# Resilient Control under Quantization and Denial-of-Service: Co-designing a
Deadbeat Controller and Transmission Protocol

Wenjie Liu, Jian Sun, , Gang Wang, ,
Francesco Bullo, , and Jie Chen This work was supported in part by the
National Natural Science Foundation of China under Grants 61925303, 62088101,
U20B2073, 61720106011, and the National Key R$\&$D Program of China under
Grant 2018YFB1700100. W. Liu and G. Wang are with the State Key Lab of
Intelligent Control and Decision of Complex Systems and the School of
Automation, Beijing Institute of Technology, Beijing 100081, China (e-mail:
<EMAIL_ADDRESS>gangwang@bit.edu.cn). J. Sun is with the State Key Lab
of Intelligent Control and Decision of Complex Systems and the School of
Automation, Beijing Institute of Technology, Beijing 100081, China, and the
Beijing Institute of Technology Chongqing Innovation Center, Chongqing 401120,
China (e-mail: sunjian@bit.edu.cn). J. Chen is with the Department of Control
Science and Engineering, Tongji University, Shanghai 201804, China, and also
with the State Key Lab of Intelligent Control and Decision of Complex Systems
and the School of Automation, Beijing Institute of Technology, Beijing 100081,
China (e-mail: chenjie@bit.edu.cn). F. Bullo is with the Mechanical
Engineering Department and the Center of Control, Dynamical Systems and
Computation, UC Santa Barbara, CA 93106-5070, USA (e-mail: bullo@ucsb.edu).

###### Abstract

This paper is concerned with the problem of stabilizing continuous-time linear
time-invariant systems subject to quantization and Denial-of-Service (DoS)
attacks. In this context, two DoS-induced challenges emerge with the design of
resilient encoding schemes, namely, the coupling between encoding strategies
of different signals, and the synchronization between the encoder and decoder.
To address these challenges, a novel structure that is equipped with a
deadbeat controller as well as a delicate transmission protocol for the input
and output channels, co-designed leveraging the controllability index, is put
forward. When both input and output channels are subject to DoS attacks and
quantization, the proposed structure is shown able to decouple the encoding
schemes for input, output, and estimated output signals. This property is
further corroborated by designing encoding schemes as well as conditions that
ensure exponential stability of the closed-loop system. On the other hand,
when only the output channel is subject to network phenomenon, the proposed
structure can achieve exponential stabilization without acknowledgment (ACK)
signals, in contrast to existing ACK-based results. Finally, a numerical
example is given to demonstrate the practical merits of the proposed approach
as well as the theory.

###### Index Terms:

Denial-of-Service attacks, quantization, deadbeat control, acknowledgment-free
protocol.

##  I Introduction

Driven by recent advances in computing, communication, and networking
technologies, modern engineering systems (e.g., [1, 2, 3]) have gradually
shifted their computing and control workload to the cloud, and even edge with
data transmitted over wired or wireless networks. Despite their flexibility,
such network-based control systems (a.k.a., networked control systems) are
known vulnerable to cyber threats [4, 5]. In fact, existing works have shown
that malicious attacks can severely disrupt the control performance and even
render the system unstable [6]. Examples of such failures in widely used
safety- and security-critical control systems nowadays could put our lives and
even national infrastructure at risk [7].

Several types of cyberattacks have been studied, including replay attacks [8,
9], false-data injection attacks [10, 11], and Denial-of-Service (DoS) attacks
[12, 13, 14]. Relative to the others, DoS attacks can cause jamming in
communication channels with little knowledge of system dynamics. They are easy
to launch and have received considerable attention [15]. For instance, the
work [16] developed a general DoS framework, under which closed-loop system
stability can be preserved via state-feedback control, provided certain DoS
attack frequency and duration conditions are met. This result has been
extended in several directions, e.g., via output-feedback control in [17], as
well as considering multiple output channels in [18].

All the aforementioned works assumed that the communication channels have an
infinite data-rate. Clearly for real-world engineering systems, this condition
is difficult to be met. Systems with digital communication channels offer a
basic paradigm. The problem of limited bandwidth have been studied by
accounting for the effect of quantization. There is a great deal of research
indicating that even without attacks, quantization can compromise system
performance [19], which is often addressed by designing suitable encoding
schemes and providing enough quantization levels. To name a few, for
stabilization of systems with quantized measurements, [20] first introduced
the so-called “zooming-in” and “zooming-out” method. Following this work, a
number of stabilization encoding schemes have been designed for systems with
quantized output feedback in [21, 22], and switched systems in [23, 24, 25,
26]. Recently, a few works have considered these two factors (i.e.,
quantization and DoS attacks) simultaneously; see [27, 28, 29, 30, 31]. The
trade-off between system resilience against DoS attacks and data-rate was
analyzed in [28]. The minimum data-rate for stabilizing a centralized system
and a multi-agent system were derived in [29] and [30], respectively.
Capitalizing on the zooming-in and -out method, the work [31] designed a
resilient output encoding scheme for systems whose output channel is subject
to DoS attacks and limited data-rate.

The goal of this paper is to stabilize systems with both input (controller-to-
plant) and output (plant-to-controller) channels subject to DoS attacks and
limited bandwidth. To this aim, the quantizer encoding schemes should be
carefully designed. In the absence of DoS attacks, the work [22] developed
encoding schemes for signals transmitted through both input and output
channels. However, their schemes cannot be applied here, due to the coupling
between encoding strategies for different signals in the presence of DoS
attacks. To overcome this challenge, we put forth a delicate structure,
including a deadbeat controller and a transmission protocol. Our protocol
requires signals transmitted through the input channel at a higher rate than
those through the output channel. Precisely, their transmission rate ratio is
exactly the controllability index of the system. Its efficacy is corroborated
by the possibility to decouple design of different encoding schemes as, well
as, establishing closed-loop stability conditions. We further apply this
structure to stabilize systems with only output channel has network
imperfections. In this scenario, it is proved that the proposed structure can
secure the synchronization between encoder and decoder even without
acknowledgments (ACKs), which are required by existing works, e.g., [31, 28].

In a nutshell, the main contributions of the present work are summarized as
follows.

  * c1)

To cope with the coupling and synchronization issues, a structure consisting
of a deadbeat controller and a transmission protocol for input and output
channels, co-designed in terms of the controllability index, is advocated.

  * c2)

Under this structure, the input, output, and estimated output encoding schemes
can be designed separately to achieve closed-loop stability when both input
and output channels are subject to DoS attacks and quantization; and,

  * c3)

When such network phenomena appear only in the output channel, an encoding
scheme is designed such that the system can be stabilized through an ACK-free
protocol, that is in sharp contrast to existing ACK-based results.

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

##  II Preliminaries and Problem Formulation

###  II-A Problem formulation

In this paper, we study the networked control architecture in Fig. 1, where a
plant is to be stabilized by a remote digital controller over a network
subject to DoS attacks. The plant is described by the following dynamics

$\displaystyle\dot{x}(t)=Ax(t)+Bu(t)$ (1a) $\displaystyle y(t)=Cx(t)$ (1b)

where $x(t)\in\mathbb{R}^{n_{x}},u(t)\in\mathbb{R}^{n_{u}}$, and
$y(t)\in\mathbb{R}^{n_{y}}$ are the state, the control input, and the output,
respectively. Here, we consider output signals and control inputs to be
transmitted through different channels over a shared network, which are
accordingly referred to as output channel and input channel. Specifically,
data transmissions over the output channel occur periodically with interval
$\Delta>0$. That is, the output encoder samples $y(t)$ and sends its quantized
version to the controller every $\Delta$ time. Likewise, the digital
controller generates control signals and transmits their quantized values to
the plant periodically with interval $\delta>0$. At the plant side, the
quantized control inputs are first decoded, then pass through a zero-order
hold (ZOH) before entering the plant. To maintain the synchronization between
input and output transmissions, we choose $\delta=\Delta/b$ for some
$b\in\mathbb{N}$. For future reference, let

$x_{q,k}:=x(q\Delta+k\delta),\qquad y_{q,k}:=y(q\Delta+k\delta)$

for every $q\in\mathbb{Z}_{\geq 0}$, and $k=0,\cdots\\!,\frac{\Delta}{b}$, and

$A_{d}:=e^{A\delta},\qquad B_{d}:=\int_{0}^{\delta}{e^{As}B\,ds}.$ (2)

Moreover, we use $x_{q}$ to denote $x_{q,0}$ for simplicity.

Figure 1: Networked control system with both input (the blue line) and output
(the green line) channels subject to DoS attacks.

We make the following assumptions on system (1).

######  Assumption 1 (Controllability and observability).

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

######  Assumption 2 (Initial state bound).

An upper bound on the initial state $|x_{0}|$ is known.

######  Remark 1.

Thanks to As. 1, it has been shown in [32] that if $\delta$ is non-
pathological, then $(A_{d},B_{d})$ in (2) is controllable. Let $\eta$ denote
its controllability index, which can be computed by evaluating ${\rm
rank}[B_{d},\cdots\\!,A_{d}^{\eta}B_{d}]=n_{x}$. Similarly, $(C,A_{d}^{\eta})$
is observable. An upper bound on the initial state in As. 2 can be derived via
the zooming-out method; see [31, Sec. 4].

###  II-B Denial-of-Service attack

In Fig. 1, since both input and output signals are transmitted periodically,
we adopt the discrete-time DoS attack model in [31]. Under this model, attacks
are launched only at output transmission instants, and each lasts for an
output transmission period $\Delta$. This model is general enough since it
only poses requirements on the frequency and duration of DoS attacks. Here,
DoS frequency is the number of DoS _off/on_ switches over a fixed time
interval, while DoS duration represents the total number of attacks.

######  Assumption 3 (DoS frequency).

There exist constants $\kappa_{f}\in\mathbb{R}_{\geq 0}$ and
$\nu_{f}\in\mathbb{R}_{\geq 2}$ such that DoS frequency satisfies

$\Phi_{f}(q)\leq\kappa_{f}+\frac{q}{\nu_{f}}$ (3)

over time interval $[0,q\Delta)$, where $q\in\mathbb{Z}_{\geq 0}$.

######  Assumption 4 (DoS duration).

There exist constants $\kappa_{d}\in\mathbb{R}_{\geq 0}$ and
$\nu_{d}\in\mathbb{Z}_{\geq 1}$ such that DoS duration satisfies

$\Phi_{d}(q)\leq\kappa_{d}+\frac{q}{\nu_{d}}$ (4)

over time interval $[0,q\Delta)$, where $q\in\mathbb{Z}_{\geq 0}$.

######  Remark 2.

Given its generality, this attack model has been widely used in e.g., [31, 30,
17, 18, 28, 16]. As pointed out in [33], $\nu_{f}\Delta$ in As. 3 can be
regarded as the average dwell-time between two consecutive DoS attacks
_off/on_ switches. On the other hand, As. 4 indicates that, the average
duration of DoS attacks does not exceed a proportion $1/\nu_{d}$ of the time
interval. Constants $\kappa_{f}$ and $\kappa_{d}$ are also known as chatter
bounds. Conditions $\nu_{d}\geq 1$ and $\nu_{f}\geq 2$ suggest that DoS
attacks are not strong enough to prevent all packets from being transmitted,
thus rendering it possible for the system to be stabilized by suitable control
strategies.

##  III Networked Phenomena at Both Input and Output Channels

This section aims to design resilient encoding schemes for stabilization of
system (1) via a remote observer-based digital controller over communication
channels subject to limited bandwidth and DoS attacks; see Fig. 1. To this
end, there are three signals that need to be quantized, i.e., the estimated
output by observer $\hat{y}_{q}$, the control input $u_{q,k}$, and the plant
output $y_{q}$, with their quantized values denote by $Q_{1}(\hat{y}_{q})$,
$Q_{2}(u_{q,k})$, and $Q_{3}(y_{q})$, respectively. In addition, since the
input and output channels share a communication network, we assume for
simplicity that, once there is a DoS attack, neither the input nor the output
signals will be received, and both of them are set to the default zero. In
this manner, the decoder and encoder at both input and output sides can infer
whether there is an attack. Further, their quantization ranges and centers are
identical at every transmission instant. As a result, they can be synchronized
even with an ACK-free protocol.

###  III-A Controller architecture

To stabilize system (1), we put forth a two-stage observer-based controller by
considering whether there is an attack or not. Specifically, in the absence of
DoS attacks, both $Q_{3}(y_{q})$ and $Q_{1}(\hat{y}_{q-1,\eta})$ are available
at the observer side, so we construct the following controller

$\displaystyle\hat{x}_{q,k+1}\\!=\\!A_{d}\hat{x}_{q,k}\\!+\\!B_{d}u_{q,k},\\!\\!$
$\displaystyle k$ $\displaystyle\leq\eta-1$ (5a)
$\displaystyle\hat{x}_{q}\\!=\\!\hat{x}_{q-1,k}\\!+\\!M\big{[}Q_{3}(y_{q})\\!-\\!Q_{1}(\hat{y}_{q-1,k})\big{]},\\!\\!$
$\displaystyle k$ $\displaystyle=\eta$ (5b)
$\displaystyle\hat{y}_{q,k}\\!=\\!C\hat{x}_{q,k}$ (5c) $\displaystyle
u_{q,k}\\!=\\!K\hat{x}_{q,k}$ (5d)

where the initial condition $\hat{x}_{0}$ is given by $\hat{x}_{0}=0$, and
$\delta$ is chosen such that

$\delta=\frac{\Delta}{\eta}.$ (6)

Matrix $M\in\mathbb{R}^{n_{x}\times n_{y}}$ can be regarded as an observer
gain such that $R:=A_{d}^{\eta}(I-MC)$ is schur stable, which always exists
since $(C,A_{d}^{\eta})$ is observable. Moreover, $(A_{d},B_{d})$ is
controllable, thus a controller gain matrix $K\in\mathbb{R}^{n_{u}\times
n_{x}}$ can be designed such that

$\bar{R}^{\eta}=(A_{d}+B_{d}K)^{\eta}=0.$ (7)

######  Remark 3.

Matrix $K$ obeying (7) is also known as a class of deadbeat controller gain,
since it assigns all the eigenvalues of $A_{d}+B_{d}K$ to the origin. Solution
of this eigenstructure assignment problem is non-unique, and can be obtained
through several approaches, e.g., [34].

On the other hand, when there is a DoS attack, none of $Q_{1}(\hat{y}_{q})$,
$Q_{2}(u_{q,k})$, or $Q_{3}(y_{q})$ can be received, thus we simply employ an
open-loop controller as follows

$\displaystyle\hat{x}_{q,k+1}=A_{d}\hat{x}_{q,k}$ (8a)
$\displaystyle\hat{y}_{q,k}=C\hat{x}_{q,k}$ (8b) $\displaystyle u_{q,k}=0$
(8c)

with the initial estimated state $\hat{x}_{0}=0$.

In addition, to apply the discrete-time signal $Q_{2}(u_{q,k})$ to the
continuous-time system (1a), a ZOH is used, and the control input is given by

$u(t)=Q_{2}(u_{q,k}),\qquad q\Delta+k\delta\leq t<q\Delta+(k+1)\delta$

where $k=0,\cdots\\!,\eta-1$.

###  III-B Quantizer

We first design quantizers at the input channel. According to (5a) and (5b),
$u_{q,k}$ is needed for feedback control, whereas $\hat{y}_{q,k}$, resetting
the estimated state, is required at each successful transmission instants.
Therefore, the controller sends $u_{q,k}$ and $\hat{y}_{q,\eta}$ to the
quantizers periodically at a different rate. In more precise terms, periods
for the former and the latter are $\delta$, and $\eta\delta$, respectively.
Let $E_{1,q}\geq 0$ and $E_{2,q,k}\geq 0$ satisfy

$|\hat{y}_{q-1,\eta}|\leq E_{1,q},\ \ \ |u_{q,k}|\leq E_{2,q,k}.$ (9)

Suppose there are $N_{1}$ ($N_{2}$) levels for quantization of
$\hat{y}_{q,\eta}$ ($u_{q,k}$). Partition the hypercubes at the encoders

$\begin{split}\\{\hat{y}\in\mathbb{R}^{n_{y}}:|\hat{y}_{q-1,\eta}|\leq
E_{1,q}\\},~{}~{}\\{u\in\mathbb{R}^{n_{u}}:|u_{q,k}|\leq
E_{2,q,k}\\}\end{split}$

into $N_{1}^{n_{y}}$, and $N_{2}^{n_{u}}$ equal-sized boxes, respectively. In
addition, each box is represented by a value in
$\\{1,\cdots\\!,N_{1}^{n_{y}}\\}$, or $\\{1,\cdots\\!,N_{2}^{n_{u}}\\}$
following a bijection mapping. Indices that denote the partitioned boxes
containing $\hat{y}_{q,\eta}$ and $u_{q,k}$ are then sent to the decoders. If
$\hat{y}_{q,\eta}$ and $u_{q,k}$ are on the boundary of several boxes, then
anyone of them can be chosen. At the decoders side, $Q_{1}(\hat{y}_{q,\eta})$
and $Q_{2}(u_{q,k})$ are recovered from the indices. This implies that the
encoder and its corresponding decoder should share the same quantization
ranges and centers. Since DoS attacks block both input and output signals from
transmitting, encoders and decoders at both sides of input and output channels
are naturally synchronized. The quantization errors of the aforementioned
encoding schemes obey

$|\hat{y}_{q-1,\eta}-Q_{1}(\hat{y}_{q-1,\eta})|\leq\frac{E_{1,q}}{N_{1}},$
(10) $|u_{q,k}-Q_{2}(u_{q,k})|\leq\frac{E_{2,q,k}}{N_{2}}.$ (11)

Since $\hat{x}_{0}=0$, we deduce that $\hat{y}_{0}=u_{0}=0$. Therefore, the
initial bounds $E_{1,0}$ and $E_{2,0,0}$ can be set by

$E_{1,0}=0,\ \ \ E_{2,0,0}=0.$

Moreover, as for the output $y_{q}$, choose $E_{3,q}\geq 0$ such that

$|y_{q}-Q_{1}(\hat{y}_{q-1,\eta})|\leq E_{3,q}.$ (12)

Let $N_{3}$ be the quantization level of $y_{q}$. The hypercube

$\\{y\in\mathbb{R}^{n_{y}}:|y_{q}-Q_{1}(\hat{y}_{q-1,\eta})|\leq E_{3,q}\\}.$
(13)

is partitioned into $N_{3}^{n_{y}}$ equal-sized boxes with the center
$Q_{1}(\hat{y}_{q-1,\eta})$. Then, following the same procedure as the above
two quantizers, $Q_{3}(y_{q})$ is transmitted to the controller every $\Delta$
time. The quantization error satisfies

$|y_{q}-Q_{3}(y_{q})|\leq\frac{E_{3,q}}{N_{3}}.$

Define error of the system $e_{q,k}:=x_{q,k}-\hat{x}_{q,k}$. Combining
$\hat{x}_{0}=0$ with As. 2, we deduce that the initial error obeys
$|e_{0}|=|x_{0}|$. Thus it suffices to set $E_{3,0}:=\|C\||x_{0}|$.

###  III-C Stability analysis

In this subsection, we start by presenting encoding schemes
$\\{E_{p,q,k}\\}(p=1,2,3)$, followed by formal stability conditions. Design
$\\{E_{1,q}:q\in\mathbb{Z}_{\geq 1}\\}$ such that

$E_{1,q}=E_{1,0},\qquad\forall q\in\mathbb{Z}_{\geq 1}$ (14)

and let $\\{E_{2,q,k}:q\in\mathbb{Z}_{\geq 1},k=0,\cdots\\!,\eta-1\\}$ be
updated by

$E_{2,q,k}=\frac{N_{3}-1}{N_{3}}\left\|K\bar{R}^{k}M\right\|E_{3,q},\qquad{\rm{~{}if~{}}}q\Delta=s_{r}.$
(15)

Moreover, the sequence $\\{E_{3,q}:q\in\mathbb{Z}_{\geq 1}\\}$ is given by

$E_{3,q+1}:=\left\\{\begin{aligned} &\hat{\theta}_{a}E_{3,q},&q\Delta\neq
s_{r}\\\ &\hat{\theta}_{0}E_{3,q},&(q-1)\Delta\neq s_{r},q\Delta=s_{r}\\\
&\hat{\theta}_{na}E_{3,q},&(q-1)\Delta=s_{r},q\Delta=s_{r}\end{aligned}\right.$
(16)

where

$\displaystyle\hat{\theta}_{a}$ $\displaystyle:=\|A_{d}^{\eta}\|$
$\displaystyle\hat{\theta}_{0}$
$\displaystyle:=a_{0}\rho+\frac{\|C\|a_{1}}{N_{3}}+\|C\|a_{2}\frac{N_{3}-1}{N_{2}N_{3}}$
$\displaystyle\hat{\theta}_{na}$
$\displaystyle:=\rho+\frac{\|C\|a_{1}}{N_{3}}+\|C\|a_{2}\frac{N_{3}-1}{N_{2}N_{3}}$

with positive constants $a_{0},a_{1},a_{2}$, and $0<\rho<1$ validating the
following for all $\ell\geq 1$

$\displaystyle\big{\|}R^{\ell}\big{\|}\leq
a_{0}\rho^{\ell},~{}~{}~{}~{}\big{\|}R^{\ell}A_{d}^{\eta}M\big{\|}\leq
a_{1}\rho^{\ell}$ (17)
$\displaystyle\sum_{i=0}^{\eta-1}{\big{\|}R^{\ell}A_{d}^{\eta-i-1}B_{d}\big{\|}\big{\|}K\bar{R}^{i}M\big{\|}}\leq
a_{2}\rho^{\ell}.$

Since $R$ is schur stable, there always exist such constants.

Next, we show that our designed schemes above are resilient to DoS attacks,
which is one of our main results too.

######  Theorem 1.

Consider system (1) with the observer-based controller in (5) and (8), with
$K$ obeying (7) and $M$ chosen such that $R$ is schur stable. Let As. 1–4
hold. If i) the input and output transmission periods adhere to (6), ii) the
number of quantization levels $N_{1}\geq 1$ is odd,

$\begin{split}N_{2}\\!>\\!\max\\!{\left\\{\frac{a_{2}\|C\|}{1-\rho},\,\frac{a_{2}}{a_{1}}\right\\}},\,{\rm{~{}and~{}}}N_{3}\\!>\\!\frac{\|C\|a_{1}-\frac{\|C\|a_{2}}{N_{2}}}{1-\rho-\frac{\|C\|a_{2}}{N_{2}}}\end{split}$
(18)

and, iii) DoS attacks satisfy

$\frac{1}{\nu_{d}}\leq\frac{\log{(1/\hat{\theta}_{na})}}{\log{(\hat{\theta}_{a}/\hat{\theta}_{na})}}-\frac{\log{(\hat{\theta}_{0}/\hat{\theta}_{na})}}{\log{(\hat{\theta}_{a}/\hat{\theta}_{na})}}\frac{1}{\nu_{f}}$
(19)

then the system is exponentially stable under the encoding scheme with error
bounds $\\{E_{p,q,k}:q\in\mathbb{Z}_{\geq 1},k=0,\cdots\\!,\eta-1\\}(p=1,2,3)$
constructed by the update rule in (14)-(16).

We begin proving Thm. 1 by giving a lemma demonstrating that the update rules
in (14)-(16) satisfy (9) and (12).

######  Lemma 1.

Consider system (1) with the controller in (5) and (8), where $K$ obeys (7).
Let As. 1–4 hold. If $\\{E_{p,q,k}:q\in\mathbb{Z}_{\geq
1},k=0,\cdots\\!,\eta-1\\}(p=1,2,3)$ obey (14)-(16), then (9) and (12) hold
true for all $q\in\mathbb{Z}_{\geq 1}$.

###### Proof.

Encoding schemes for systems with quantized inputs and outputs in the absence
of DoS attacks have been discussed in [22]. However, their methods cannot be
directly applied due to the DoS-induced coupling between these schemes. This
challenge is addressed through our carefully designed controller structure in
(5)-(7). According to (5) and (7),

$\hat{y}_{q-1,\eta}=C\hat{x}_{q-1,\eta}=C\bar{R}^{\eta}\hat{x}_{q-1}=0$

holds true irrespective of DoS attacks, which implies
$E_{1,q}\geq|\hat{y}_{q-1,\eta}|=E_{1,0}$ for all $q\in\mathbb{Z}_{\geq 1}$,
so $E_{1,q}$ remains unchanged. This result further indicates that
$Q_{1}(\hat{y}_{q-1,\eta})=0$. Hence, it follows from (13) that the
quantization center of $Q_{3}(y_{q})$ is at the origin. On the other hand, if
no DoS attacks occur within $[q_{1}\Delta,(q_{1}+1)\Delta)$, then

$\displaystyle\hat{x}_{q_{1},k}=$ $\displaystyle\
(A_{d}+B_{d}K)^{k}\hat{x}_{q_{1}}$ (20) $\displaystyle=$ $\displaystyle\
\bar{R}^{k}(\hat{x}_{q_{1}-1,\eta}+M[Q_{3}(y_{q_{1}})-Q_{1}(\hat{y}_{q_{1}-1,\eta})])$
$\displaystyle=$ $\displaystyle\
\bar{R}^{k}M[Q_{3}(y_{q_{1}})-Q_{1}(\hat{y}_{q_{1}-1,\eta})]$

hence $u_{q_{1},k}$ can be expressed by
$Q_{3}(y_{q_{1}})-Q_{1}(\hat{y}_{q_{1}-1,\eta})$. In addition, since

$|Q_{3}(y_{q})-Q_{1}(\hat{y}_{q-1,\eta})|\leq\frac{N_{3}-1}{N_{3}}E_{3,q}$

it follows that, in the absence of DoS attacks, $u_{q,k}$ satisfies

$|u_{q,k}|\leq\frac{N_{3}-1}{N_{3}}\big{\|}K\bar{R}^{k}M\big{\|}E_{3,q}=:E_{2,q,k}$
(21)

for all $q\geq 1,k=0,\cdots\\!,\eta-1$. When DoS attacks occur, the plant
cannot receive inputs from controller. In other words, $E_{2,q,k}$ only
depends on the latest $E_{3,q}$, thus $E_{2,q,k}$ can remain unchanged during
DoS attacks.

Following the definitions of $E_{1,q}$ and $E_{2,q,k}$, we are able to design
sequence $\\{E_{3,q}:q\in\mathbb{Z}_{\geq 1}\\}$. First, in the absence of DoS
attacks, the error just before each transmission instant,
$e_{q-1,\eta}=x_{q}-\hat{x}_{q-1,\eta}$, satisfies

$\displaystyle e_{q-1,\eta}=$ $\displaystyle\
A_{d}^{\eta}(I-MC)e_{q-1}-A_{d}^{\eta}M[Q_{3}(y_{q})-y_{q}]$
$\displaystyle+\sum_{i=0}^{\eta-1}{A_{d}^{\eta-i-1}B_{d}[Q_{2}(u_{q-1,i})-u_{q-1,i}]}$
$\displaystyle-
A_{d}^{\eta}\big{[}\hat{y}_{q-1,\eta}-Q_{1}(\hat{y}_{q-1,\eta})\big{]}$

which implies that $e_{q-1,\eta}$ generally relies on
$\hat{y}_{q-1,\eta},u_{q,k}$, and itself, thus introducing coupling in
$E_{3,q}$ design. Here, this issue is addressed by (7). To see this, recalling
(21), $\hat{y}_{q-1,\eta}=0$, and $Q_{1}(\hat{y}_{q-1,\eta})=0$, we have that

$\displaystyle
e_{q+{\ell}-1,\eta}=R^{\ell}e_{q-1}+\sum_{j=0}^{{\ell}-1}R^{j}A_{d}^{\eta}M(Q_{3}(y_{q-j})-y_{q-j})$
$\displaystyle+\sum_{j=0}^{{\ell}-1}R^{j}\sum_{i=0}^{\eta\\!-\\!1}A_{d}^{\eta\\!-\\!i-1}B_{d}\big{[}Q_{2}(u_{q+\ell\\!-\\!j\\!-\\!1,i})\\!-\\!u_{q+{\ell}-j-1,i}\big{]}.$
(22)

Define $E_{3,q}$ as follows

$\displaystyle E_{3,q+{\ell}}:=$ $\displaystyle\ a_{0}\rho^{\ell}E_{3,q}$
$\displaystyle+\sum_{i=0}^{{\ell}-1}\left(\frac{(N_{3}-1)a_{2}\|C\|}{N_{2}N_{3}}+\frac{\|C\|a_{1}}{N_{3}}\right)\rho^{i}E_{3,q-i}.$

Hence, combining (21) with (III-C) yields

$\displaystyle|y_{q+1}-Q_{1}(\hat{y}_{q,\eta})|$
$\displaystyle\leq|y_{q+1}-\hat{y}_{q,\eta}|+|\hat{y}_{q,\eta}-Q_{1}(\hat{y}_{q,\eta})|$
$\displaystyle\leq\|C\||x_{q+1}-\hat{x}_{q,\eta}|$
$\displaystyle\leq\hat{\theta}_{na}E_{3,q}\leq E_{3,q+1}.$ (23)

Moreover, since both the input and output channels are blocked in the presence
of DoS attacks, and $\hat{y}_{q,\eta}=0$, due to the property of $\bar{R}$, it
follows that

$|y_{q+1}-Q_{1}(\hat{y}_{q,\eta})|\leq\hat{\theta}_{a}E_{3,q}\leq E_{3,q+1}$

and we complete the proof. ∎

Next, we establish upper bounds on the sequences
$\\{E_{p,q,k}:q\in\mathbb{Z}_{\geq 1},k=0,\cdots\\!,\eta-1\\}(p=1,2,3)$, whose
existence will imply the boundness of state trajectory.

######  Lemma 2.

Consider system (1) with controller in (5) and (8), where $K$ satisfies (7)
and $M$ is chosen such that $R$ is schur stable. Let the assumptions and
conditions in Thm. 1 hold. If further $\\{E_{p,q,k}:q\in\mathbb{Z}_{\geq
1},k=0,\cdots\\!,\eta-1\\}(p=1,2,3)$ obey (14)-(16), there exist
$\Omega_{1}\geq 1$ and $\gamma\in(0,1)$ such that

$E_{3,q}\leq\Omega_{1}\gamma^{q}|x_{0}|,\qquad\forall k\in\mathbb{Z}_{\geq 1}$
(24)

and $E_{1,q}=E_{1,0}$, and $E_{2,q,k}\leq\Omega_{2}\gamma^{q}|x_{0}|$.

###### Proof.

Using (14), $E_{1,q}$ remains unchanged within the considered interval,
therefore, $E_{1,q}=E_{1,0}$ holds for all $q\in\mathbb{Z}_{\geq 1}$. The
proof for $E_{3,q}\leq\Omega\gamma^{q}|x_{0}|,\forall q\in\mathbb{Z}_{\geq 1}$
follows directly from that of Lemma 3.9 in [31], where
$\Omega_{1}:=\frac{\hat{\theta}_{0}^{\Pi_{f}+1}\hat{\theta}_{a}^{\Pi_{d}}}{\hat{\theta}_{na}^{\Pi_{f}+\Pi_{d}+1}}\big{(}\hat{\theta}_{na}\big{(}\frac{\hat{\theta}_{0}}{\hat{\theta}_{na}}\big{)}^{\nu_{f}}\big{(}\frac{\hat{\theta}_{a}}{\hat{\theta}_{na}}\big{)}^{\nu_{d}}\big{)}$.
Moreover, applying (21), $E_{2,q,k}\leq\Omega_{2}\gamma^{q}|x_{0}|$ can be
verified with
$\Omega_{2}:=\frac{N_{3}-1}{N_{3}}\|K\bar{R}^{\eta-1}\|\Omega_{1}$. ∎

We are now in a position to prove Thm. 1.

###### Proof of Theorem 1.

We first establish the bound of the state $x$ at the transmission instants,
i.e., $|x(q\Delta)|$, then derive its bound at the sampling instants, i.e.,
$|x(q\Delta+k\delta)|$. Finally, combining these two bounds to yield bound
$|x(t)|$ in the considered horizon.

First, according to (1), (5), and (8), one has

$\displaystyle x_{q,\eta}=$
$\displaystyle~{}\bar{R}^{\eta}x_{q,k}+\sum_{i=0}^{\eta-1}\bar{R}^{i}B_{d}K(x_{q,\eta-i-1}-\hat{x}_{q,\eta-i-1})$
$\displaystyle+\sum_{i=0}^{\eta-1}\bar{R}^{i}B_{d}(Q_{2}(u_{q,\eta-i-1})-K\hat{x}_{q,\eta-i-1})$
(25)

and

$\displaystyle|x_{q,\eta}|\leq$
$\displaystyle~{}\|\bar{R}^{\eta}\||x_{q}|+\sum_{i=0}^{\eta-1}\|\bar{R}^{i}B_{d}K\||(x_{q,\eta-i-1}-\hat{x}_{q,\eta-i-1})|$
$\displaystyle+\sum_{i=0}^{\eta-1}\|\bar{R}^{i}B_{d}\||(Q_{2}(u_{q,\eta-i-1})-K\hat{x}_{q,\eta-i-1})|.$
(26)

Since (III-C), it follows that

$\|x_{q}-\hat{x}_{q-1,\eta}\|\leq\frac{E_{3,q}}{\|C\|}.$ (27)

Noticing that $\bar{R}^{\eta}=0$, substituting (15) and (27) into (III-C),

$\displaystyle|x_{q,\eta}|\leq$ $\displaystyle\
\sum_{i=0}^{\eta-1}\frac{N_{3}-1}{N_{2}N_{3}}\|\bar{R}^{i}B_{d}\|\|K\bar{R}^{\eta-i-1}M\|E_{3,q}$
$\displaystyle+\sum_{i=0}^{\eta-1}\frac{\|\bar{R}^{i}B_{d}KA_{d}^{\eta-i-1}\|}{\|C\|}E_{3,q}$
$\displaystyle\leq$ $\displaystyle\ \Omega_{x}\Omega_{1}\gamma^{q}|x_{0}|$
(28)

where
$\Omega_{x}:=\sum_{i=0}^{\eta-1}\big{\\{}\frac{N_{3}-1}{N_{2}N_{3}}\|\bar{R}^{i}B_{d}\|\|K\bar{R}^{\eta-i-1}M\|+\frac{\|\bar{R}^{i}B_{d}KA_{d}^{\eta-i-1}\|}{\|C\|}\big{\\}}$,
and the last inequality holds due to (24).

Since $x_{q,k+1}=A_{d}x_{q,k}+B_{d}Q_{2}(u_{q,k})$, we have that

$\displaystyle|x_{q,{\ell}}|\leq$ $\displaystyle\
\|\bar{R}^{{\ell}}\||x_{q}|+\sum_{i=0}^{{\ell}-1}\frac{\|\bar{R}^{i}B_{d}KA_{d}^{{\ell}-i-1}\|}{\|C\|}E_{3,q}$
$\displaystyle+\sum_{i=0}^{{\ell}-1}\frac{N_{3}-1}{N_{2}N_{3}}\|\bar{R}^{i}B_{d}\|\|K\bar{R}^{{\ell}-i-1}M\|E_{3,q}$
(29) $\displaystyle\leq$ $\displaystyle\
\Omega_{x}\Omega_{1}\gamma^{q}|x_{0}|+\Omega_{3}\gamma^{q}|x_{0}|\leq\bar{\Omega}_{x}\gamma^{q}|x_{0}|$

where
$\Omega_{3}:=\Omega_{1}\max_{{\ell}}\sum_{i=0}^{{\ell}}\big{\\{}\frac{N_{3}-1}{N_{2}N_{3}}\|\bar{R}^{i}B_{d}\|\|K\bar{R}^{{\ell}-1}M\|+\frac{\|\bar{R}^{i}B_{d}KA_{d}^{\eta-i-1}\|}{\|C\|}\big{\\}},{\ell}\in\\{1,\cdots\\!,\eta-1\\}$,
and $\bar{\Omega}_{x}:=\Omega_{3}+\|\bar{R}^{{\ell}}\|\Omega_{x}$.

Finally, abiding by (1), $x(t)$ satisfies

$\displaystyle
x(t)=e^{A(t-q\Delta-k\delta)}+\int_{q\Delta+k\delta}^{t}e^{As}BQ_{2}(u_{q,k})\,ds$

for all $t\in[q\Delta+k\delta,q\Delta+(k+1)\delta)$. Combining Lem. 2 and
(III-C), it follows that

$\displaystyle|x(t)|\leq\big{(}\|A_{d}\|\bar{\Omega}_{x}+\frac{N_{2}+1}{N_{2}}\|B_{d}\|\Omega_{2}\big{)}\gamma^{q}|x_{0}|\leq\tilde{\Omega}_{x}e^{-\sigma
t}|x_{0}|$

where $\sigma:=\frac{1}{\eta\delta}\log\frac{1}{\gamma}$ and
$\tilde{\Omega}_{x}:=\|A_{d}\|\bar{\Omega}_{x}+\frac{N_{2}+1}{N_{2}}\|B_{d}\|\Omega_{2}$.
This implies exponential convergence of the state. ∎

######  Remark 4.

Leveraging the same technique as in Rmk. 3, one can also design $M$ to nullify
$R^{\mu}=0$, where $\mu$ is the observability index of $(C,A_{d}^{\eta})$. A
direct benefit from using the deadbeat observer gain is that the encoding
schemes can be simplified, since $R^{\ell}=0$ holds for all $\ell\geq\mu$.
However, the results in [22] indicate that despite exhibiting faster
convergence and fewer quantization levels, due to the deadbeat property of
matrices $R$ and $\bar{R}$, the quantization step size $E_{p,q}/N_{p}$ is
large, which leads to large quantization errors. Moreover, it was shown in
[31] that if the quantization step size $E_{p,q}/N_{p}$ grows slower during
DoS attacks, then the overshoot from an attack is smaller, and the level of
system robustness is stronger. Therefore, instead of a deadbeat observer gain,
a general one that can make $R$ schur stable is employed in the present work.

##  IV Network Phenomena at Output Channel

In this section, we consider stabilizing linear systems over a communication
network, where only the output channel is subject to DoS attacks, i.e., the
input channel is assumed ideal; see Fig. 2. The transmission policy in the
previous section is considered here; that is, the digital controller receives
quantized output $Q(y_{q})$ from the plant with period $\Delta$ and generates
control input $u_{q,k}$ with period $\delta$. Notice that the decoder can
recover the correct quantized value from the index sent by the encoder only if
they share the same quantization ranges and centers. It is thus necessary to
ensure that the encoder and the decoder are _synchronized_ before designing
encoding schemes. A direct way to maintain synchronization is through using an
ACK-based protocol; see Fig. 3, which has been adopted in previous studies,
such as, [28, 31]. Nevertheless, in real-time applications, protocols without
ACKs, e.g., UDP, are often preferred since the resulting implementation is
simpler as well as saves the additional energy required for sending ACKs [35].
Hence, in the following, we first show that method for stabilizing systems
with ACK-based protocols can no longer be used under ACK-free protocols. Then,
we demonstrate that our proposed methods can inform the encoder of DoS attacks
from zero inputs, thus the decoder and the encoder can be synchronized even
without ACKs.

Figure 2: Closed-loop system with an ACK-free protocol.
Figure 3: Closed-loop system with an ACK-based protocol. The black dashed line
represents the ACKs sent from the decoder to the encoder.

###  IV-A Controller under an acknowledgment-based protocol

Recall that $\\{s_{r}\\}_{r\in\mathbb{N}_{0}}$ collects the sequence of
successful transmission instants. Let $\delta=\Delta$, and choose $K$ such
that $\bar{R}=A_{d}+B_{d}K$ is schur stable.

We consider an observer-based controller described by

$\displaystyle\hat{x}_{q+1}=A_{d}\hat{x}_{q}+B_{d}u_{q}+L(Q(y_{q})-\hat{y}_{q}),$
$\displaystyle q\Delta=s_{r}$ (30a)
$\displaystyle\hat{x}_{q+1}=A_{d}\hat{x}_{q}+B_{d}u_{q},$ $\displaystyle
q\Delta\neq s_{r}$ (30b) $\displaystyle\hat{y}_{q}=C\hat{x}_{q}$ (30c)
$\displaystyle u_{q}=K\hat{x}_{q}$ (30d)

where
$\hat{x}_{q}\\!\in\\!\mathbb{R}^{n_{x}},\hat{y}_{q}\\!\in\\!\mathbb{R}^{n_{y}}$,
and $Q(y_{q})\\!\in\\!\mathbb{R}^{n_{y}}$ are the estimated state, the
estimated output, and the quantized output, respectively. The initial
condition is set to be $\hat{x}_{0}=0$. Since the input channel is ideal, it
follows that

$u(t)=u_{q},\quad q\Delta\leq t<(q+1)\Delta,\quad q\in\mathbb{Z}_{\geq 0}.$

To design an encoding scheme such that the output $y_{q}$ can be quantized
without saturation, an error bound between the estimated output and the actual
output, i.e., $|e_{q}|:=|x_{q}-\hat{x}_{q}|\leq E_{q}$, should be derived.
Based on (1b) and (30b), it can be deduced that

$|y_{q}-\hat{y}_{q}|=|C(x_{q}-\hat{x}_{q})|=|Ce_{q}|\leq\|C\|E_{q}.$ (31)

Let $N$ denote the number of quantization levels of $y_{q}$. Similar to the
previous section, we partition the hypercube
$\\{y\in\mathbb{R}^{n_{y}}:|y_{q}-\hat{y}_{q}|\leq\|C\|E_{q}\\}$ into
$N^{n_{y}}$ equal-sized boxes. The quantization error obeys
$|Q(y_{q})-y_{q}|\leq\frac{\|C\|}{N}E_{q}$. According to As. 2, the initial
value $E_{0}$ is given by

$|e_{0}|=|x_{0}|\stackrel{{\scriptstyle\triangle}}{{=}}E_{0}.$ (32)

Sequence $\\{E_{q},q\in\mathbb{Z}_{\geq 1}\\}$ will be specified latter.
Notice that the hypercube center is $\hat{y}_{q}$, which is generated by the
predictor-based observer in (30). Therefore, this predictor should also be
equipped at the encoder side. Under ACK-based protocol, the decoder sends ACKs
to the encoder without delay at successful transmission instants; and when the
encoder does not receive the ACKs, it infers that there is a DoS attack. In
this manner, synchronization between these two predictors is ensured, which
consequently implies that the quantization ranges and the centers at the
encoder are identical to that of the decoder.

Before giving stability condition for ACK-based protocol case, we present an
output encoding scheme. Let

$E_{q+1}:=\left\\{\begin{aligned} &\theta_{a}E_{q},&q\Delta\neq s_{r}\\\
&\theta_{0}E_{q},&(q-1)\Delta\neq s_{r},q\Delta=s_{r}\\\
&\theta_{na}E_{q},&(q-1)\Delta=s_{r},q\Delta=s_{r}\end{aligned}\right.$ (33)

with

$\displaystyle\theta_{a}$ $\displaystyle:=\left\|A_{d}\right\|$ (34a)
$\displaystyle\theta_{0}$
$\displaystyle:=H_{0}\rho+\frac{H_{1}\left\|C\right\|}{N}$ (34b)
$\displaystyle\theta_{na}$
$\displaystyle:=\rho+\frac{H_{1}\left\|C\right\|}{N}$ (34c)

where constants $H_{0}$, $H_{1}$, and $0<\rho<1$ satisfy

$\|(A_{d}-LC)^{\ell}\|\leq H_{0}\rho^{\ell},\quad\|(A_{d}-LC)^{\ell}L\|\leq
H_{1}\rho^{\ell}.$

######  Theorem 2.

Consider system (1) with controller (30), where $M$ and $K$ are chosen such
that $A_{d}-LC$ and $A_{d}+B_{d}K$ are schur stable. Under As. 1–4, if i) the
quantization levels

$\begin{split}N>\frac{H_{1}\|C\|}{1-\rho}\end{split}$ (35)

and, ii) DoS attacks satisfy

$\frac{1}{\nu_{d}}\leq\frac{\log{(1/{\theta}_{na})}}{\log{({\theta}_{a}/{\theta}_{na})}}-\frac{\log{({\theta}_{0}/{\theta}_{na})}}{\log{(\theta_{a}/{\theta}_{na})}}\frac{1}{\nu_{f}}$
(36)

then the system is exponentially stable under the encoding scheme with error
bounds $\\{E_{q}:q\in\mathbb{Z}_{\geq 1}\\}$ constructed by the update rule
(33).

The proof is similar to that of [31, Thm. 3.4] and is thus omitted here due to
space limitations.

###  IV-B Controller under an acknowledgment-free protocol

In this subsection, we show that the aforementioned controller and encoding
scheme cannot stabilize the system when the ACK-based protocol is replaced by
an ACK-free protocol. This is because synchronization between the encoder and
decoder is no longer guaranteed. To see this, consider controller (30) with
the encoding scheme in (33) employing an ACK-free protocol. In this setting,
predictors at the encoder and decoder sides may become asynchronized, since no
matter whether DoS attacks happen or not, the decoder does not send ACKs to
the encoder. When a DoS attack occurs, the predictor at the controller side
switches to (30b), whereas the predictor at the encoder side sticks to (30a).
Moreover, the update rule of sequence $E_{q}$ at the decoder switches to
(34a), while adhering to (34b)-(34c) at the encoder. As a result, their
quantization ranges and centers may deviate, and the correct output value
cannot be recovered by the decoder. We prove that even if one DoS attack
occurs (i.e., decoder and encoder are asynchronized for only one transmission
period), the state may diverge eventually.

To distinguish between predictors at the encoder and decoder, let
$\hat{x}_{q}$, $\hat{y}_{q}$, and $\hat{Q}(y_{q})$ denote the estimated state,
estimated output, and quantized output at the controller side, and
$\tilde{x}_{q},\tilde{y}_{q}$, and $Q(y_{q})$ denote their counterparts at the
encoder side. In addition, let $u_{q}$ stand for the input sent by the
controller, and $\tilde{u}_{q}$ the estimated input generated by the predictor
at the encoder side. Predictor at the controller side can be expressed by

$\displaystyle\hat{x}_{q+1}=A_{d}\hat{x}_{q}+B_{d}u_{q}+L(\hat{Q}(y_{q})-\hat{y}_{q}),$
$\displaystyle q\Delta=s_{r}$ (37a)
$\displaystyle\hat{x}_{q+1}=A_{d}\hat{x}_{q}+B_{d}u_{q},$ $\displaystyle
q\Delta\neq s_{r}$ (37b) $\displaystyle\hat{y}_{q}=C\hat{x}_{q}$ (37c)
$\displaystyle u_{q}=K\hat{x}_{q}$ (37d)

and predictor at the encoder side is described by

$\displaystyle\tilde{x}_{q+1}=A_{d}\tilde{x}_{q}+B_{d}\tilde{u}_{q}+L({Q}(y_{q})-\tilde{y}_{q})$
(38a) $\displaystyle\tilde{y}_{q}=C\tilde{x}_{q}$ (38b)
$\displaystyle\tilde{u}_{q}=K\tilde{x}_{q}$ (38c)

where $q\in\mathbb{Z}_{\geq 0}$. Similarly, let $E_{d,q}$, and $E_{e,q}$
denote the error bound at the decoder, and the encoder side, respectively

$\displaystyle E_{d,q+1}:=\left\\{\begin{aligned}
&\theta_{a}E_{d,q},&q\Delta\neq s_{r}\\\ &\theta_{0}E_{d,q},&(q-1)\Delta\neq
s_{r},q\Delta=s_{r}\\\
&\theta_{na}E_{d,q},&(q-1)\Delta=s_{r},q\Delta=s_{r}\end{aligned}\right.$
$\displaystyle E_{e,q+1}:=\left\\{\begin{aligned}
&\theta_{0}E_{e,q},&q\Delta=0\\\
&\theta_{na}E_{e,q},&q\Delta>0\end{aligned}\right.$

where $\theta_{0}$, $\theta_{a}$, and $\theta_{na}$ are defined in (34).
Accordingly, the errors at the encoder and decoder sides are
$e_{e,q}:=x_{q}-\tilde{x}_{q}$, and $e_{d,q}:=x_{q}-\hat{x}_{q}$. Moreover,
the quantized outputs in (37b) and (38a) are

$\displaystyle\hat{Q}(y_{q})=\hat{y}_{q}+Q^{i}_{q}\frac{\|C\|E_{d,q}}{N}$ (39)
$\displaystyle Q(y_{q})=\tilde{y}_{q}+Q^{i}_{q}\frac{\|C\|E_{e,q}}{N}$ (40)

where $Q_{q}^{i}$ denotes the quantization index transmitted from the encoder
to the decoder.

Suppose that a DoS attack is launched at $q_{a}\Delta$ and no attacks happen
before or after $q_{a}\Delta$. It follows that $\hat{Q}(y_{q})=Q(y_{q})$ for
all $q\leq q_{a}$, and

$\displaystyle\hat{x}_{q_{a}}=\tilde{x}_{q_{a}}$ (41a)
$\displaystyle\hat{x}_{q_{a}\\!+\\!1}=(A_{d}\\!+\\!B_{d}K)\hat{x}_{q_{a}}$
(41b)
$\displaystyle\tilde{x}_{q_{a}\\!+\\!1}=(A_{d}\\!+\\!B_{d}K)\tilde{x}_{q_{a}}\\!+\\!LQ^{i}_{q_{a}}\frac{\|C\|E_{e,q_{a}}}{N}$
(41c)
$\displaystyle\hat{x}_{q_{a}+2}=(A_{d}\\!+\\!B_{d}K)^{2}\hat{x}_{q_{a}}\\!+\\!LQ^{i}_{q_{a}\\!+\\!1}\frac{\|C\|E_{d,q_{a}\\!+\\!1}}{N}$
(41d)
$\displaystyle\begin{split}&\tilde{x}_{q_{a}+2}=(A_{d}\\!+\\!B_{d}K)^{2}\tilde{x}_{q_{a}}\\!+\\!LQ^{i}_{q_{a}+1}\frac{\|C\|E_{e,q_{a}+1}}{N}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\!+\\!(A_{d}\\!+\\!B_{d}K)LQ^{i}_{q_{a}}\frac{\|C\|E_{e,q_{a}}}{N}\\\
\end{split}$ (41e) $\displaystyle\cdots$

Notice that the quantizer operates normally without saturation only if
$E_{e,q}\geq|e_{e,q}|=|x_{q}-\tilde{x}_{q}|$ and
$E_{d,q}\geq|e_{d,q}|=|x_{q}-\hat{x}_{q}|$ hold for all $q\in\mathbb{Z}_{\geq
1}$. If the quantizer saturates, the error between the actual output and the
quantized output maybe large, which consequently renders the system unstable.
In the following, we assume that the quantizer is not saturated; that is
$E_{e,q}\geq|e_{e,q}|$ and $E_{d,q}\geq|e_{d,q}|$ for all
$q\in\mathbb{Z}_{\geq 1}$, and reach a contradiction. Since
$E_{e,q+1}=\theta_{na}E_{e,q},q>0$, and $\theta_{na}<1$, sequence
$\\{|x_{q}-\tilde{x}_{q}|\\}$ is decreasing. Let $\Pi_{L}:=A_{d}-LC$ and
$\Pi_{K}:=A_{d}+B_{d}K$. Combining (39) and (41) yields

$\displaystyle|x_{q_{a}+1}-\tilde{x}_{q_{a}+1}|$
$\displaystyle=|\Pi_{L}(x_{q_{a}}-\tilde{x}_{q_{a}})-L(Q(y_{q_{a}})-y_{q_{a}})|$
$\displaystyle\leq
E_{e,q_{a}+1}\stackrel{{\scriptstyle\triangle}}{{=}}\tilde{E}_{e,q_{a}+1}.$

Likewise,

$\displaystyle~{}|x_{q_{a}+2}-\tilde{x}_{q_{a}+2}|$ $\displaystyle=$
$\displaystyle\
\big{|}\Pi_{L}(x_{q_{a}+1}-\tilde{x}_{q_{a}+1})-L(Q(y_{q_{a}+1})-y_{q_{a}+1})$
$\displaystyle-BKLQ^{i}_{q_{a}}\frac{\|CR^{-1}\|E_{e,q_{a}}}{N}\big{|}$
$\displaystyle\leq$ $\displaystyle\
E_{q_{a}+2}+\frac{1}{\theta_{na}^{2}}\frac{\|BKLQ^{i}_{q_{a}}\|\|CR^{-1}\|}{N}E_{e,q_{a}+2}$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}$ $\displaystyle\
\tilde{E}_{e,q_{a}+2}.$

Iteratively, for $\ell\geq 3$, it follows that

$\displaystyle~{}|x_{q_{a}+\ell}-\tilde{x}_{q_{a}+\ell}|$ $\displaystyle\leq$
$\displaystyle\
E_{e,q_{a}+\ell}+\frac{1}{\theta_{na}^{\ell}}\frac{\|B_{d}K\Pi_{K}^{\ell-1}LQ^{i}_{q_{a}}\|\|C\|E_{e,q_{a}}}{N}$
$\displaystyle+\frac{1}{\theta_{na}^{\ell}}\frac{\|B_{d}K\Pi_{K}^{\ell-2}LQ^{i}_{q_{a}+1}\|\|C\|(\theta_{a}-\theta_{na})E_{e,q_{a}}}{N}$
$\displaystyle+\sum_{i=0}^{\ell-3}\frac{1}{\theta_{na}^{i+3}}\frac{\|B_{d}K\Pi_{K}^{i}LQ^{i}_{q_{a}+\ell-i-1}\|\|CR^{-1}\|}{N}$
$\displaystyle~{}~{}\times(\theta_{0}\theta_{a}-\theta_{na}^{2})E_{e,q_{a}}$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle~{}\tilde{E}_{e,q_{a}+\ell}.$

Since ${1}/{\theta_{na}}>1$, $\\{\tilde{E}_{e,q}\\}$ is an increasing
sequence, which contradicts the assumption that $\\{|x_{q}-\tilde{x}_{q}|\\}$
is a decreasing sequence. Therefore, it can be concluded that without ACKs,
predictors at the encoder and controller sides may get asynchronized even if
there is a single DoS attack. This causes mismatches on their quantization
centers and ranges, and there exists $\hat{q}\geq q_{a}$ such that
$E_{e,q}<|x_{q}-\tilde{x}_{q}|$ holds for all $q\geq\hat{q}$, and the state
diverges eventually.

We have just shown that the synchronization between decoder and encoder is
essential. However, ACK-based protocol is not the only way to achieve this
goal. In the absence of ACKs, this challenge can be overcome by using a
deadbeat controller, and the prove will be given in the following. Let the
number of the quantization level $N$ to be even. We adopt the same quantizer
as in (31)-(32), with $\hat{x}_{q}$, $e_{q}$, and $\hat{y}_{q}$ replaced by
$\hat{x}_{q-1,\eta}$, $e_{q-1,\eta}$, and $\hat{y}_{q-1,\eta}$, respectively.
The observer-based controller is employed only at the decoder side

$\displaystyle\hat{x}_{q,k+1}=A_{d}\hat{x}_{q,k}+B_{d}u_{q,k},$ $\displaystyle
k$ $\displaystyle\leq\eta-1$ (42a)
$\displaystyle\hat{x}_{q}=\hat{x}_{q-1,\eta}+M_{q}[Q(y_{q})-\hat{y}_{q-1,\eta}],$
$\displaystyle k$ $\displaystyle=\eta$ (42b)
$\displaystyle\hat{y}_{q,k}=C\hat{x}_{q,k}$ (42c) $\displaystyle
u_{q,k}=K\hat{x}_{q,k}$ (42d)

Thanks to the ideal input channel,

$u(t)=u_{q,k},\qquad q\Delta+k\delta\leq t<q\Delta+(k+1)\delta$

for every $q\in\mathbb{Z}_{\geq 0}$, and $k=0,\cdots\\!,\eta-1$.

Consider an arbitrary transmission interval $[q\Delta,(q+1)\Delta)$. From the
property (7), one gets that
$\hat{x}_{q,\eta}=(A_{d}+B_{d}K)^{\eta}\hat{x}_{q}=0$, and
$\hat{y}_{q,\eta}=C\hat{x}_{q,\eta}=0$. It is thus sufficient to choose the
quantization center to be the origin, and predictor (42) is not needed at the
encoder side. This saves computational resources.

If an attack is launched at $(q+1)\Delta$, the decoder is not going to receive
the quantized output $Q(y_{q+1})$, and instead it will use a default zero.
Then, it follows from (42a)-(42b) that $\hat{x}_{q+1}=\hat{x}_{q,\eta}=0$, and
$u_{q+1}=K\hat{x}_{q+1}=0$. On the other hand, in the absence of DoS attacks,
since the quantization center is zero and $N$ is even, the quantized value is
nonzero. Therefore, the decoder receives a quantized output $Q(y_{q+1})\neq
0$. As a result,
$\hat{x}_{q+1}=\hat{x}_{q,\eta}+M(Q(y_{q+1})-\hat{y}_{q,\eta})=MQ(y_{q+1})\neq
0$, and $u_{q+1}=K\hat{x}_{q+1}\neq 0$. This suggests that the encoder can
infer whether there is an attack or not from the input signals, thus its
quantization ranges can be updated following the same scheme with the decoder.

We have secured synchronization between the encoder and decoder. Now, what is
left behind is the system stability analysis. Recall that $A_{d}^{\eta}(I-MC)$
is schur stable, there exist constants $G_{0},G_{1}$, and $0<\rho<1$ such that

$\displaystyle\big{\|}R^{\ell}\big{\|}\leq
G_{0}\rho^{\ell},\quad\big{\|}{R}^{\ell}A_{d}^{\eta}M\big{\|}\leq
G_{1}\rho^{\ell}.$ (43)

Define constants

$\displaystyle\tilde{\theta}_{a}:=\|A_{d}^{\eta}\|,~{}~{}\tilde{\theta}_{0}:=G_{0}\rho+\frac{G_{1}\|C\|}{N},~{}~{}\tilde{\theta}_{na}:=\rho+\frac{G_{1}\|C\|}{N}$

and the error bound $\\{E_{q}:q\in\mathbb{Z}_{\geq 1}\\}$ is updated by

$E_{q+1}:=\left\\{\begin{aligned} &\tilde{\theta}_{a}E_{q},&q\Delta\neq
s_{r}\\\ &\tilde{\theta}_{0}E_{q},&(q-1)\Delta\neq s_{r},q\Delta=s_{r}\\\
&\tilde{\theta}_{na}E_{q},&(q-1)\Delta=s_{r},q\Delta=s_{r}\end{aligned}\right..$
(44)

The following result is an extension of Thm. 2 under an ACK-free protocol,
whose proof follows from that of Thm. 2.

######  Theorem 3.

Consider system (1) equipped with controller in (42), where $M$ and $K$ are
chosen such that $R$ is schur stable and (7) is met. Let As. 1–4 hold. If i)
the output and input transmission periods satisfy (6), ii) the quantization
levels $N$ is even, and obey

$N>\frac{G_{1}\|C\|}{1-\rho}$ (45)

and, iii) DoS attacks satisfy

$\frac{1}{\nu_{d}}\leq\frac{\log{(1/\tilde{\theta}_{na})}}{\log{(\tilde{\theta}_{a}/\tilde{\theta}_{na})}}-\frac{\log{(\tilde{\theta}_{0}/\tilde{\theta}_{na})}}{\log{(\tilde{\theta}_{a}/\tilde{\theta}_{na})}}\frac{1}{\nu_{f}}$
(46)

then the system is exponentially stable under the encoding scheme with error
bound $\\{E_{q}:q\in\mathbb{Z}_{\geq 1}\\}$ constructed by (44).

Figure 4: Maximum norm of state $x$ and its estimate $\hat{x}$ with controller
(5).
Figure 5: Relationship between normalized quantization range $E_{3,k}/N_{3}$
and actual error $|y_{q}-Q_{1}(\hat{y}_{q})|$ with controller (5).
Figure 6: Normalized quantization range $E_{2,q,k}/N_{2}$ with controller (5).
Figure 7: Normalized quantization ranges $E_{3,q}/N_{3}$ from using general
observer gain and deadbeat observer gain in log space.
Figure 8: Maximum norm of state $x$ and its estimate $\hat{x}$ with controller
(5) and deadbeat observer gain.

##  V Numerical Example

A linearized model of the unstable batch reactor in [31] is given by
$\dot{x}(t)=Ax(t)+Bu(t)$ and $y=Cx(t)$, where

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

This system $(A,B,C)$ is observable and controllable with $\eta=\mu=2$. Let
the output transmission period $\Delta=0.2$, so $\delta=\Delta/\eta=0.1$.
Choosing matrix $K$, such that (7) is met, i.e.,

$\displaystyle K:=\left[\begin{matrix}1.0106&-1.5661&0.0385&-4.0366\\\
8.1074&-0.0347&4.3337&-3.6241\end{matrix}\right].$

Calculating the gain of the steady-state Kalman filter

$\displaystyle M:=\left[\begin{matrix}0.5534&-0.0249\\\ -0.0287&0.0396\\\
0.1489&0.0892\\\ 0.0810&0.0931\end{matrix}\right].$

We first present the time responses when both input and output channels suffer
from the network phenomena. Applying Thm. 1, when both the quantization levels
$N_{2}$ and $N_{3}$ go to infinity, the duration bound $1/\nu_{d}$ and the
frequency bound $1/\nu_{f}$ of DoS attacks approach to the line
$\frac{1}{\nu_{d}}\approx-0.5544\frac{1}{\nu_{f}}+0.2707$. According to (19),
if $\frac{1}{\nu_{d}}<-2.0380\frac{1}{\nu_{f}}+0.2269$, then the closed-loop
system with encoding schemes (14)-(16) is stabilized. Over a simulation
horizon of $160$s ($800$ time-step), DoS attacks (the gray shades) are
generated randomly with $\Phi_{d}=47$ and $\Phi_{f}=44$. Setting
$\kappa_{d}=3,\nu_{d}=18,\kappa_{f}=2,\nu_{f}=19$, condition (19) holds, i.e.,
$1/\nu_{d}=0.056<0.119$. Figs. 4 and 5 illustrate the time response in this
situation. Since the condition in Thm. 1 is satisfied, the maximum norm of the
state converges, and the bound $E_{3,q}$ exponentially decreases. Fig. 5
depicts that $E_{3,q}$ shares the same trend with
$|y_{q}-Q_{1}(\hat{y}_{q})|$, and Fig. 6 demonstrates the evolution of the
quantization step size $E_{2,q,k}/N_{2}$, which jumps up and down within an
output transmission period, and decreases in general. Difference between the
trend of $E_{3,q}/N_{3}$ and $E_{2,q,k}/N_{2}$ lies in the property of $\|R\|$
and $\|\bar{R}\|$. Fig. 7 compares the quantization step size $E_{3,q}/N_{3}$
of a general observer gain (blue line), such that $R$ is schur stable, and the
deadbeat observer gain (dot marked green line), namely $R^{\mu}=0$. This panel
illustrates that although $E_{3,q}$ responds faster under deadbeat observer,
the large quantization step size results in large overshoot of the state; see
Fig. 8, which confirms Rmk. 4.

Figure 9: Maximum norm of state $x$ and its estimate $\hat{x}$ with controller
(42).
Figure 10: Relationship between normalized quantization range $E_{q}/N$ and
actual error $|e_{q}|$ with controller (42).
Figure 11: Input signal $u_{q,k}$ with controller (42).

Next, consider network phenomena only at output channel. From (45), the
quantization levels satisfies $N>6.957$, also since $N$ is even, we set
$N=100$. Over a simulation horizon of $60$s ($300$ time-step), generating DoS
attacks randomly with $\Phi_{d}=27$ and $\Phi_{f}=25$. Setting
$\kappa_{d}=1,\nu_{d}=11,\kappa_{f}=1,\nu_{f}=11$, so condition (46) is met
with $1/\nu_{d}=0.01<0.198$, and convergence of the state is presented in
Figs. 9 and 10. Further, Fig. 11 shows that when a DoS attack happens, the
control input is set to zero immediately, which verifies the effectiveness of
our method.

##  VI Conclusions

This paper considered the problem of stabilizing networked control systems in
the presence of DoS attacks and limited data rates. To overcome the network-
induced challenges, a structure consisting of a deadbeat controller and a
transmission protocol which are carefully co-designed based on the system
controllability index, was proposed to address the network-induced challenges.
Specifically, when both input and output channels are subject to the network
phenomena, it was shown that the proposed structure can decouple and thus
allow for separate design of encoding schemes for the input, output, and
estimated output signals. Furthermore, easy-to-check conditions were derived
such that exponential stability of the closed-loop system under this structure
is ensured. On the other hand, when only the output channel is subject to the
network phenomena, the proposed structure was shown able to guarantee
synchronization between the encoder and decoder under an ACK-free protocol.
Finally, a numerical example was presented to verify the effectiveness of our
approach as well as the correctness of our theory. Future developments will
focus on generalizing the results to more general systems and controllers
under ACK-free protocols.

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|  Wenjie Liu received the bachelor’s degree in Automation from Tianjin
University, Tianjin, China, in 2019. She is currently pursuing the Ph.D.
degree in control science and engineering with the School of Automation,
Beijing Institute of Technology, Beijing, China. Her current research
interests include cyber–physical systems and network control under
communication constraints.
---|---
|  Jian Sun received the bachelor’s degree from the Department of Automation
and Electric Engineering, Jilin Institute of Technology, Changchun, China, in
2001, the master’s degree from the Changchun Institute of Optics, Fine
Mechanics and Physics, Chinese Academy of Sciences (CAS), Changchun, China, in
2004, and the Ph.D. degree from the Institute of Automation, CAS, Beijing,
China, in 2007. He was a Research Fellow with the Faculty of Advanced
Technology, University of Glamorgan, Pontypridd, U.K., from 2008 to 2009. He
was a Post-Doctoral Research Fellow with the Beijing Institute of Technology,
Beijing, from 2007 to 2010. In 2010, he joined the School of Automation,
Beijing Institute of Technology, where he has been a Professor since 2013. His
current research interests include networked control systems, time-delay
systems, and security of cyber-physical systems. Dr. Sun is an Editorial Board
Member of the IEEE Transactions on Systems, Man and Cybernetics: Systems, the
Journal of Systems Science & Complexity, and Acta. Automatica Sinica.
---|---
|  Gang Wang (M’18) received a B.Eng. degree in Automatic Control in 2011, and
a Ph.D. degree in Control Science and Engineering in 2018, both from the
Beijing Institute of Technology, Beijing, China. He also received a Ph.D.
degree in Electrical and Computer Engineering from the University of
Minnesota, Minneapolis, USA, in 2018, where he stayed as a postdoctoral
researcher until July 2020. Since August 2020, he has been a professor with
the School of Automation at the Beijing Institute of Technology. His research
interests focus on the areas of signal processing, control, and reinforcement
learning with applications to cyber-physical systems and multi-agent systems.
He was the recipient of the Excellent Doctoral Dissertation Award from the
Chinese Association of Automation in 2019, the Best Student Paper Award from
the 2017 European Signal Processing Conference, and the Best Conference Paper
at the 2019 IEEE Power & Energy Society General Meeting. He is currently on
the editorial board of Signal Processing.
---|---
|  Francesco Bullo (IEEE S’95-M’99-SM’03-F’10) is a Professor with the
Mechanical Engineering Department and the Center for Control, Dynamical
Systems and Computation at the University of California, Santa Barbara. He was
previously associated with the University of Padova, the California Institute
of Technology, and the University of Illinois. His research focuses on
modeling, dynamics and control of multi-agent network systems, with
applications to robotic coordination, power systems, distributed computing and
social networks. Previous work includes contributions to geometric control,
Lagrangian systems, vehicle routing, and motion planning. He has published
more than 300 papers in international journals, books, and refereed
conferences. He is the coauthor, with Andrew D. Lewis, of the book “Geometric
Control of Mechanical Systems” (Springer, 2004, 0-387-22195-6), with Jorge
Cortés and Sonia Martínez, of the book “Distributed Control of Robotic
Networks” (Princeton, 2009, 978-0-691-14195-4), with Stephen L. Smith of the
book “Lectures on Robotics Planning and Kinematics” (SIAM, 2019, under
contract); and of the book “Lectures on Network Systems” (Kindle Direct
Publishing, 2020, v1.4, 978-1986425643). He received best paper awards for his
work in IEEE Control Systems, Automatica, SIAM Journal on Control and
Optimization, IEEE Transactions on Circuits and Systems, and IEEE Transactions
on Control of Network Systems. He is a Fellow of IEEE, IFAC, and SIAM. He has
served on the editorial boards of IEEE, SIAM, and ESAIM journals, and serves
as 2018 IEEE CSS President. He is serving as Chair of the SIAM Activity Group
on Control and Systems Theory, for 2020-2021.
---|---
|  Jie Chen (F’19) received his B.Sc., M.Sc., and the Ph.D. degrees in control
theory and control engineering from the Beijing Institute of Technology,
Beijing, China, in 1986, 1996, and 2001, respectively. From 1989 to 1990, he
was a visiting scholar at the California State University, Long Beach,
California, USA. From 1996 to 1997, he was a research fellow in the School of
Engineering at the University of Birmingham, Birmingham, UK. He is a Professor
with the School of Automation, Beijing Institute of Technology, where he
serves as the Director of the Key Laboratory of Intelligent Control and
Decision of Complex Systems. He also serves as the President of Tongji
University, Shanghai, China. His research interests include complex systems,
multiagent systems, multiobjective optimization and decision, and constrained
nonlinear control. Prof. Chen is currently the Editor-in-Chief of Unmanned
Systems and the Journal of Systems Science and Complexity. He has served on
the editorial boards of several journals, including the IEEE Transactions on
Cybernetics, International Journal of Robust and Nonlinear Control, and
Science China Information Sciences. He is a Fellow of IEEE, IFAC, and a member
of the Chinese Academy of Engineering.
---|---