L\mathcal{Q}(y_{s_{\ell}})\big{]}$
$\displaystyle=A^{p}x_{s_{\ell}}\\!+\\!\sum_{i=0}^{p-1}A^{i}BK(A\\!+\\!BK\\!-\\!LC)^{p-i-1}\hat{x}_{s_{\ell}}$
$\displaystyle\quad+\\!\\!\sum_{i=0}^{p-1}\\!\\!A^{i}BK\\!\\!\\!\sum_{j=0}^{p-i-2}\\!\\!\\!(\\!A\\!+\\!BK\\!-\\!LC)^{j}\\!L\mathcal{Q}(y_{s_{\ell}}).$
(11)

Moreover,

$\displaystyle x_{s_{\ell}+p}-x_{s_{\ell}}$
$\displaystyle=(A^{p}-I)(x_{s_{\ell}}-\hat{x}_{s_{\ell}})+\Big{[}(A^{p}-I)$
$\displaystyle\quad+\sum_{i=0}^{p-1}A^{i}BK(A+BK-
LC)^{p-i-1}\Big{]}\hat{x}_{s_{\ell}}$
$\displaystyle\quad+\\!\sum_{i=0}^{p-1}\\!A^{i}BK\\!\\!\\!\sum_{j=0}^{p-i-2}\\!\\!\\!(A\\!+\\!BK\\!-\\!LC)^{j}L\mathcal{Q}(y_{s_{\ell}}).$

Therefore, the output error satisfies

$\displaystyle\quad~{}\|y_{s_{\ell}+p}-\mathcal{Q}(y_{s_{\ell}})\|_{\infty}$
(12a)
$\displaystyle=\|C(x_{s_{\ell}+p}-x_{s_{\ell}})+y_{s_{\ell}}-\mathcal{Q}(y_{s_{\ell}})\|_{\infty}$
(12b)
$\displaystyle\leq\Big{(}\frac{\|C(A^{p}-I)\|_{\infty}}{\|C\|_{\infty}}+\frac{1}{N}\Big{)}E_{s_{\ell}}$
(12c)
$\displaystyle\quad+\Big{\|}C\Big{(}\\!A^{p}\\!-\\!I\\!+\\!\sum_{i=0}^{p-1}\\!A^{i}BK(A\\!+\\!BK\\!-\\!LC)^{p-i-1}\Big{)}\hat{x}_{s_{\ell}}$
$\displaystyle\quad+C\sum_{i=0}^{p-1}A^{i}BK\\!\\!\sum_{j=0}^{p-j-2}\\!\\!(A\\!+BK\\!-LC)^{i}L\mathcal{Q}(y_{s_{\ell}})\Big{\|}_{\infty}$

where (12c) is derived by substituting (4) and (5) into (12b). Furthermore,
according to (7), it can be deduced that
$\|\mathcal{Q}(y_{s_{\ell}})-y_{s}\|_{\infty}\leq
g(\mathcal{Q}(y_{s_{\ell}}),E_{s_{\ell}},\hat{x}_{s_{\ell}},p)$. Finally,
noticing from (6) that
$g(\mathcal{Q}(y_{s_{\ell}}),E_{s_{\ell}},\hat{x}_{s_{\ell}},p)\leq\sigma
E_{s_{\ell}}$, we have that $\|\mathcal{Q}(y_{s_{\ell}})-y_{s}\|_{\infty}\leq
g(\mathcal{Q}(y_{s_{\ell}}),E_{s_{\ell}},\hat{x}_{s_{\ell}},p)\leq\sigma
E_{s_{\ell}}$, thus completing the proof.

####  3.1.2 Encoding scheme

When adopting a dynamic quantizer, the quantization center
$\hat{y}_{s_{\ell}}$ and the range $E_{s_{\ell}}$ should evolve with time to
ensure that the hypercube always contains the actual output $y_{s_{\ell}}$.
Recall that the center $\hat{y}_{s_{\ell}}$ is obtained by running the
controller (2) at the encoder side in parallel. We here design an update rule
for the sequence $\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ such that (4) is
satisfied.

Since matrix $L$ is chosen such that $A_{cl}:=A-LC$ is Schur stable, there
exist constants $\Gamma\geq 1$ and $\gamma\in(0,1)$ such that

$\displaystyle\|(A-LC)^{s}\|_{\infty}\leq\Gamma\gamma^{s},\quad\forall
s\in\mathbb{N}_{0}.$ (13)

Choosing $\alpha:={\Gamma\|L\|_{\infty}\|C\|_{\infty}}/(1-\gamma)$, we define
$\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ as follows

$\begin{cases}E_{s_{\ell}}:=\|C\|_{\infty}E^{x}_{s_{\ell}}\\\
E^{x}_{0}:=\Gamma E_{in}\\\
E^{x}_{s_{\ell+1}}\\!:=\\!(\gamma^{s_{\ell+1}\\!-\\!s_{\ell}}(1-\alpha\sigma)\\!+\\!\alpha\sigma)E^{x}_{s_{\ell}},~{}\forall\ell\\!\in\\!\mathbb{N}_{0}\end{cases}$
(14)

where $\sigma$ is the threshold parameter in (6).

The following result proves that the output quantization error is upper-
bounded by the sequence (14), and it converges to the origin exponentially
fast.

######  Lemma 3.2.

Let Assumptions 2.1—2.2 hold. Consider system (1) with the standard controller
(2). Suppose that the sampling times $\\{s_{\ell}\\}_{\ell\in\mathbb{N}_{0}}$
are generated by (6) with i) the self-triggering function $g$ in (7), ii) the
sequence $\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ in (14), and iii) the
threshold $\sigma>0$ satisfying

$\frac{1}{N}\leq\sigma<\frac{1}{\alpha}=\frac{1-\gamma}{\Gamma\|L\|_{\infty}\|C\|_{\infty}}.$
(15)

Then, the following statements hold:

  * s1)

$\|x_{s_{\ell}+p}-\hat{x}_{s_{\ell}+p}\|_{A_{cl}}\leq(\gamma^{p}(1-\alpha\sigma)+\alpha\sigma)E^{x}_{s_{\ell}}$,
where $p\in[0,s_{\ell+1}-s_{\ell}]$;

  * s2)

For $\tau\in\mathbb{R}_{\geq 0}$, function
$\left(\gamma^{\tau}(1-\alpha\sigma)+\alpha\sigma\right)^{1/\tau}$ is strictly
increasing, and
$\tau_{\max}=\underset{1\leq\tau\leq\tau_{\max}}{\operatorname{argmax}}\left(\gamma^{\tau}(1-\delta\sigma)+\delta\sigma\right)^{1/\tau}$;

  * s3)

$\|y_{s_{\ell}}-\hat{y}_{s_{\ell}}\|_{\infty}\leq
E_{s_{\ell}}\leq\Gamma_{1}E_{in}\omega_{1}^{s_{\ell}}$ where
$\ell\in\mathbb{N}_{0}$,
$\omega_{1}:=(\gamma^{\tau_{\max}}(1-\alpha\sigma)+\alpha\sigma)^{1/\tau_{\max}}<1$,
and $\Gamma_{1}:=\|C\|_{\infty}\Gamma$. In particular, the sequence
$\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ converges exponentially to the
origin.

We begin by proving statement (s1). Based on the norm defined in Lem. 2.1,
$e^{x}_{s_{\ell}}=x_{s_{\ell}}-\hat{x}_{s_{\ell}}$ satisfies

$\displaystyle\|e^{x}_{s_{\ell}}\|_{\infty}\leq\|e^{x}_{s_{\ell}}\|_{A_{cl}}\leq\Gamma\|e^{x}_{s_{\ell}}\|_{\infty}$
(16)

and

$\|A_{cl}^{q}e^{x}_{s_{\ell}}\|_{A_{cl}}\leq\gamma^{q}\|e^{x}_{s_{\ell}}\|_{A_{cl}},~{}\forall
q\in\mathbb{N}_{0}.$ (17)

According to Assumption 2.2 and (14), we have that
$\|x_{0}\|_{A_{cl}}\leq\Gamma\|x_{0}\|_{\infty}=E^{x}_{0}$. Supposing that
$\|e^{x}_{s_{\ell}}\|_{A_{cl}}\leq E^{x}_{\ell}$ holds for $s_{\ell}$, we use
induction to prove that it holds for $s_{\ell+1}$. It follows from (1) and (2)
that
$x_{s_{\ell}+1}-\hat{x}_{s_{\ell}+1}=A_{cl}(x_{s_{\ell}}-\hat{x}_{s_{\ell}})-L(\mathcal{Q}(y_{s_{\ell}})-\hat{y}_{s_{\ell}})$.
Let $p\in[0,s_{\ell+1}-s_{\ell}]$. Recursively, it can be deduced that

$\displaystyle\quad~{}\|x_{s_{\ell}+p}-\hat{x}_{s_{\ell}+p}\|_{A_{cl}}$
$\displaystyle=\Big{\|}A_{cl}^{p}(x_{s_{\ell}}\\!-\\!\hat{x}_{s_{\ell}})\\!-\\!\sum_{i=0}^{p-1}A_{cl}^{i}L(\mathcal{Q}(y_{s_{\ell}})\\!-\\!y_{s_{\ell}+p-i-1})\Big{\|}_{A_{cl}}$
$\displaystyle\leq\gamma^{p}\|x_{s_{\ell}}-\hat{x}_{s_{\ell}}\|_{A_{cl}}$
$\displaystyle~{}~{}~{}+\Gamma\sum_{i=0}^{p-1}\|A_{cl}^{i}L\|_{\infty}\|\mathcal{Q}(y_{s_{\ell}})-y_{s_{\ell}+p-i-1}\|_{\infty}$
(18a)
$\displaystyle\leq\gamma^{p}E^{x}_{s_{\ell}}+\sigma\Gamma\|L\|_{\infty}\|C\|_{\infty}\sum_{i=0}^{p-1}\gamma^{i}E^{x}_{s_{\ell}}$
(18b)
$\displaystyle\leq(\gamma^{p}(1-\alpha\sigma)+\alpha\sigma)E^{x}_{s_{\ell}}$
(18c)

where inequality (18a) follows from (16) and (17). According to Lem. 3.1, we
arrive at (18b). When $p=s_{\ell+1}-s_{\ell}$, inequality
$\|x_{s_{\ell+1}}-\hat{x}_{s_{\ell+1}}\|_{A_{cl}}\leq E^{x}_{s_{\ell+1}}$
arises by substituting (14) into (18c). Therefore,
$\|e^{x}_{s_{\ell}}\|_{A_{cl}}\leq E^{x}_{s_{\ell}}$ holds for all
$\ell\in\mathbb{N}_{0}$. Noticing that
$\|e^{x}_{s_{\ell}}\|_{\infty}\leq\|e^{x}_{s_{\ell}}\|_{A_{cl}}$, hence
$\|e^{x}_{s_{\ell}}\|_{\infty}\leq E^{x}_{s_{\ell}}$ holds for all
$\ell\in\mathbb{N}_{0}$.

Statement (s2) can be derived from the proof of (Wakaiki, 2022, Thm. 3.5) and
is thus omitted here.

Next, since
$\|y_{s_{\ell}}-\hat{y}_{s_{\ell}}\|_{\infty}=\|C(x_{s_{\ell}}-\hat{x}_{s_{\ell}})\|_{\infty}=\|Ce^{x}_{s_{\ell}}\|_{\infty}$
and $E_{s_{\ell}}=\|C\|_{\infty}E^{x}_{s_{\ell}}$, it can be deduced that
$\|y_{s_{\ell}}-\hat{y}_{s_{\ell}}\|_{\infty}\leq E_{s_{\ell}}$ is met. In
addition, although $\tau\in\mathbb{N}$ in (6), the self-triggering function
$g$ should always satisfy
$g(\mathcal{Q}(y_{s_{\ell}}),E_{s_{\ell}},\hat{x}_{s_{\ell}},0)=(1/N)E_{s_{\ell}}\leq\sigma
E_{s_{\ell}}$; otherwise,
$g(\mathcal{Q}(y_{s_{\ell}}),E_{s_{\ell}},\hat{x}_{s_{\ell}},\tau)\leq\sigma
E_{s_{\ell}}$ does not hold even for $\tau=1$. Therefore, parameter $\sigma$
is chosen such that $\sigma\geq 1/N$. Based on (14), it follows that

$\displaystyle E_{s_{\ell+1}}$ $\displaystyle=\|C\|_{\infty}\Gamma
E_{in}\prod_{i=0}^{\ell}[\gamma^{s_{i+1}-s_{i}}(1-\alpha\sigma)+\alpha\sigma]$
$\displaystyle\leq\Gamma_{1}E_{in}\omega_{1}^{s_{\ell+1}}$ (19)

where
$\omega_{1}:=(\gamma^{\tau_{\max}}(1-\alpha\sigma)+\alpha\sigma)^{1/\tau_{\max}}$
and $\Gamma_{1}:=\|C\|_{\infty}\Gamma$. It follows from (15) that
$\alpha\sigma<1$, and since $\gamma\in(0,1)$, it can be deduced that
$\gamma^{\tau}(1-\alpha\sigma)+\alpha\sigma<\gamma^{0}(1-\alpha\sigma)+\alpha\sigma=1$
Therefore, $\omega_{1}<1$, and it follows from (19) that sequence
$\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ converges to the origin, which
completes the proof. Based on Lems. 3.1 and 3.2, the stability condition is
now ready to be presented.

######  Theorem 3.1.

Consider system (1) with the controller (2), where the matrices $L$ and $K$
are chosen such that $A-LC$ and $A+BK$ are Schur stable. Under Assumptions
2.1—2.2, if the sampling times $\\{s_{\ell}\\}_{\ell\in\mathbb{N}_{0}}$ are
generated by (6) with i) the self-triggering function $g$ in (7), ii) the
sequence $\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ in (14), and iii) the
threshold parameter $\sigma>0$ satisfying (15), then the system (1) is
exponentially stable. In particular, the decay parameter
$\omega\in(\hat{\omega},1)$ satisfies Def. 2.1 for some constant $\Omega>0$
where $\hat{\omega}>\max\\{\omega_{1},\omega_{2}\\}$ with $\omega_{2}\in(0,1)$
obeying $\|(A+BK)^{s}\|_{\infty}\geq\Omega_{K}\omega_{2}^{s}$ for some
constant $\Omega_{K}>0$ and all $s\in\mathbb{N}_{0}$.

It follows from (1a) that

$\displaystyle\|x_{s_{\ell}}\|_{\infty}$
$\displaystyle=\Big{\|}(A\\!+\\!BK)^{s_{\ell}}x_{0}$ (20a)
$\displaystyle~{}~{}~{}-\sum_{i=0}^{s_{\ell}-1}(A\\!+\\!BK)^{s_{\ell}-i-1}BKe^{x}_{i}\Big{\|}_{\infty}$
$\displaystyle\leq\|(A\\!+\\!BK)^{s_{\ell}}\|_{\infty}\|x_{0}\|_{\infty}$
(20b)
$\displaystyle~{}~{}~{}+\sum_{i=0}^{s_{\ell}-1}\|(A\\!+\\!BK)^{s_{\ell}-i-1}BK\|_{\infty}\|e^{x}_{i}\|_{\infty}.$

Since $A+BK$ is Schur stable, there exist constants $\Omega_{K}>0$, and
$\omega_{2}\in(0,1)$ such that
$\|(A+BK)^{s}\|_{\infty}\leq\Omega_{K}\omega_{2}^{s}$ for all
$s\in\mathbb{N}_{0}$. In addition, noticing that $\gamma\in(0,1)$, it can be
deduced from the statement (s1) of Lem. 3.2 that for all
$\ell\in\mathbb{N}_{0}$, and $p\in[0,s_{\ell+1}-s_{\ell})$

$\|e^{x}_{s_{\ell}+p}\|_{\infty}\leq
E^{x}_{s_{\ell}}\leq\omega_{1}^{-\tau_{\max}}\Gamma
E_{in}\omega_{1}^{s_{\ell}+p}.$

Hence,

$\displaystyle\sum_{i=0}^{s_{\ell}-1}\|(A+BK)^{s_{\ell}-i-1}BK\|_{\infty}\|e^{x}_{i}\|_{\infty}$
$\displaystyle\leq\sum_{i=0}^{s_{\ell}-1}\\!\big{(}\|BK\|_{\infty}\Omega_{K}\omega_{2}^{s_{\ell}-i-1}\big{)}\\!\cdot\\!\big{(}\omega_{1}^{-\tau_{\max}}\Gamma
E_{in}\omega_{1}^{i}\big{)}$ (21a)
$\displaystyle\leq\big{(}\underbrace{\|BK\|_{\infty}\Omega_{K}\omega_{1}^{-\tau_{\max}}\Gamma}_{\stackrel{{\scriptstyle\triangle}}{{=}}\Omega_{1}}E_{in}\big{)}\sum_{i=0}^{s_{\ell}-1}\omega_{2}^{s_{\ell}-i-1}\omega_{1}^{i}$
(21b)
$\displaystyle\leq\Omega_{1}E_{in}\sum_{i=0}^{s_{\ell}-1}\omega^{s_{\ell}-1},\quad\omega:=\max\\{\omega_{1},\omega_{2}\\},$
(21c) $\displaystyle\leq\Omega_{1}E_{in}(s_{\ell}\omega^{s_{\ell}-1}).$ (21d)

Notice that for all $\hat{\omega}>\omega$, there exists a constant $c_{1}>0$
such that $s\omega^{s}\leq c_{1}\hat{\omega}^{s}$ for all
$s\in\mathbb{N}_{0}$. Moreover,

$\|(A\\!+\\!BK)^{s_{\ell}}\|_{\infty}\|x_{0}\|_{\infty}\leq\Omega_{K}E_{in}\omega^{s_{\ell}}$
(22)

thus we have from (20) that

$\|x_{s_{\ell}}\|_{\infty}\leq\hat{\Omega}_{1}E_{in}\hat{\omega}^{s_{\ell}}$
(23)

for some $\hat{\Omega}_{1}>0$.

Having shown the exponential convergence for $s=s_{\ell}$, we next extend (23)
to all $s\in\mathbb{N}_{0}$. It can be deduced from (11) that, for all
$\ell\in\mathbb{N}_{0}$ and $p\in[0,s_{\ell+1}-s_{\ell})$

$\|x_{s_{\ell}+p}\|_{\infty}\leq
c_{2}\|x_{s_{\ell}}\|_{\infty}+c_{3}\|\hat{x}_{s_{\ell}}\|_{\infty}+c_{4}\|\mathcal{Q}(y_{s_{\ell}})\|_{\infty}$
(24)

where constants $c_{2},c_{3},c_{4}>0$ are finite because
$s_{\ell+1}-s_{\ell}\leq\tau_{\max}$. Moreover, since

$\displaystyle\|\mathcal{Q}(y_{s_{\ell}})\|_{\infty}$ $\displaystyle\leq
E_{s_{\ell}}+\|C\|_{\infty}\|\hat{x}_{s_{\ell}}\|_{\infty}$
$\displaystyle\|\hat{x}_{s_{\ell}}\|$
$\displaystyle\leq\|x_{s_{\ell}}\|_{\infty}+\|e^{x}_{s_{\ell}}\|_{\infty}$

there exists $c_{5}>0$ such that

$\|x_{s_{\ell}+p}\|_{\infty}\\!\\!\leq\\!c_{5}E_{in}\hat{\omega}^{s_{\ell}}\\!=\\!\\!\big{(}c_{5}\hat{\omega}^{\\!-\\!\tau_{\max}}\\!\big{)}\\!E_{in}\hat{\omega}^{s_{\ell}+p}\\!=\\!\hat{\Omega}_{2}E_{in}\hat{\omega}^{s_{\ell}+p}$

for all $\ell\\!\in\\!\mathbb{N}_{0}$ and
$p\\!\in\\![0,s_{\ell+1}\\!-\\!s_{\ell})$. Hence, based on Def. 2.1, system
(1) converges exponentially to the origin.

######  Remark 3.1 (_Computational resource_).

Although (2) achieves exponential stabilization for the self-triggered control
system (1) using quantized output, the quantizer requires running the
controller (2) at the encoder side. In addition, the self-triggering module
using the estimated state $\hat{x}$ also needs to run the controller (2). This
certainly challenges the practical implementation and increases the
computational burden. As a remedy, one can simply choose the origin to be the
quantization center. In that case, it has been shown that computational
savings can be achieved by sacrificing the convergence rate in Wakaiki et al.
(2020). Specifically, the origin quantization center may decrease the accuracy
of the quantized data under the same data rate. In addition, coarser
quantization may lead to smaller inter-sampling times, and consequently
increase the communication overhead. To balance between the computational and
communication loads as well as maintain a fast convergence rate, a deadbeat
control scheme is advocated in the ensuing section.

###  3.2 Deadbeat Control

In the previous section, the observer-based state feedback controller should
be copied at both the encoder side and the self-triggering module, which
inevitably increases the computational overload. Moreover, compared with the
self-triggering mechanism suggested for quantized state feedback control in
Wakaiki (2022), condition (7) entails an additional term, namely the estimated
state $\hat{x}$. Involving $\hat{x}$ in the self-triggering condition appears
natural, because only quantized output is available here. Nonetheless, we next
show that this information can be eliminated from the condition to facilitate
implementation of the self-triggering mechanism, by means of a simple deadbeat
control protocol design.

We assume in this section that the C-A channel can afford a higher
communication rate. Specially, the discrete-time system in (1) is replaced by

$\displaystyle x_{s,k+1}$ $\displaystyle=\tilde{A}x_{s,k}+\tilde{B}u_{s,k}$
(25a) $\displaystyle y_{s,k}$ $\displaystyle=Cx_{s,k}$ (25b)

where $s\in\mathbb{N}_{0}$ denotes the time index of the S-C channel (S-C time
step), and $k=0,\,1,\,\cdots\\!,\,\eta-1$ is the index of the time step at the
C-A channel (C-A time step) during one S-C time step; see Fig. 2 for an
illustration. The integer $\eta$ is the controllability index of system (25),
which can be obtained by computing the smallest integer $\eta$ such that ${\rm
rank}\big{[}\tilde{B},\,\tilde{A}\tilde{B},\cdots,\tilde{A}^{\eta}\tilde{B}\big{]}=n_{x}$.
The relationship between matrices $A$, $B$, and $\tilde{A}$, $\tilde{B}$ are
$A=\tilde{A}^{\eta}$, and $B=\sum_{i=0}^{\eta-1}\tilde{A}^{i}\tilde{B}$. For
simplicity, we use $x_{s}$ to denote $x_{s,0}$, and the same for
$\hat{x}_{s}$, $\hat{y}_{s}$ and $y_{s}$. In addition, assuming the lengths of
one time step of the S-C channel and the C-A channel are $\Delta$ and
$\delta$, respectively, we work with

######  Remark 3.2.

In the sampling-data setting, the discrete-time system matrices are calculated
using the continuous-time system matrices and the sampling period, i.e.,
$\Delta$ and $\delta$; see, e.g., Remark 2.4 in Wakaiki (2022). Hence, it is
easy to construct matrices $\tilde{A}$ and $\tilde{B}$ such that
$A=\tilde{A}^{\eta}$ and $B=\sum_{i=0}^{\eta-1}\tilde{A}^{i}\tilde{B}$ hold
true.

Figure 2: Relationship between the C-A time step and the S-C time step with
$\eta=3$.

Similar to Assumption 2.1, matrices $\tilde{A}$, $\tilde{B}$, and $C$ obey the
following assumption.

######  Assumption 3.1 (_Controllability and observability_).

The pairs $(\tilde{A},\tilde{B})$ and $(C,\tilde{A}^{\eta})$ of the linear
discrete-time system (25) are controllable and observable.

The controller (2) is now replaced by the observer-based deadbeat controller

$\displaystyle\hat{x}_{s,k+1}=\tilde{A}\hat{x}_{s,k}+\tilde{B}u_{s,k},$
$\displaystyle k$ $\displaystyle\leq\eta-1$ (26a)
$\displaystyle\hat{x}_{s}=\hat{x}_{s-1,\eta}+M[\mathcal{Q}(y_{s_{\ell}})-\hat{y}_{s-1,\eta}],$
(26b) $\displaystyle\hat{y}_{s,k}=C\hat{x}_{s,k}$ (26c) $\displaystyle
u_{s,k}=K\hat{x}_{s,k}$ (26d)

with $\hat{x}_{0}=0$. The control input is generated by (26d). Matrix
$M\in\mathbb{R}^{n_{x}\times n_{y}}$ can be seen as the observer gain such
that $\bar{A}_{cl}:=\tilde{A}^{\eta}(I-MC)$ is Schur stable, which is always
feasible for observable systems $(C,\tilde{A}^{\eta})$. In particular, one can
choose suitable $\bar{M}:=\tilde{A}^{\eta}M$ such that
$\tilde{A}^{\eta}-\bar{M}C$ is Schur stable, and
$M=(\tilde{A}^{\eta})^{-1}\bar{M}$. In addition, since $(\tilde{A},\tilde{B})$
is controllable, the controller gain $K\in\mathbb{R}^{n_{u}\times n_{x}}$ can
be designed such that

$(\tilde{A}+\tilde{B}K)^{\eta}=0.$ (27)

Let
$\|e^{x}_{s-1,\eta}\|_{\infty}:=\|x_{s-1,\eta}-\hat{x}_{s-1,\eta}\|_{\infty}$,
and construct the sequence $\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$
obeying $E_{s_{\ell}}\geq\|C\|_{\infty}\|e^{x}_{s_{\ell}-1,\eta}\|_{\infty}$
for all $\ell\in\mathbb{N}_{0}$. Adopt the same quantizer described in Sec.
3.1 with the quantization level $N$, the center $\hat{y}_{s_{\ell}-1,\eta}$,
and the range $E_{s_{\ell}}$. It can be similarly shown that the quantization
error $\|y_{s_{\ell}-1,\eta}-\mathcal{Q}(y_{s_{\ell}-1,\eta})\|_{\infty}$
satisfies (5). Noticing from (26a) that
$\hat{x}_{s,\eta}=(\tilde{A}+\tilde{B}K)\hat{x}_{s,\eta-1}=(\tilde{A}+\tilde{B}K)^{\eta}\hat{x}_{s}=0$,
one gets that $\hat{y}_{s,\eta}=C\hat{x}_{s,\eta}=0$ for all
$s\in\mathbb{N}_{0}$. Therefore, it is sufficient to choose zero as the
quantization center, and the observer-based deadbeat controller (26) does not
need to be present at the encoder side. The updated self-triggering mechanism
and associated output encoding scheme are presented as follows. We consider
the following self-triggering mechanism

$\bigg{\\{}\begin{aligned}
&s_{\ell+1}:=s_{\ell}+\min\\{\tau_{\max},\tau_{\ell}\\},~{}~{}\ell\in\mathbb{N}_{0},~{}~{}s_{0}:=0\\\
&\tau_{\ell}:=\min\\{\tau\in\mathbb{N}:\bar{g}(\mathcal{Q}(y_{s_{\ell}}),E_{s_{\ell}},\tau)>\sigma
E_{s_{\ell}}\\}.\end{aligned}$ (28)

In contrast to the self-triggering function $g$ in (6), function $\bar{g}$
here does not require the quantization center $\hat{x}_{s_{\ell}}$, but only
the quantized output $\mathcal{Q}(y_{s_{\ell}})$ and the quantization range
$E_{s_{\ell}}$. This simplifies implementation and saves computational
resources. The following lemma presenting the self-triggering function
$\bar{g}$ in (28) is a counterpart of Lem. 3.1.

######  Lemma 3.3.

Let Assumptions 2.2, and 3.1 hold. Consider system (1) adopting the observer-
based deadbeat controller in (26), and a quantizer such that condition (4) is
satisfied with a sequence $\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ for all
$\ell\in\mathbb{N}_{0}$. For every $s\in[s_{\ell},s_{\ell+1})$ with
$\ell\in\mathbb{N}_{0}$, if the self-triggering function $\bar{g}$ in (28) is
chosen as

$\displaystyle\bar{g}(q,\\!E,\\!1)\\!:=$
$\displaystyle\bigg{[}\\!\frac{\|C\Phi_{A}(I\\!-\\!MC)\|_{\infty}}{\|C\|_{\infty}}\\!+\\!\frac{\|C\Phi_{A}M\|_{\infty}}{N}\\!\bigg{]}E\\!+\\!\|q\|_{\infty}$
(29a) $\displaystyle\bar{g}(q,\\!E,\\!\tau)\\!:=$
$\displaystyle\bigg{[}\\!\frac{\|C\Phi_{A}\\!\tilde{A}^{(\tau-2)\eta}\\!\bar{A}_{cl}\|_{\infty}}{\|C\|_{\infty}}\\!+\\!\frac{\|C\Phi_{A}\\!\tilde{A}^{(\tau-1)\eta}\\!M\|_{\infty}}{N}\\!\bigg{]}\\!E$
$\displaystyle+\Big{\|}\Big{[}C\Phi_{A}\sum_{i=0}^{\tau-2}\tilde{A}^{i\eta}M+I\Big{]}q\Big{\|}_{\infty}$
(29b)

where
$\Phi_{A}=-\sum_{i=0}^{\eta-1}(\tilde{A}+\tilde{B}K)^{i}\tilde{B}K\tilde{A}^{\eta-i-1}$
and $\tau\geq 2$, then the output error obeys

$\|\mathcal{Q}(y_{s_{\ell}})-y_{s}\|_{\infty}\leq\sigma E_{{s_{\ell}}}.$ (30)

Based on the self-triggering mechanism above, we construct a sequence
$\\{E_{s_{\ell}}\\}$ to validate the condition in Lem. 3.3. Select constants
$\bar{\Gamma}\geq 1$ and $\bar{\gamma}\in(0,1)$ such that
$\|\bar{A}_{cl}^{s}\|_{\infty}=\|(\tilde{A}^{\eta}(I-MC))^{s}\|_{\infty}\leq\bar{\Gamma}\bar{\gamma}^{s}$
for all $s\in\mathbb{N}_{0}$. Update the sequence
$\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ as follows

$\left\\{\begin{aligned} &E_{s_{\ell}}:=\|C\|_{\infty}E^{x}_{s_{\ell}}\\\
&E^{x}_{0}:=\bar{\Gamma}E^{x}_{in}\\\
&E^{x}_{s_{\ell+1}}=(\bar{\gamma}^{s_{\ell+1}-s_{\ell}}(1-\bar{\alpha}\sigma)+\bar{\alpha}\sigma)E^{x}_{s_{\ell}},~{}\ell\in\mathbb{N}_{0}\end{aligned}\right.$
(31)

where
$\bar{\alpha}:=(\bar{\Gamma}\|\tilde{A}^{\eta}M\|_{\infty}\|C\|_{\infty})/(1-\bar{\gamma})$.
The next result generalizes Thm. 3.1 by adopting the deadbeat controller in
(26).

Following the proof of Lem. 3.1, we give an upper bound on the output error
$\|\mathcal{Q}(y_{s_{\ell}})-y_{s}\|_{\infty}$. Substituting (26a) into (25),
one gets that

$\displaystyle x_{s_{\ell},k+1}$
$\displaystyle=\tilde{A}x_{s_{\ell},k}+\tilde{B}K\hat{x}_{s_{\ell},k}$
$\displaystyle=(\tilde{A}+\tilde{B}K)x_{s_{\ell},k}-\tilde{B}K(x_{s_{\ell},k}-\hat{x}_{s_{\ell},k})$

and hence

$\displaystyle x_{s_{\ell},\eta}=~{}$
$\displaystyle(\tilde{A}+\tilde{B}K)^{\eta}x_{s_{\ell}}$
$\displaystyle-\sum_{i=0}^{\eta-1}(\tilde{A}+\tilde{B}K)^{i}\tilde{B}K(x_{s_{\ell},\eta-i-1}-\hat{x}_{s_{\ell},\eta-i-1})$
$\displaystyle=$
$\displaystyle-\sum_{i=0}^{\eta-1}(\tilde{A}+\tilde{B}K)^{i}\tilde{B}K(x_{s_{\ell},\eta-i-1}-\hat{x}_{s_{\ell},\eta-i-1})$

where the second equation holds because $(\tilde{A}+\tilde{B}K)^{\eta}=0$.
Recursively, it can be deduced that for $p\in[0,s_{\ell+1}\\!-\\!s_{\ell})$

$\displaystyle x_{s_{\ell}+p,\eta}=$ (32)
$\displaystyle\\!-\\!\\!\sum_{i=0}^{\eta-1}(\tilde{A}\\!+\\!\tilde{B}K)^{i}\\!\tilde{B}K(x_{s_{\ell}+p,\eta-i-1}\\!-\\!\hat{x}_{s_{\ell}+p,\eta-i-1})$

and we only need to calculate
$x_{s_{\ell}+p,\eta-i-1}-\hat{x}_{s_{\ell}+p,\eta-i-1}$.

(c1) For every $s\in[s_{\ell},s_{\ell+1})$ and $k\in[1,\eta-1]$, it follows
from (25) and (26a) that
$\hat{x}_{s,k+1}=\tilde{A}\hat{x}_{s,k}+\tilde{B}K\hat{x}_{s,k}$, and
$x_{s,k+1}=\tilde{A}x_{s,k}+\tilde{B}K\hat{x}_{s,k}$. Therefore,

$x_{s,k+1}-\hat{x}_{s,k+1}=\tilde{A}^{k+1}(x_{s}-\hat{x}_{s}).$ (33)

(c2) For $s=s_{\ell}$, it can be obtained from (26b) that,

$\displaystyle x_{s_{\ell}}$ $\displaystyle=x_{s_{\ell}-1,\eta}$
$\displaystyle\hat{x}_{s_{\ell}}$
$\displaystyle=\hat{x}_{s_{\ell}-1,\eta}+M[\mathcal{Q}(y_{s_{\ell}})-\hat{y}_{s_{\ell}-1,\eta}].$

Hence, one gets that

$\displaystyle x_{s_{\ell}}\\!-\\!\hat{x}_{s_{\ell}}$
$\displaystyle=x_{s_{\ell}}\\!-\\!\hat{x}_{s_{\ell}-1,\eta}\\!-\\!M\big{[}\mathcal{Q}(y_{s_{\ell}})\\!-\\!y_{s_{\ell}}\\!+\\!y_{s_{\ell}}\\!-\\!\hat{y}_{s_{\ell}-1,\eta}\big{]}$
$\displaystyle=(I\\!-\\!MC)(x_{s_{\ell}}\\!-\\!\hat{x}_{s_{\ell}-1,\eta})\\!-\\!M\big{[}\mathcal{Q}(y_{s_{\ell}})\\!-\\!y_{s_{\ell}}\big{]}.$

(c3) For $s>s_{\ell}$, it can be deduced that

$\displaystyle x_{s_{\ell}+p}$ $\displaystyle=x_{s_{\ell}+p-1,\eta}$ (35a)
$\displaystyle\hat{x}_{s_{\ell}+p}$
$\displaystyle=\hat{x}_{s_{\ell}+p-1,\eta}+M\mathcal{Q}(y_{s_{\ell}})$ (35b)

where (35b) holds because
$\hat{y}_{s_{\ell}+p-1,\eta}=C\hat{x}_{s_{\ell}+p-1,\eta}=C(\tilde{A}+\tilde{B}K)^{\eta}\hat{x}_{s_{\ell}+p-1}=0$.
Hence,

$x_{s_{\ell}+p}-\hat{x}_{s_{\ell}+p}=(x_{s_{\ell}+p-1,\eta}-\hat{x}_{s_{\ell}+p-1,\eta})-M\mathcal{Q}(y_{s_{\ell}}).$

Substituting (33) into (32) yields

$x_{s_{\ell}+p,\eta}=\Phi_{A}(x_{s_{\ell}+p}-\hat{x}_{s_{\ell}+p})$

where
$\Phi_{A}=-\sum_{i=0}^{\eta-1}(\tilde{A}+\tilde{B}K)^{i}\tilde{B}K\tilde{A}^{\eta-i-1}$
is defined in Lem. 3.3. In addition, noticing from (c1) and (c3) that

$\displaystyle
x_{s_{\ell}+p}-\hat{x}_{s_{\ell}+p}=\tilde{A}^{\eta}(x_{s_{\ell}+p-1}-\hat{x}_{s_{\ell}+p-1})-M\mathcal{Q}(y_{s_{\ell}}).$

Combining this with (c2), recursively one deduces that

$\displaystyle x_{s_{\ell}+p}-\hat{x}_{s_{\ell}+p}$ $\displaystyle=$
$\displaystyle~{}\tilde{A}^{\eta}\big{[}\tilde{A}^{\eta}(x_{s_{\ell}+p-2}-\hat{x}_{s_{\ell}+p-2})-M\mathcal{Q}(y_{s_{\ell}})\big{]}-M\mathcal{Q}(y_{s_{\ell}})$
$\displaystyle=$
$\displaystyle~{}\tilde{A}^{(p-1)\eta}\big{[}\bar{A}_{cl}(x_{s_{\ell}}-\hat{x}_{s_{\ell}-1,\eta})\big{]}$
$\displaystyle-\tilde{A}^{p\eta}M\big{[}\mathcal{Q}(y_{s_{\ell}})-y_{s_{\ell}}\big{]}-\sum_{i=0}^{p-1}\tilde{A}^{i\eta}M\mathcal{Q}(y_{s_{\ell}}).$
(36)

Substituting (3.2) into (32), one has that

$\displaystyle
x_{s_{\ell},\eta}=\Phi_{A}[(I\\!-\\!MC)(x_{s_{\ell}}\\!-\\!\hat{x}_{s_{\ell}-1,\eta})\\!-\\!M(\mathcal{Q}(y_{s_{\ell}})\\!-\\!y_{s_{\ell}})]$

and for $p\geq 1$,

$\displaystyle
x_{s_{\ell}+p,\eta}=\Phi_{A}\tilde{A}^{(p-1)\eta}\bar{A}_{cl}(x_{s_{\ell}}-\hat{x}_{s_{\ell}-1,\eta})$
$\displaystyle-\Phi_{A}\tilde{A}^{p\eta}M(\mathcal{Q}(y_{s_{\ell}})-y_{s_{\ell}})\\!-\\!\Phi_{A}\sum_{i=0}^{p-1}\tilde{A}^{i\eta}M\mathcal{Q}(y_{s_{\ell}}).$

Hence, the output error obeys,

$\displaystyle\|y_{s_{\ell},\eta}-\mathcal{Q}(y_{s_{\ell}})\|_{\infty}$
$\displaystyle=\Big{\|}C\Phi_{A}(I-MC)(x_{s_{\ell}}-\hat{x}_{s_{\ell}-1,\eta})$
$\displaystyle~{}~{}~{}-C\Phi_{A}M[\mathcal{Q}(y_{s_{\ell}})-y_{s_{\ell}}]-\mathcal{Q}(y_{s_{\ell}})\Big{\|}_{\infty}$
$\displaystyle\leq\bigg{[}\frac{\|C\Phi_{A}(I\\!-\\!MC)\|_{\infty}}{\|C\|_{\infty}}\\!+\\!\frac{\|C\Phi_{A}M\|_{\infty}}{N}\bigg{]}E_{s_{\ell}}\\!+\\!\|\mathcal{Q}(y_{s_{\ell}})\|_{\infty}$

and for $p\geq 1$,

$\displaystyle\|y_{s_{\ell}+p,\eta}-\mathcal{Q}(y_{s_{\ell}})\|_{\infty}$
$\displaystyle=\Big{\|}C\Phi_{A}\tilde{A}^{(p-1)\eta}\bar{A}_{cl}(x_{s_{\ell}}-\hat{x}_{s_{\ell}-1,\eta})$
$\displaystyle~{}~{}~{}-C\Phi_{A}\tilde{A}^{p\eta}M[\mathcal{Q}(y_{s_{\ell}})-y_{s_{\ell}}]$
$\displaystyle~{}~{}~{}-\Big{[}C\Phi_{A}\sum_{i=0}^{p-1}\tilde{A}^{i\eta}M+I\Big{]}\mathcal{Q}(y_{s_{\ell}})\Big{\|}_{\infty}$
$\displaystyle\leq\bigg{[}\frac{\|C\Phi_{A}\tilde{A}^{(p-1)\eta}\bar{A}_{cl}\|_{\infty}}{\|C\|_{\infty}}+\frac{\|C\Phi_{A}\tilde{A}^{p\eta}M\|_{\infty}}{N}\bigg{]}E_{s_{\ell}}$
$\displaystyle~{}~{}~{}+\Big{\|}\Big{[}C\Phi_{A}\sum_{i=0}^{p-1}\tilde{A}^{i\eta}M+I\Big{]}\mathcal{Q}(y_{s_{\ell}})\Big{\|}_{\infty}.$
(37)

Eventually, according to the inequality
$\bar{g}(\mathcal{Q}(y_{s_{\ell}}),E_{s_{\ell}},p)\leq\sigma E_{s_{\ell}}$ in
(28) and noticing (37), if $\bar{g}$ is defined as in (29), then
$\|\mathcal{Q}(y_{s_{\ell}})-y_{s}\|_{\infty}\leq\bar{g}(\mathcal{Q}(y_{s_{\ell}}),E_{s_{\ell}},p)\leq\sigma
E_{s_{\ell}}$ is obtained using (30). This completes the proof.

######  Theorem 3.2.

Consider system (1) with the deadbeat controller (26), where the gain matrices
$M$ and $K$ are chosen such that $\tilde{A}^{\eta}(I-MC)$ is Schur stable and
$(\tilde{A}+\tilde{B}K)^{\eta}=0$. Under Assumptions 2.2 and 3.1, if the
sampling times $\\{s_{\ell}\\}_{\ell\in\mathbb{N}_{0}}$ are generated by (28)
with i) the self-triggering function $\bar{g}$ in (29), ii) the sequence
$\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ in (31), and iii) the threshold
$\sigma>0$ obeying

$\frac{1}{N}\leq\sigma<\frac{1}{\bar{\alpha}}=\frac{1-\bar{\gamma}}{\bar{\Gamma}\|\tilde{A}^{\eta}M\|_{\infty}\|C\|_{\infty}}$
(38)

then the system (1) is exponentially stable. In particular, the decay
parameter
$\omega=\bar{\gamma}^{\tau}_{\max}(1-\bar{\alpha}\sigma)+\bar{\alpha}\sigma\in(0,1)$
satisfies Def. 2.1 for some constant $\Omega>0$.

We begin by showing that
$\|y_{s_{\ell+1}-1,\eta}-\hat{y}_{s_{\ell+1}-1,\eta}\|_{\infty}\leq
E_{s_{\ell}+1}$ for $\ell\in\mathbb{N}_{0}$.

It follows from (25) and (26) that

$\displaystyle
x_{s_{\ell},\eta}\\!-\\!\hat{x}_{s_{\ell},\eta}\\!=\\!\bar{A}_{cl}(x_{s_{\ell}-1,\eta}\\!-\\!\hat{x}_{s_{\ell}-1,\eta})\\!-\\!\tilde{A}^{\eta}M(\mathcal{Q}(y_{s_{\ell}})\\!-\\!y_{s_{\ell}}).$

Assuming that $\|x_{s_{\ell}-1,\eta}-\hat{x}_{s_{\ell}-1,\eta}\|_{A_{cl}}\leq
E_{s_{\ell}}^{x}$, recursively, one gets for $p\in[0,s_{\ell+1}-s_{\ell})$
that

$\displaystyle~{}~{}~{}\|x_{s_{\ell}+p,\eta}-\hat{x}_{s_{\ell}+p,\eta}\|_{\bar{A}_{cl}}$
(39a)
$\displaystyle=\Big{\|}\bar{A}_{cl}^{p+1}(x_{s_{\ell}-1,\eta}-\hat{x}_{s_{\ell}-1,\eta})$
$\displaystyle~{}~{}~{}-\sum_{i=0}^{p}\bar{A}_{cl}^{i}\tilde{A}^{\eta}M(\mathcal{Q}(y_{s_{\ell}})-y_{s_{\ell}+p-i})\Big{\|}_{\bar{A}_{cl}}$
(39b)
$\displaystyle\leq\bar{\gamma}^{p+1}\|x_{s_{\ell}-1,\eta}-\hat{x}_{s_{\ell}-1,\eta}\|_{A_{cl}}$
$\displaystyle~{}~{}~{}+\bar{\Gamma}\sum_{i=0}^{p}\|\bar{A}_{cl}^{i}\tilde{A}^{\eta}M\|_{\infty}\|\mathcal{Q}(y_{s_{\ell}})-y_{s_{\ell}+p-i}\|_{\infty}$
(39c)
$\displaystyle\leq\gamma^{p+1}E^{x}_{s_{\ell}}+\sigma\bar{\Gamma}\|\tilde{A}^{\eta}M\|_{\infty}\|C\|_{\infty}\sum_{i=0}^{p}\bar{\gamma}^{i}E^{x}_{s_{\ell}}$
(39d)
$\displaystyle\leq(\bar{\gamma}^{p+1}(1-\bar{\alpha}\sigma)+\bar{\alpha}\sigma)E^{x}_{s_{\ell}}$
(39e)
$\displaystyle\leq\bar{\omega}_{1}^{p+1}E^{x}_{s_{\ell}},\quad\bar{\omega}_{1}:=\bar{\gamma}^{\tau_{\max}}(1-\bar{\alpha}\sigma)+\bar{\alpha}\sigma$
(39f)

where
$\bar{\alpha}:=(\bar{\Gamma}\|\tilde{A}^{\eta}M\|_{\infty}\|C\|_{\infty})/(1-\bar{\gamma})$
is defined in (31). Since $\sigma<1/\bar{\alpha}$, and $\bar{\gamma}<1$, we
deduce that
$\bar{\gamma}^{p+1}(1-\bar{\alpha}\sigma)+\bar{\alpha}\sigma\leq\bar{\gamma}^{0}(1-\bar{\alpha}\sigma)+\bar{\alpha}\sigma=1$.
Inequality (39f) follows from the statement (s2) in Lem. 3.2. Therefore,
letting $p=s_{\ell+1}-s_{\ell}-1$ in (39f) it follows from (31) that

$\displaystyle\|y_{s_{\ell+1}-1,\eta}-\hat{y}_{s_{\ell+1}-1,\eta}\|_{\infty}$
$\displaystyle\leq E_{s_{\ell+1}}.$ (40)

This is a counterpart of the statement (s3) in Lem. 3.2.

Next, leveraging (40), we are able to prove the convergence of the state.
Noticing from (27) that $\hat{x}_{s,\eta}=0$ for every $s\in\mathbb{N}_{0}$.
Hence, for $p\in[0,s_{\ell+1}-s_{\ell})$, according to (25) and (39f), we
arrive at

$\displaystyle\|x_{s_{\ell}+p}\|_{\infty}$
$\displaystyle=\|x_{s_{\ell}+p-1,\eta}-\hat{x}_{s_{\ell}+p-1,\eta}\|_{\infty}$
$\displaystyle\leq\bar{\Gamma}E_{in}\bar{\omega}_{1}^{s_{\ell}+p}.$ (41)

In addition, for $k\in[0,\eta-1]$, one has from Lem. 3.3 that

$\displaystyle\|x_{s_{\ell}+p,k+1}\|_{\infty}$
$\displaystyle=\Big{\|}(\tilde{A}+\tilde{B}K)^{k+1}x_{s_{\ell}+p}$
$\displaystyle~{}~{}~{}-\\!\\!\sum_{i=0}^{k}(\tilde{A}\\!+\\!\tilde{B}K)^{i}\tilde{B}K\tilde{A}^{k-i}(x_{s_{\ell}+p}\\!-\\!\hat{x}_{s_{\ell}+p})\Big{\|}_{\infty}$
(42a)
$\displaystyle=\Big{\|}(\tilde{A}+\tilde{B}K)^{k+1}x_{s_{\ell}+p}\\!-\\!\\!\sum_{i=0}^{k}(\tilde{A}+\tilde{B}K)^{i}\tilde{B}K\tilde{A}^{k-i}$
$\displaystyle~{}~{}~{}\times\big{[}(I-MC)(x_{s_{\ell}+p-1,\eta}-\hat{x}_{s_{\ell}+p-1,\eta})$
$\displaystyle~{}~{}~{}-M(\mathcal{Q}(y_{s_{\ell}})-y_{s_{\ell}+p})\big{]}\Big{\|}_{\infty}$
(42b)
$\displaystyle\leq\bar{\Omega}_{K}\|x_{s_{\ell}+p}\|_{\infty}\\!+\\!\big{(}\bar{\Omega}_{2}\bar{\Gamma}\\!+\\!\bar{\Omega}_{3}\sigma\bar{\Gamma}_{1}\bar{\omega}_{1}^{-\tau_{\max}}\big{)}E_{in}\bar{\omega}_{1}^{s_{\ell}+p}$
(42c) $\displaystyle\leq\bar{\Omega}E_{in}\bar{\omega}_{1}^{s_{\ell}+p}$ (42d)

where $\bar{\Gamma}_{1}:=\|C\|_{\infty}\bar{\Gamma}$, and

$\displaystyle\bar{\Omega}_{K}:=\max_{1\leq
k\leq\eta}\|(\tilde{A}+\tilde{B}K)^{k}\|_{\infty}$
$\displaystyle\bar{\Omega}_{2}:=\max_{0\leq
k\leq\eta-1}\|\sum_{i=0}^{k}(\tilde{A}+\tilde{B}K)^{i}\tilde{B}K\tilde{A}^{k-i}(I-MC)\|_{\infty}$
$\displaystyle\bar{\Omega}_{3}:=\max_{0\leq
k\leq\eta-1}\|\sum_{i=0}^{k}(\tilde{A}+\tilde{B}K)^{i}\tilde{B}K\tilde{A}^{k-i}M\|_{\infty}$
$\displaystyle\bar{\Omega}:=(\bar{\Omega}_{K}+\bar{\Omega}_{2}+\bar{\Omega}_{3}\sigma\|C\|_{\infty}\bar{\omega}_{1}^{-\tau_{\max}})\bar{\Gamma}.$

Equation (42a) holds by adopting (33). In addition, equation (42b) holds by
using (35). Finally, substituting (41) into (42c), the system is exponentially
stable according to Def. 2.1, which completes the proof.

######  Remark 3.3.

(_Comparison between the controllers ( 2) and (26)_). Relative to the standard
observer-based controller in (2), the deadbeat controller in (26) offers at
least two advantages.

  * i)

The self-triggering mechanism requires only the quantized output
$\mathcal{Q}(y_{s_{\ell}})$ and the quantization range $E_{s_{\ell}}$ to
compute the next sampling time $s_{\ell+1}$, whose implementation in practice
is as simple as that for quantized state feedback control studied in Wakaiki
(2022).

  * ii)

At every sampling time $s_{\ell}$, the quantization center,
$\hat{y}_{s_{\ell}-1,\eta}$ is zero because of the deadbeat controller. This
eliminates the need for running the controller at the encoder side and at the
self-triggering module to compute the quantization center $\hat{y}_{s_{\ell}}$
for the standard observer-based controller, which is both computationally and
implementation-wise more appealing.

Certainly, these two merits come at the price of a more complicated
communication protocol between the S-C and C-A channels, increased
communication overhead in the C-A channel as well as requiring a deadbeat
controller gain matrix $K$. The deadbeat condition (27) is more conservative
compared with the Schur stable one, since only a few matrices $K$ obey
$(\tilde{A}+\tilde{B}K)^{\eta}=0$. To increase the number of candidate
matrices $K$, this condition can be relaxed by choosing $K$ such that
$(\tilde{A}+\tilde{B}K)^{i}=0$ with $i\in[\eta,n_{x}]$, and setting
$\delta=\Delta/i$. Under this condition, it has been shown in Fahmy and
O’Reilly (1983) that a large number of matrices $K$ can be constructed. This
decreases the conservativeness of using the deadbeat controller.

##  4 Self-triggered Control: DoS Case

Besides the communication cost, it has been reported that cyber-physical
networked systems are often vulnerable to cyber-attacks, consisting of e.g.,
false-data injection attacks, switching attacks, replay attacks, and denial-
of-service (DoS) attacks Pasqualetti et al. (2013); Wu et al. (2020); Guo et
al. (2023); Liu et al. (2022b). For this reason, research on designing
resilient control strategies against cyber-attacks has attracted lots of
attention recently; see e.g., Yuan et al. (2020); Li et al. (2021); Franze et
al. (2021). In this section, we consider that the S-C channel is exposed to
DoS attacks, and develop a DoS-resilient self-triggering mechanism and
associated encoding scheme as well as stability analysis.

###  4.1 Denial-of-Service Attack

A brief introduction on the DoS attack is outlined. Launched by adversaries,
DoS attacks are intended to affect the timeliness of the information, and
result in packet losses. To characterize DoS attacks, we here employ the
‘duration-frequency’ model initially studied in De Persis and Tesi (2015) and
later in De Persis and Tesi (2016); Wakaiki et al. (2020); Feng and Tesi
(2017); Liu et al. (2022a).

Before proceeding, let us define at each time $s\in\mathbb{N}_{0}$ an attack-
indicator function

$h(s):=\begin{cases}0,&{\text{no DoS attack at}}~{}s,\\\ 1,&{\text{DoS attack
at}}~{}s.\end{cases}$ (44)

For an arbitrary constant $\tau^{a}\in\mathbb{N}$, the DoS duration within the
interval $[s,s+\tau^{a})$ is defined as
$\Phi_{d}(s,\tau^{a})=\sum_{i=s}^{s+\tau^{a}-1}h(i)$. Here, we only place an
assumption on the DoS duration, since the result in this section demonstrates
that the DoS frequency does not explicitly affect system stability in the
considered setup.

######  Assumption 4.1 (_DoS duration_).

There exist constants $\kappa_{d}\in\mathbb{R}_{\geq 0}$ and
$\nu_{d}\in\mathbb{R}_{\geq 1}$, also known as chatter bound and average
duration ratio, respectively, such that the DoS duration obeys

$\Phi_{d}(0,s)\leq\kappa_{d}+{s}/{\nu_{d}}$ (45)

over the interval $[0,s)$ for all $s\in\mathbb{N}_{0}$.

######  Remark 4.1 (_Implication of DoS parameters_).

Assumption 4.1 indicates that, the average duration of DoS attacks does not
exceed a proportion $1/\nu_{d}$ of the entire interval. This assumption is
general enough, since DoS attacks with higher strength can be modeled with a
smaller $\nu_{d}$. In addition, as the parameter $\nu_{d}$ approaches zero,
this model characterizes a type of attacks preventing all packets from
transmission. However, if no packet can be received successfully, then no
controller can be constructed to stabilize open-loop unstable plants. To
prevent such a situation, the condition $\nu_{d}\geq 1$ is placed.

###  4.2 Resilient Control

According to the previous subsection, no packet can be received at the decoder
side if the current sampling time belongs to a DoS interval. Therefore, our
proposed self-triggering mechanism and encoding scheme for the DoS-free case
cannot be directly employed and do not necessarily ensure system stability. In
the following, we focus on the standard observer-based state feedback
controller in Sec. 3.1. The results can be generalized to the deadbeat
controller (26).

Some assumptions are placed to show how the system reacts to the DoS attacks.

######  Assumption 4.2.

If there is a DoS attack at the transmission time $s_{\ell}$, i.e.,
$h(s_{\ell})=1$, then not quantized output $\mathcal{Q}(y_{s_{\ell}})$ is
received, and a default zero will be used.

######  Assumption 4.3.

An ACK-based protocol is implemented by the S-C channel. That is, the decoder
sends an ACK to the encoder without delay when it receives a quantized output.

Assumption 4.3 enables the encoder to infer a DoS attack if no ACK is
received. This assumption however can be removed if the deadbeat controller
(26) is adopted, which resembles that in (Liu et al., 2022a, Sec. IV). Under
Assumption 4.2, the observer-based state feedback controller (2) is replaced
by

$\displaystyle\hat{x}_{s+1}$
$\displaystyle=A\hat{x}_{s}\\!+\\!\tilde{B}u_{s}\\!+\\!L[\mathcal{Q}(y_{s_{\ell}})\\!-\\!\hat{y}_{s}],~{}h(s_{\ell})=0$
(46a) $\displaystyle\hat{x}_{s+1}$
$\displaystyle=A\hat{x}_{s}+\tilde{B}u_{s},~{}~{}h(s_{\ell})=1$ (46b)
$\displaystyle\hat{y}_{s}$ $\displaystyle=C\hat{x}_{s}$ (46c) $\displaystyle
u_{s}$ $\displaystyle=K\hat{x}_{s}$ (46d)

with the initial condition $\hat{x}_{0}=0$. Moreover, if a DoS attack occurs
at a triggering time, that is, $h(s_{\ell})=1$, then the inequality (8) is no
longer valid. To enhance system resilience against DoS attacks, we consider a
mixed sampling scheme. To be specific, i) during DoS attacks, the quantized
output is communicated periodically with interval $\Delta$; and, ii) when
there is no DoS attack, the next sampling time is calculated by (6). In this
manner, a successful transmission occurs as soon as an attack stops.

$s_{\ell+1}=\begin{cases}\eqref{eq:lqselftri},~{}~{}&h(s_{\ell})=0\\\
s_{\ell}+1,~{}~{}&h(s_{\ell})=1\end{cases}.$ (47)

The update rule of $\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ in (14) is
modified as follows

$E_{s_{\ell+1}}:=\begin{cases}\eqref{eq:lqE},&h(s_{\ell})=0\\\
\omega_{a}E_{s_{\ell}},&h(s_{\ell})=1\end{cases}$ (48)

where $\omega_{a}>1$ is chosen such that
$\|A^{s}\|_{\infty}\leq(1/\Gamma)\omega_{a}^{s}$ holds for all
$s\in\mathbb{N}_{0}$ with $\Gamma$ defined in (13).

Based on (46)–(48), our stability result comes as follows.

######  Theorem 4.1.

Consider system (1) with the controller (46), where matrices $L$ and $K$ are
chosen such that $A-LC$ and $A+BK$ are Schur stable. Under Assumptions 2.1—2.2
and 4.1—4.3, if the sampling times $\\{s_{\ell}\\}_{\ell\in\mathbb{N}_{0}}$
are generated by (47), with i) the bound sequence
$\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ in (48), ii) the threshold
parameter ${\sigma}>0$ obeying (15), and iii) DoS attacks satisfying

${1}/{\nu_{d}}<\big{(}\log(1/\omega_{1})\big{)}/\big{(}\log(\omega_{a}/\omega_{1})\big{)}$
(49)

with $\omega_{1}$ defined in (19), then the system (1) is exponentially
stable. In particular, the decay parameter
$\omega=\min\\{\hat{\omega}_{1},1\\}$ satisfies Def. 2.1 for some constant
$\Omega>0$ where
$\hat{\omega}_{1}>\max\\{\hat{\omega},\omega_{1}(\omega_{a}/\omega_{1})^{1/\nu_{d}}\\}$
and $\hat{\omega}$ is defined in Thm. 3.1.

We begin by constructing the sequence
$\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ presented in (48). Based on
Assumption 4.2, if a DoS attack occurs at a sampling time, denoted by
$s^{a}\in\mathbb{N}_{0}$, and lasts for $\tau^{a}$, then the error between the
estimated state and the actual state obeys

$\displaystyle
x_{s^{a}+i}-\hat{x}_{s^{a}+i}=A_{d}^{i}(x_{s^{a}}-\hat{x}_{s^{a}}),~{}i\in[0,\tau^{a}).$
(50)

Based on the definition of $\omega_{a}$, and assuming that
$\|x_{s^{a}}-\hat{x}_{s^{a}}\|_{\infty}\leq E_{s^{a}}^{x}$ holds true, one
gets that

$\displaystyle\left\|x_{s^{a}+i}-\hat{x}_{s^{a}+i}\right\|_{A_{cl}}$
$\displaystyle\leq\Gamma\left\|x_{s^{a}+i}-\hat{x}_{s^{a}+i}\right\|_{\infty}$
(51a)
$\displaystyle\leq\Gamma\Big{(}\frac{1}{\Gamma}\omega_{a}^{i}\left\|x_{s^{a}}-\hat{x}_{s^{a}}\right\|_{\infty}\Big{)}$
(51b)
$\displaystyle\leq\omega_{a}^{i}\left\|x_{s^{a}}-\hat{x}_{s^{a}}\right\|_{A_{cl}}$
(51c) $\displaystyle\leq\omega_{a}^{i}E_{s^{a}}^{x}$ (51d)

where equations (51a) and (51c) are derived following the definition of the
norm $\|\cdot\|_{A_{cl}}$ in Lem. 2.1. Inequality (51b) follows from the
definition of $\omega_{a}$ in (48). Let
$E^{x}_{s^{a}+1}:=\omega_{a}E^{x}_{s^{a}}$, and hence
$E_{s^{a}+1}=\|C\|_{\infty}E^{x}_{s^{a}+1}=\|C\|_{\infty}\omega_{a}E^{x}_{s^{a}}$.
Therefore, sequence $\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ adheres to
(4) with

$\displaystyle\|y_{s^{a}+i}-\hat{y}_{s^{a}+i}\|_{\infty}$
$\displaystyle=\|C(x_{s^{a}+i}-\hat{x}_{s^{a}+i})\|_{\infty}$
$\displaystyle\leq\|C\|_{\infty}E^{x}_{s^{a}+i}=E_{s^{a}+i}.$

Capitalizing on the self-triggering mechanism in (47) and the encoding scheme
in (48), additional conditions on DoS attacks such that the system can achieve
stability are derived in the following.

For interval $[0,s_{e})$ with arbitrary $s_{e}\in\mathbb{N}_{0}$, assume that
there are $p\in\mathbb{N}_{0}$ DoS attacks launched within this interval; that
is, there are $p$ DoS _off/on_ switches within this interval. To be specific,
let $s^{a}_{m}$ and $\tau^{a}_{m}$ denote the beginning time and the duration
time of the $m$-th DoS attack, respectively, where $m=1,\cdots\\!,p$. For
simplicity, let $s^{a}_{1}>0$ and $s^{a}_{p}+\tau^{a}_{p}<s_{e}$. Hence, it
follows from the definition of the DoS duration that
$\Phi_{d}(s_{e})=\sum_{m=1}^{p}\tau^{a}_{m}$. Consider that the adversary is
aware of the self-triggering mechanism, and thus launches attacks at only the
self-triggering times, namely
$\left[s_{m}^{a},s_{m}^{a}+\tau_{m}^{a}\right)\subset\left\\{s_{\ell}\right\\}_{\ell\in\mathbb{N}_{0}}$
for all $m=1,\cdots\\!,p$. This can be the worst case, since each attack
causes a packet loss and is effective.

Since there is no attacks before $s^{a}_{1}$, one gets from (48) that

$E_{s^{a}_{1}}^{x}=\Gamma E_{in}\omega_{1}^{s^{a}_{1}}.$ (52)

In addition, since no attacks happen within
$[s^{a}_{m}+\tau^{a}_{m},s^{a}_{m+1})\cup(s_{p}^{a}+\tau_{p}^{a},s_{e}]$ for
$m=1,\cdots\\!,p-1$, it follows from (48) that

$E_{s_{m+1}^{a}}^{x}\\!\leq\omega_{1}^{s_{m+1}^{a}-s_{m}^{a}-\tau_{m}^{a}}E_{s_{m}^{a}+\tau_{m}^{a}}^{x}\\!=\\!\omega_{1}^{s_{m+1}^{a}-s_{m}^{a}-\tau_{m}^{a}}\omega_{a}^{\tau_{m}^{a}}E_{s_{m}^{a}}^{x}$

where $\omega_{1}<1$ is defined in (19). Combining the estimates (18) and
(51), we have from (48) again that

$\|x_{s_{p}^{a}}-\hat{x}_{s_{p}^{a}}\|_{A_{cl}}\leq E^{x}_{s_{p}^{a}}.$

This implies that

$\|x_{s_{p}^{a}+\tau_{p}^{a}+q}-\hat{x}_{s_{p}^{a}+\tau_{p}^{a}+q}\|_{A_{cl}}\leq\omega_{1}^{q}\omega_{a}^{\tau_{p}^{a}}E_{s_{p}^{a}}$

for $q\in(0,s_{e}-s_{p}^{a}-\tau_{p}^{a}]$. Recursively,

$\displaystyle\|e_{s_{e}}^{i}\|_{\infty}$
$\displaystyle=\|x_{s_{e}}-\hat{x}_{s_{e}}\|_{\infty}$
$\displaystyle\leq\Gamma\omega_{1}^{s_{e}-\Phi_{d}(0,s_{e})}\omega_{a}^{\Phi_{d}(0,s_{e})}E_{in}$
$\displaystyle\leq\Gamma\Big{(}\frac{\omega_{a}}{\omega_{1}}\Big{)}^{\kappa_{d}}\Big{(}\omega_{1}\Big{(}\frac{\omega_{a}}{\omega_{1}}\Big{)}^{1/\nu_{d}}\Big{)}^{s_{e}}E_{in}$
$\displaystyle\leq\Gamma_{2}\omega_{3}^{s_{e}}E_{in}$

where $\Gamma_{2}:=\Gamma(\omega_{a}/\omega_{1})^{\kappa_{d}}$, and
$\omega_{3}:=\omega_{1}(\omega_{a}/\omega_{1})^{1/\nu_{d}}$. Moreover, it
follows from (49) that $\omega_{3}<1$. Therefore, the error between the actual
state $x_{s_{e}}$ and the estimated output $\hat{x}_{s_{e}}$, i.e.,
$\|e_{s_{e}}^{x}\|=\|x_{s_{e}}-\hat{x}_{s_{e}}\|_{\infty}$, converges to the
origin.

Now, we are ready to derive the convergence bound for the state. Noticing from
(1) that

$\displaystyle\|x_{s_{e}}\|_{\infty}$
$\displaystyle=\Big{\|}(A\\!+\\!BK)^{s_{e}}x_{0}$
$\displaystyle~{}~{}~{}-\sum_{i=0}^{s_{e}-1}(A\\!+\\!BK)^{s_{e}-i-1}BKe^{x}_{i}\Big{\|}_{\infty}$
$\displaystyle\leq\|(A\\!+\\!BK)^{s_{e}}\|_{\infty}\|x_{0}\|_{\infty}$
$\displaystyle~{}~{}~{}+\sum_{i=0}^{s_{e}-1}\|(A\\!+\\!BK)^{s_{e}-i-1}BK\|_{\infty}\|e^{x}_{i}\|_{\infty}.$

Combining with (22), similar from (21) in the proof of Thm. 3.1, we have that

$\displaystyle\|x_{s_{e}}\|_{\infty}$
$\displaystyle\leq\Omega_{K}E_{in}\omega^{s_{e}}$
$\displaystyle~{}~{}~{}+\underbrace{(\|BK\|_{\infty}\Omega_{K}\Gamma_{2})}_{\Omega_{4}}E_{in}\sum_{i=0}^{s_{e}-1}\omega^{s_{e}-i-1}\omega_{3}^{i}$
$\displaystyle\leq\Omega_{K}E_{in}\omega^{s_{e}}+\Omega_{4}E_{in}\sum_{i=0}^{s_{e}-1}\omega_{4}^{s_{e}-1}$
(54a)
$\displaystyle\leq\Omega_{K}E_{in}\omega^{s_{e}}+\Omega_{4}E_{in}(s_{e}-1)\omega_{4}^{s_{e}}/\omega_{4}$
(54b) $\displaystyle\leq\Omega_{5}E_{in}\hat{\omega}_{4}^{s_{e}}$ (54c)

where $\omega_{4}:=\max\\{\omega,\omega_{3}\\}$,
$\hat{\omega}_{4}>\max\\{\omega,\omega_{4}\\}$ and
$\Omega_{5}=\Omega_{K}+\Omega_{4}c_{6}/\hat{\omega}_{4}$ with
$(s_{e}-1)\omega_{4}^{s_{e}-1}\leq c_{6}\hat{\omega}_{4}^{s_{e}-1}$. This
completes the proof according to Def. 2.1.

######  Remark 4.2 (_Conservativeness of DoS bound_).

Condition (49) is derived by only taking into account the DoS attacks at self-
triggering times. Since attacks that occur between two consecutive self-
triggering times $(s_{\ell},s_{\ell+1})$ are ignored, this bound could be the
worst, where the adversary knows the self-triggering mechanism. Therefore, to
save energy during inter-sampling times, attacker launches attacks only at the
sampling times. Hence, if more information on DoS attacks is provided, less
conservative bounds can be obtained.

######  Remark 4.3.

(_Connection with the work Wakaiki et al. (2020)_) Condition (49) indicates
that under the proposed encoding schemes and self-triggering mechanism, system
achieves exponential stability without any DoS frequency assumption. In
addition, instead of (48), if we let i) $E_{s+1}=\omega_{1}E_{s}$ for
$h(s)=0$, and ii) $E_{s+1}=\omega_{a}E_{s}$ for $h(s)=1$, then the bound
sequence $\\{E_{s_{\ell}}\\}_{\ell\in\mathbb{N}_{0}}$ is updated periodically.
In this setting, the condition (49) coincides with the one derived in Wakaiki
et al. (2020). Moreover, to enhance the system resilience against DoS attacks,
one can choose smaller $\sigma$, which results in smaller inter-sampling times
and more samplings.

##  5 Numerical Example

To numerically test the performance of the proposed self-triggered quantized
controllers, we call for the unstable batch reactor in Walsh and Ye (2002),
for which a linearized model has been studied in e.g., Wakaiki et al. (2020);
Liu et al. (2022a). The system matrices are given by

$\displaystyle A:=\left[\begin{matrix}1.38&-0.2077&6.715&-5.676\\\
-0.5814&-4.29&0&0.675\\\ 1.067&4.273&-6.654&5.893\\\
0.048&4.273&-1.343&-2.104\end{matrix}\right]$ $\displaystyle
B:=\left[\begin{matrix}0&0\\\ 5.679&0\\\ 1.136&-3.146\\\
1.136&0\end{matrix}\right],C:=\left[\begin{matrix}1&0&1&-1\\\
0&1&0&0\end{matrix}\right].$

The system $(A,B,C)$ is observable and controllable with $\eta=2$. For the
feedback controller (2) in Sec. 3.1, we discretize the system with a sampling
period $\Delta=0.005$; and for the observer-based deadbeat controller (26) in
Sec. 3.2, we have $\delta=\Delta/\eta=0.005$. In this setting, the C-A channel
communication costs incurred by the controllers (2) and (26) are the same.
Choose the gain matrices $K_{1}$ and $K_{2}$, such that $A+BK_{1}$ is Schur
Stable and that $(A+BK_{2})^{\eta}=0$ holds, yielding

$\displaystyle K_{1}:=\left[\begin{matrix}1.4110&-3.5708&-0.6385&-4.1134\\\
6.0726&-0.0486&4.6801&-2.5005\end{matrix}\right]$ $\displaystyle
K_{2}:=\left[\begin{matrix}288.3&233.4&1000.0&-1429.4\\\
1945.2&-1.9&94.7&-84.7\end{matrix}\right].$

The gain observers $L$ and $M$ are chosen such that $A-LC$ and
$\tilde{A}^{\eta}(I-MC)$ are both Schur stable. Based on (15) and (38), take
the parameters of the self-triggering mechanism to be $\sigma=0.0343$, and the
maximum inter-sampling time to be $\tau_{\max}=20$.

Figure 3: System performance with different controllers and different
quantization levels. Panels (a1) and (a2): controller (2) under $N=31$ and
$N=101$. Panels (b1) and (b2): controller (26) under $N=11$ and $N=101$.
Figure 4: System performance under different $\sigma$. Panels (a1) and (a2):
controller (2) under $N=31$ with $\sigma=0.0193$ and $\sigma=0.0343$. Panels
(b1) and (b2): controller (26) under $N=11$ with $\sigma=0.0193$ and
$\sigma=0.0343$.

First, system performance in the DoS-free case using different controllers
under different levels of quantization is reported in Fig. 3. It is evident in
(7) that a larger quantization level results in a larger inter-sampling time
and thus less samples. However, in Fig. 3, change in the quantization level
has only little effect on the convergence of state. In addition, compared with
the standard controller, system under the deadbeat controller not only incurs
less samplings but also converges in a faster rate.

For a given quantization level, Fig. 4 compares the performance under
different values of $\sigma$. It is clear that smaller $\sigma$ leads to
faster convergence rate of the quantization range $E$, whereas yielding
smaller inter-sampling interval $s_{\ell+1}-s_{\ell}$. This can be observed
from the definition of $\omega_{1}$ in (19).

Figure 5: System performance with controller under DoS attacks with $N=101$:
controller (2) (top two) vs. controller (26) (bottom two).

Moreover, consider DoS attacks at the S-C channel, which are generated
randomly and represented by gray shades. The red cross marked lines represent
the sampling times that fail due to DoS attacks. Set the quantization level
$N=101$. Fig. 5 depicts the system performance under DoS attacks with
controllers in Secs. 3.1 (top two) and 3.2 (bottom two). Collecting only the
effective DoS attacks, denoted by $\Phi_{d}^{e}(0,s)$ (i.e., attacks that
happen at sampling times, represented by red cross marked line) in the top
panel of Fig. 5, we have that $\Phi_{d}^{e}(0,400)=10$. Let $\kappa_{d}=1$ and
$\Phi_{d}^{e}(0,400)=10$ obey Assumption 4.1 by setting $\nu_{d}=44$.
Similarly, for the bottom panel in Fig. 5, the number of effective DoS attacks
is $\Phi_{d}^{e}(0,400)=12$, and Assumption 4.1 is met with $\kappa_{d}=1$ and
$\nu_{d}=36$. In addition, according to Thm. 4.1, system with controller (2)
achieves stability if DoS attacks obey $1/\nu_{d}\leq 0.0042$ ($\nu_{d}\geq
238$) and system adopts the deadbeat controller in (26) achieves stability if
DoS attacks obey $1/\nu_{d}\leq 0.0056$ ($\nu_{d}\geq 179$). This reveals the
conservativeness of the result (49).

Figure 6: Relationship between the self-triggering parameters and DoS
duration. The maximum inter-sampling time $\tau_{\max}$: controller (2) (top)
vs. controller (26) (middle).

Finally, the relationship between the self-triggering parameters $\sigma$,
$\tau_{\max}$, and the upper bound on DoS duration $1/\nu_{d}$ under which the
state converges, is depicted in Fig. 6. It can be seen that to achieve better
resilience against DoS attacks, one can choose smaller $\sigma$ (see the
bottom panel in Fig. 6), and the choice of maximum inter-sampling time has
minimal influence on the system resilience (see the top two panels in Fig. 6).
In addition, the bottom panel in Fig. 6 illustrates that under the same
$\sigma$, system with controller (26) achieves better resilience against DoS
attacks while requiring less samplings. Furthermore, (15) and (38) indicate
that the lower bound of $\sigma$ depends on $N$. This is depicted in Fig. 6 by
the axis $N$. As $N\rightarrow\infty$, the lower bound of $\sigma$ approaches
zero, and one can choose smaller $\sigma$ to enhance system resilience against
DoS attacks.

##  6 Conclusions

This paper investigated the self-triggered stabilization problem of linear
control systems with quantized output. Employing a standard observer-based
state feedback control law, a self-triggering mechanism and an output encoding
scheme were designed such that exponential system stability is achieved. When
faster communication rates can be afforded at the C-A channel, an observer-
based deadbeat controller was developed which can considerably simplify the
self-triggering mechanism as well as save the computational and communication
resources. Indeed, the resulting self-triggering mechanism for quantized
output control is as simple as that for quantized state feedback control. To
further endow the system with resilience against DoS attacks, some
modifications were made to the proposed self-triggering mechanism and to the
output encoding scheme. It was shown that as long as the DoS duration time
does not exceed a certain bound, the closed-loop system is guaranteed to be
exponentially stable. Moreover, a trade-off was revealed between the maximum
inter-sampling time and the resilience. Finally, numerical tests were provided
to showcase the effectiveness of the proposed self-triggered quantized
controllers.

This article also opens up several avenues for future research. To name a few,
since the self-triggering parameters are related with the controller gain $K$,
the results in Sec. 3.1 can be extended by using a model predictive
controller. Moreover, for the deadbeat controller in Sec. 3.2, designing an
optimal observer gain $L$ to further reduce the required samplings is another
interesting direction.

## References

  * Åarzén (1999) K. Åarzén.  A simple event-based PID controller.  _Proc. IFAC_ , 32(2):8687–8692, June, 7-9, 1999\.
  * Anta and Tabuada (2010) A. Anta and P. Tabuada.  To sample or not to sample: Self-triggered control for nonlinear systems.  _IEEE Trans. Autom. Control_ , 55(9):2030–2042, Sept. 2010.
  * Brockett and Liberzon (2000) R. W. Brockett and D. Liberzon.  Quantized feedback stabilization of linear systems.  _IEEE Trans. Autom. Control_ , 45(7):1279–1289, July, 2000.
  * Bullo and Liberzon (2006) F. Bullo and D. Liberzon.  Quantized control via locational optimization.  _IEEE Trans. Autom. Control_ , 51(1):2–13, Jan. 2006.
  * Chen et al. (2022) J. Chen, J. Sun, and G. Wang.  From unmanned systems to autonomous intelligent systems.  _Eng._ , (5):16–19, 2022.
  * De Persis and Frasca (2013) C. De Persis and P. Frasca.  Robust self-triggered coordination with ternary controllers.  _IEEE Trans. Autom. Control_ , 58(12):3024–3038, July, 2013.
  * De Persis and Tesi (2015) C. De Persis and P. Tesi.  Input-to-state stabilizing control under Denial-of-Service.  _IEEE Trans. Autom. Control_ , 60(11):2930–2944, Nov. 2015.
  * De Persis and Tesi (2016) C. De Persis and P. Tesi.  Networked control of nonlinear systems under Denial-of-Service.  _Syst. Control Lett._ , 96:124–131, Oct. 2016.
  * Fahmy and O’Reilly (1983) M. M. Fahmy and J. O’Reilly.  Dead-beat control of linear discrete-time systems.  _Int. J. Control_ , 37(4):685–705, Sept. 1983\.
  * Feng and Tesi (2017) S. Feng and P. Tesi.  Resilient control under Denial-of-Service: Robust design.  _Automatica_ , 79:42–51, May, 2017.
  * Franze et al. (2021) G. Franze, W. Lucia, and F. Tedesco.  Resilient model predictive control for constrained cyber-physical systems subject to severe attacks on the communication channels.  _IEEE Trans. Autom. Control_ , 67(4):1822–1836, Apr. 2021.
  * Gommans and Heemels (2015) T. M. P. Gommans and W. P. M. H. Heemels.  Resource-aware MPC for constrained nonlinear systems: A self-triggered control approach.  _Syst. Control Lett._ , 79:59–67, May, 2015.
  * Guo et al. (2023) H. Guo, J. Sun, and Z.-H. Pang.  Residual-based false data injection attacks against multi-sensor estimation systems.  _IEEE/CAA J. Autom. Sinica_ , pages 1–10, Jan. 2023.  10.1109/JAS.2023.123441.
  * Heemels (2013) M. C. F. Heemels, W. P. M. H .and Donkers.  Model-based periodic event-triggered control for linear systems.  _Automatica_ , 49(3):698–711, Mar. 2013.
  * Heemels et al. (2012) W. P. M. H. Heemels, K. H. Johansson, and P. Tabuada.  An introduction to event-triggered and self-triggered control.  In _Proc. IEEE Conf. Decis. Control_ , pages 3270–3285, Maui, HI, USA, Dec. 10-13, 2012.
  * Ikeda et al. (2016) T. Ikeda, M. Nagahara, and D. E. Quevedo.  Quantized self-triggered control by sum-of-absolute-values optimization.  In _Proc. Int. Symp. Math. Theory Netw. Syst._ , Minneapolis, MN, USA, July, 12-15, 2016.
  * Li et al. (2021) T. Li, B. Chen, L. Yu, and W. A. Zhang.  Active security control approach against DoS attacks in cyber-physical systems.  _IEEE Trans. Autom. Control_ , 66(9):4303–4310, Sept. 2021.
  * Liberzon (2003) D. Liberzon.  On stabilization of linear systems with limited information.  _IEEE Trans. Autom. Control_ , 48(2):304–307, Feb. 2003.
  * Liberzon and Hespanha (2005) D. Liberzon and J.P. Hespanha.  Stabilization of nonlinear systems with limited information feedback.  _IEEE Trans. Autom. Control_ , 50(6):910–915, June, 2005.
  * Liu et al. (2022a) W. Liu, J. Sun, G. Wang, F. Bullo, and J. Chen.  Resilient control under quantization and denial-of-service: Co-designing a deadbeat controller and transmission protocol.  _IEEE Trans. Autom. Control_ , 67(8):3879–3891, Aug. 2022a.
  * Liu et al. (2022b) W. Liu, J. Sun, G. Wang, F. Bullo, and J. Chen.  Data-driven resilient predictive control under denial-of-service.  _IEEE Trans. Autom. Control_ , pages 1–16, Sept. 2022b.  10.1109/TAC.2022.3209399.
  * Liu et al. (2023) W. Liu, J. Sun, G. Wang, F. Bullo, and J. Chen.  Data-driven self-triggered control via trajectory prediction.  _IEEE Trans. Autom. Control_ , pages 1–8, 2023.  10.1109/TAC.2023.3244116.
  * Matsume et al. (2021) H. Matsume, Y. Wang, and H. Ishii.  Resilient self/event-triggered consensus based on ternary control.  _Nonlinear Anal. Hyb._ , 42:101091, Nov. 2021.
  * O’Reilly (1983) J. O’Reilly.  _Mathematics in Science and Engineering, Observers for Linear Systems_.  London: Academic Press, 1983.
  * Pasqualetti et al. (2012) F. Pasqualetti, R. Carli, and F. Bullo.  Distributed estimation via iterative projections with application to power network monitoring.  _Automatica_ , 48(5):747–758, May, 2012.
  * Pasqualetti et al. (2013) F. Pasqualetti, F. Dörfler, and F. Bullo.  Attack detection and identification in cyber-physical systems.  _IEEE Trans. Autom. Control_ , 58(11):2715–2729, June, 2013.
  * Wakaiki (2022) M. Wakaiki.  Self-triggered stabilization of discrete-time linear systems with quantized state measurements.  _IEEE Trans. Autom. Control_ , pages 1–8, Mar. 2022.  10.1109/TAC.2022.3159262.
  * Wakaiki and Sano (2020) M. Wakaiki and H. Sano.  Event-triggered control of infinite-dimensional systems.  _SIAM J. Control Optim._ , 58(2):605–635, Mar. 2020.
  * Wakaiki and Yamamoto (2017) M. Wakaiki and Y. Yamamoto.  Stabilization of switched linear systems with quantized output and switching delays.  _IEEE Trans. Autom. Control_ , 62(6):2958–2964, June, 2017.
  * Wakaiki et al. (2020) M. Wakaiki, A. Cetinkaya, and H. Ishii.  Stabilization of networked control systems under DoS attacks and output quantization.  _IEEE Trans. Autom. Control_ , 65(8):3560–3575, Aug. 2020.
  * Walsh and Ye (2002) G. C. Walsh and H. Ye.  Scheduling of networked control systems.  _IEEE Control Syst. Mag._ , 21(1):57–65, Feb. 2002.
  * Wu et al. (2020) G. Wu, G. Wang, J. Sun, and J. Chen.  Optimal partial feedback attacks in cyber-physical power systems.  _IEEE Trans. Autom. Control_ , 65(9):3919–3926, Mar. 2020.
  * Yuan et al. (2020) Y. Yuan, H. Yuan, D. W. C. Ho, and L. Guo.  Resilient control of wireless networked control system under denial-of-service attacks: A cross-layer design approach.  _IEEE Trans. Cybern._ , 50(1):48–60, Oct. 2020\.
  * Zhou et al. (2018) T. Zhou, Z. Zuo, and Y. Wang.  Self-triggered and event-triggered control for linear systems with quantization.  _IEEE Trans. Syst. Man Cybern.: Syst._ , 50(9):3136–3144, Sept. 2018.