# Data-Driven Resilient Predictive Control under Denial-of-Service

Wenjie Liu, Jian Sun, , Gang Wang, ,
Francesco Bullo, , and Jie Chen This work was supported in part by the
National Key R&D Program of China under Grant 2018YFB1700100, and the National
Natural Science Foundation of China under Grants 61925303, 62088101, U20B2073,
61720106011, 62173034. 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). 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). 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).

###### Abstract

The study of resilient control of linear time-invariant (LTI) systems against
denial-of-service (DoS) attacks is gaining popularity in emerging cyber-
physical applications. In previous works, explicit system models are required
to design a predictor-based resilient controller. These models can be either
given _a priori_ or obtained through a prior system identification step.
Recent research efforts have focused on data-driven control based on pre-
collected input-output trajectories (i.e., without explicit system models). In
this paper, we take an initial step toward data-driven stabilization of
stochastic LTI systems under DoS attacks, and develop a resilient model
predictive control (MPC) scheme driven purely by data-dependent conditions.
The proposed data-driven control method achieves the same level of resilience
as the model-based control method. For example, local input-to-state stability
(ISS) is achieved under mild assumptions on the noise and the DoS attacks. To
recover global ISS, two modifications are further suggested at the price of
reduced resilience against DoS attacks or increased computational complexity.
Finally, a numerical example is given to validate the effectiveness of the
proposed control method.

###### Index Terms:

Denial-of-service attack, data-driven control, model predictive control,
input-to-state stability.

##  I Introduction

Thanks to recent advances in computing and networking technologies, recent
years have witnessed rapid developments in cyber-physical systems (CPSs),
e.g., [1, 2, 3, 4, 5]. Nonetheless, it has been reported that such systems are
often vulnerable to cyber-attacks [6], including false-data injection attacks
[7, 8], replay attacks [9, 10], and denial-of-service (DoS) attacks [11]. For
instance, on February 8, 2020, the telecommunication network of Iran suffered
from DoS attacks for about an hour [12]. As a consequence, $25\%$ of the
national internet connection dropped, leading to severe damage of critical
infrastructure as well as significant economic loss. In general, DoS attacks
require little knowledge about the system and are therefore easy to be
implemented. Moreover, DoS attacks are destructive. If an unstable open-loop
process adopts a remote controller, then a long duration of DoS may render
irreparable damages on physical systems as well as associated components.
These observations motivate the need for effective mechanisms to defend
against DoS attacks and/or to mitigate the associated effect on physical
entities.

To this aim, a possible remedy is to design control strategies such that
satisfied performance can be maintained regardless of the DoS attack
strategies, which is referred to as resilient control. This problem was first
addressed by the work [13], in which a transmission policy along with some
conditions on DoS attacks were developed for resilient stabilization of LTI
systems. Since then, a multitude of publications have studied resilient
control of different systems, including, e.g., systems using an output-
feedback controller in [14, 15, 16], multi-agent systems in [17], and
nonlinear systems in [18].

It is worth emphasizing that all the aforementioned resilient control
strategies build upon the model-based control approach developed in [19]. In
other words, they require an explicit system model, or need to perform a
system identification step _a priori_. Nonetheless, accurate system models may
be challenging to acquire in real-world applications, while system
identification of large-scale systems could be data hungry and/or
computationally cumbersome. Data-driven methods, on the other hand, offer a
new avenue for the control of unknown dynamic systems and pursue the design of
a control law directly from data (i.e., from pre-collected input-output
trajectories), e.g., [20]. This method also comes with rigorous theoretical
guarantees thanks to the celebrated _Fundamental Lemma_ [21]. Indeed, there
has been a recent line of research works exploring the fundamental lemma for
data-driven control of unknown LTI systems. Following the state-space
description perspective, a parameterized model of linear feedback systems was
developed such that state-feedback controllers can be directly obtained from
solving data-dependent linear matrix inequalities in [22, 23, 24, 25, 26]. Due
to the popularity of model predictive controllers in industrial applications
(e.g., see [27, 28, 29]), another line of work dealt with data-driven MPC
schemes (e.g., [30, 31]). Stability analysis and robustness guarantees of
data-driven MPC against measurement noise have recently been studied in [32].

The goal of this present paper is to stabilize unknown LTI systems under DoS
attacks based solely on input-output system trajectories acquired _a priori_
through some off-line experiments. It has been shown in [14] that LTI systems
equipped with a predictor-based controller based on an explicit system model
can achieve maximum resilience against DoS attacks. Specifically, an observer-
based predictor is employed to maintain a reasonable estimate of the state
even in the presence of DoS attacks; a feedback controller is then developed
based upon this estimate. Nevertheless, both designs of the observer and of
the feedback controller require an explicit system model, rendering such
control methods difficult to apply when only input-output trajectories are
available. To overcome this challenge, a data-driven resilient MPC scheme is
developed in this paper. Resembling the data-driven MPC scheme presented in
[32], our resilient data-driven MPC generates a sequence of control inputs and
associated predicted outputs by solving a convex optimization problem with
purely data-dependent constraints, but only at time instants when new
measurements are received (i.e., no DoS attack occurs). On the other hand, at
time instants when there is a DoS attack, the corresponding measurement
transmission fails. Our scheme takes the predicted control input from the most
recent solution of the optimization problem, or simply a zero control input,
depending upon whether the duration of DoS attacks exceeds the prediction
horizon of the data-driven MPC.

It is worth pointing out that there are a couple of notable differences
between the data-driven MPC scheme in [32] and the one developed here. To
start with, in addition to DoS attacks, both process noise and measurement
noise are modeled and accounted for in this paper, whereas only measurement
noise was considered in [32]. Since both pre-collected measurements as well as
the measurements received in the execution phase are corrupted by noise in
[32], the resultant data-driven MPC involves solving a non-convex constrained
optimization problem which can be computationally challenging (NP-hard in
general). In contrast, we consider only process noise in the data-collecting
phase, while addressing both process and network-induced measurement noise in
the execution phase. Interestingly, this noise modeling approach leads to a
convex data-driven MPC scheme in our paper, that is computationally appealing.
Although a convex data-driven MPC scheme was also developed in [30], no
stability analysis and robustness guarantees were provided. To the best of our
knowledge, this is the first model-free (a.k.a. data-driven) controller that
achieves the maximum resilience performance against DoS attacks as the model-
based controller in [14].

In a nutshell, the main contribution of the present work is threefold,
summarized as follows.

  * c1)

Addressing both process and measurement noise, a data-driven MPC scheme is
proposed for the control of LTI systems. Our scheme generates a series of
control inputs and associated predicted outputs through solving a convex
optimization problem driven by purely input-output data;

  * c2)

Leveraging the idea behind the predictor-based controller [14], the proposed
data-driven MPC scheme is further modified to accommodate DoS attacks, thereby
becoming a data-driven resilient controller; and,

  * c3)

Under standard conditions on DoS attacks and noise, local input-to-state
stability (ISS) of the closed-loop system is established for the proposed
data-driven resilient MPC, along with modifications to recover global ISS.

_Notation:_ Denote the set of real numbers by $\mathbb{R}$, and the set of
integers by $\mathbb{Z}$. 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 define
$\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$. For a matrix $M$, its left pseudo-
inverse and right pseudo-inverse are denoted by $M^{\dagger}$ and
$M^{\ddagger}$, respectively. Given a vector $x\in\mathbb{R}^{n_{x}}$, $\|x\|$
is its Euclidean norm, and for a positive definite matrix $P=P^{T}\succ 0$,
define the weighted norm $\|x\|_{P}=\sqrt{x^{\top}Px}$. Let further $\|M\|$ be
the spectral norm of matrix $M$. For discrete-time signals, define
$\|w\|_{[t_{1},t_{2}]}:=\max_{t\in[t_{1},t_{2}]}\\{\|w_{t}\|\\}$, and
$\|w\|_{\infty}:=\|w\|_{[0,\infty)}$ Let $\underline{\lambda}_{P}$
($\overline{\lambda}_{P}$) represent the minimum (maximum) eigenvalue of
matrix $P$. Let $I$ represent the identity matrix with suitable dimensions,
with $I_{q}$ denoting its $q$-th row, and
$I_{[q,q+i]}=[I_{q}^{\top},I_{q+1}^{\top},\cdots\\!,I_{q+i}^{\top}]^{\top}$.
For $\delta\in\mathbb{R}_{>0}$, let
$\mathbb{B}_{\delta}=\\{x\in\mathbb{R}^{n_{x}}~{}|~{}\|x\|\leq\delta\\}$. Let
$[t_{1},t_{2}]$ denote the time interval from $t_{1}$ to $t_{2}$ with discrete
time instants.

The Hankel matrix associated to the sequence $x_{N}:=\\{x_{t}\\}_{t=0}^{N-1}$,
is denoted by

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

A stacked window of the sequence is given by

$x_{[t_{1},t_{2}]}=\left[\begin{matrix}x_{t_{1}}\\\ \vdots\\\
x_{t_{2}}\end{matrix}\right].$ (2)

For notational convenience, we will use $x$ to denote both the sequence itself
and the stacked vector $x_{[0,N-1]}$ containing all of its components,
whenever clear from the context.

Figure 1: Closed-loop system using data-driven MPC.

##  II Preliminaries and Problem Formulation

###  II-A Networked control systems

Consider the stochastic discrete-time linear time-invariant (LTI) system

$\displaystyle x_{t+1}$ $\displaystyle=Ax_{t}+Bu_{t}+w_{t}$ (3a)
$\displaystyle y_{t}$ $\displaystyle=Cx_{t}+Du_{t}$ (3b)

where $x_{t}\in\mathbb{R}^{n_{x}}$, $u_{t}\in\mathbb{R}^{n_{u}}$,
$y_{t}\in\mathbb{R}^{n_{y}}$, and $w_{t}\in\mathbb{R}^{n_{x}}$ are the state,
control input, output, and noise of the plant, respectively. The initial state
$x_{0}$ is arbitrary. We make the following assumptions on system (3).

######  Assumption 1 (_Stabilizability and observability_).

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

######  Remark 1.

The observability index of the system (3) is denoted by $\eta$, i.e.,
$\eta:=\min_{1\leq i\leq n_{x}}\\{{\rm rank}[C^{\top},(CA)^{\top},\cdots,$
$(CA^{i})^{\top}]^{\top}=n_{x}\\}$.

######  Assumption 2 (_Unknown system model_).

The system matrices $(A,B,C,D)$ in (3) are unknown, and only some input-output
trajectories, i.e., $\\{u^{s}_{t},y^{s}_{t}\\}_{t=0}^{N-1}$, obtained by some
offline experiments are available.

At each time instant $t$, an output packet containing the past $\eta$ output
measurements, i.e., $y_{[t-\eta,t-1]}$, is sent to the remote controller
through a communication channel subject to DoS attacks. As a result, not all
packets can be received at the controller side, and details about this attack
will be presented in the next section. Moreover, we consider that during
transmission, the output is corrupted by an additive network-induced noise
$n_{t}$, i.e.,

$\zeta_{t}=y_{t}+n_{t}.$ (4)

######  Assumption 3 (Noise bound).

The process noise $w_{t}$ and the network-induced noise $n_{t}$ are bounded by
a known constant $\bar{v}:=\max_{t}\\{\|w_{t}\|,\|n_{t}\|\\}$ for all
$t\in\mathbb{N}_{0}$.

On the other hand, the input $u_{t}$ generated by the controller goes through
an ideal channel, i.e., the controller-to-plant channel is not affected by DoS
attacks. See Fig. 1 for the depicted networked control system architecture.

Due to the presence of noise $w_{t}$ and $n_{t}$, rather than the asymptotic
stability, a weaker notion of stability, referred to as input-to-state
stability (ISS), is explored in this paper. The definition of ISS adapted from
[33, Definition 2.2] is presented as follows.

######  Definition 1 ( _[33, Definition 2.2]_).

The system (3) achieves input-to-state stability (ISS) if its solution
satisfies

$\|x_{t}\|\leq\beta\left(\|x_{0}\|,t\right)+\alpha\left(\|v\|_{\infty}\right),\quad\forall
t\in\mathbb{N}_{\geq 0}$ (5)

where $\alpha$ is a function of class $\mathcal{K}_{\infty}$ 111A function
$\alpha:[0,\infty)\rightarrow[0,\infty)$ is said to be of class $\mathcal{K}$
if it is continuous, strictly increasing, and $\alpha(0)=0$. A function
$\alpha:[0,\infty)\rightarrow[0,\infty)$ is said to be of class
$\mathcal{K}_{\infty}$ if it is of class $\mathcal{K}$ and also unbounded.,
and $\beta$ is a function of class $\mathcal{KL}$ 222A function
$\beta:[0,\infty)\times[0,\infty)\rightarrow[0,\infty)$ is said to be of class
$\mathcal{KL}$ if $\beta(\cdot,t)$ is of class $\mathcal{K}$ for each fixed
$t\geq 0$ and $\beta(s,t)$ decreases to $0$ as $t\rightarrow\infty$ for each
fixed $s\geq 0$.. If (5) holds with $v=0$, then the system is said to be
globally asymptotically stable (GAS).

Finally, we review the so-called uniform input-output-to-state stability
(UIOSS) of a system introduced in [34].

######  Proposition 1 (_Uniform input-output-to-state stability_).

If the system (3) is observable, there exist constants $c_{1},c_{2},c_{3}>0$,
and $\epsilon\in(0,1)$, such that for every $x\in\mathbb{R}^{n_{x}}$,
$u\in\mathbb{R}^{n_{u}}$, and $w\in\mathbb{B}_{\bar{v}}$, the system is UIOSS
with the entire trajectory obeying

$\|x_{t}\|\\!\leq\\!c_{1}\epsilon^{t+1}\|x_{0}\|+c_{2}\|u\|_{[0,j-1]}+c_{3}\|y\|_{[0,j-1]},~{}\forall
t\\!\in\\!\mathbb{N}_{0}$ (6)

for all $t\in\mathbb{N}_{0}$. In addition, there exists a matrix $P=P^{T}\succ
0$ such that for the UIOSS-Lyapunov function $V=x^{T}Px$, the following
inequality holds for some constants
$\sigma_{1},\,\sigma_{2},\,\sigma_{3}\\!>\\!0$

$V(x_{t+1})-V(x_{t})\leq-\sigma_{1}\|x_{t}\|^{2}+\sigma_{2}\|u_{t}\|^{2}+\sigma_{2}\|y_{t}\|^{2}$
(7)

for all $t\in\mathbb{N}_{0}$.

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

Denial-of-Service (DoS) attacks are attacks that are launched by malicious
routers and jammers to block communication channels, which can lead to data
packet losses, and are thus destructive. Evidently for an open-loop unstable
plant, a long-duration of DoS can degrade the closed-loop system performance
and even incur instability and divergence. Therefore, defensive structure
should be judiciously designed and equipped with the system to achieve
stability in the presence of as well as resilience against DoS attacks.

To rigorously evaluate the effectiveness of a defense method, a mathematical
model characterizing DoS attacks should be introduced. Several models have
been studied, and they can be roughly summarized as stochastic model and
deterministic model. For instance, Bernoulli processes [35] and Markov
processes [36] are often used to model DoS attacks. Another line along the
stochastic model is using game-theoretic approaches to designing the attack
strategy as well as the defense strategy simultaneously [37]. However, it is
difficult to justify the incentive of an attacker using the stochastic model.
To this aim, we call for a general deterministic model capitalizing on the DoS
duration and the DoS frequency, that was initially presented in [13]. For each
$t\in\mathbb{N}_{0}$, upon introducing the following DoS indicator

$\ell_{t}:=\left\\{\begin{array}[]{ll}0,&{\text{no DoS attack happens
at}}~{}t\\\ 1,&{\text{a DoS attack happens at}}~{}t\end{array}\right.,$ (8)

the DoS duration during the time interval $[t_{1},t_{2})$ is defined by
$\Phi_{d}(t_{1},t_{2})=\sum_{i=t_{1}}^{t_{2}-1}\ell_{i}$. In addition,
defining for each $t\in\mathbb{N}$

$d_{t}:=\left\\{\begin{array}[]{ll}1,&\ell_{t}=1~{}{\text{and}}~{}\ell_{t-1}=0\\\
0,&{\text{otherwise}}\end{array}\right.,$ (9)

then the DoS frequency during interval $[t_{1},t_{2})$ is expressed by
$\Phi_{f}(t_{1},t_{2})=\sum_{i=t_{1}}^{t_{2}-1}d_{i}$.

It is self-evident from the definitions that, if both the DoS duration and the
DoS frequency are considerably sizable to prevent all packets from being
transmitted, no meaningful control input can be constructed to stabilize the
plant. As a consequence, some assumptions on the DoS frequency and DoS
duration should be made for practical investigation purpose of system
stability in the presence of DoS attacks.

######  Assumption 4 (_DoS frequency_).

There exist constants $\kappa_{f}\in\mathbb{R}_{\geq 0}$, and
$\nu_{f}\in\mathbb{R}_{\geq 2}$, also known as chatter bound and average
dwell-time, respectively, such that the DoS frequency satisfies

$\Phi_{f}(t_{1},t_{2})\leq\kappa_{f}+\frac{t_{2}-t_{1}}{\nu_{f}}$ (10)

over every time interval $[t_{1},t_{2})$, where $t_{1}\leq
t_{2}\in\mathbb{N}_{0}$.

######  Assumption 5 (_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 satisfies

$\Phi_{d}(t_{1},t_{2})\leq\kappa_{d}+\frac{t_{2}-t_{1}}{\nu_{d}}$ (11)

over every time interval $[t_{1},t_{2})$, where $t_{1}\leq
t_{2}\in\mathbb{N}_{0}$.

In fact, the original version of modeling DoS attacks in terms of DoS
frequency and duration was established in [13] to address the stabilization
problem of a continuous-time system under DoS attacks. These notions have been
popularly used in the literature; see e.g., [14, 17, 38, 13, 18]. When
studying discrete-time systems, a discrete-time DoS model that is similar to
the present one, was discussed in [15]. In their work, LTI systems without
noise were considered, and thus weaker assumptions were used. To be specific,
they constrained the DoS frequency and duration only on the interval
$[0,t),t\in\mathbb{N}_{0}$, rather than on every sub-interval $[t_{1},t_{2})$
of $[0,t)$.

######  Remark 2 (_Implications of DoS parameters_).

Taking a switching systems perspective, the quantity $\nu_{f}$ in Assumption 4
can be interpreted as the average dwell-time [39] between two consecutive DoS
attacks _off/on_ switches over the time interval $[t_{1},t_{2})$. On the other
hand, Assumption 5 requires that the average duration of DoS attacks not
exceed a fraction $1/\nu_{d}$ of the entire interval. The constants
$\kappa_{f}$ and $\kappa_{d}$ are additional regularization parameters that
can be chosen to attain tighter bounds on the DoS duration and on the DoS
frequency, respectively. To see this, suppose a DoS attack happens at time
$t_{1}$. Then, it is obvious for the time interval $[t_{1},t_{1}+1)$, that
$\Phi_{f}(t_{1},t_{2})=1$, and $t_{2}-t_{1}=1$. By virtue of the fact that
$\nu_{f}\in\mathbb{R}_{\geq 2}$, we deduce that $(t_{1}+1-t_{1})/\nu_{f}\leq
1/2<1$, which suggests that a constant $\kappa_{f}\geq 1$ is required to
ensure that (10) holds true for every time interval. The same reasoning can be
conducted for $\kappa_{d}$.

As a direct implication of Assumptions 4 and 5, it can be deduced that the
number of time steps between two successful transmissions is upper bounded.
Let $\\{s_{r}\\}_{r\in\mathbb{N}_{0}}$ collect the successful transmission
time instants, at which $\ell_{s_{r}}=0$.

######  Lemma 1 ( [14, Lemma 3]).

Suppose that the DoS attacks satisfy Assumptions 4 and 5 with

$\frac{1}{\nu_{f}}+\frac{1}{\nu_{d}}<1.$ (12)

Then it holds that $s_{0}\leq T-1$, and $s_{r+1}-s_{r}\leq T$ for all
$r\in\mathbb{N}_{0}$ with

$T:=(\kappa_{d}+\kappa_{f})\Big{(}1-\frac{1}{\nu_{d}}-\frac{1}{\nu_{f}}\Big{)}^{-1}+1.$
(13)

######  Remark 3 (_Maximum resilience_).

Condition (12) is referred to as the _maximum resilience_ against DoS attacks
one can achieve for an open-loop unstable system [14, Lemma 3]. In other
words, if (12) does not hold, i.e., ${1}/{\nu_{f}}+{1}/{\nu_{d}}\geq 1$, then
one can always design a sequence of DoS attacks to render the system unstable
whatever control strategy is used. For example, consider a sequence of DoS
attacks under which $l_{t}=1$ for all $t\in\mathbb{N}_{0}$. It is easy to
check that this sequence of attacks satisfies Assumptions 4 and 5 with
$\nu_{f}=1$, $\nu_{d}=\infty$, $\kappa_{d}=1$, and $\kappa_{f}=1$. In
addition, it can be obtained that ${1}/{\nu_{f}}+{1}/{\nu_{d}}=1$. However, it
is impossible to stabilize an open-loop unstable system under such attacks as
there will be no successful transmissions from the system to construct
meaningful control input signals.

###  II-C Fundamental Lemma

The objective of this present paper is to design resilient controllers for
stabilization of an unknown LTI system under DoS attacks and additive noise
using only measured input-output data. Commonly, if the system matrices
$(A,B,C,D)$ were precisely known, a number of proposals have been presented
for resilient control of LTI systems (3) under DoS attacks obeying Assumptions
4 and 5. A natural approach is to endow the closed-loop control system with
prediction capabilities such that the missing measurements can be
reconstructed (predicted) during DoS [14]. To the best of our knowledge, no
previous works have dealt with the stabilization problem of _unknown_ systems
in the presence of DoS attacks. There are three emerging challenges: i) how to
characterize an LTI system as well as infer the wanted quantity from observed
noisy input-output data; ii) how to design a resilient controller against DoS
attacks based only on data; and, iii) the associated stability analysis and
robustness guarantees.

To address the first challenge, we invoke the well-known Fundamental Lemma,
that was initially discovered in [21] and subsequently generalized in [40] and
[41]. Before formally presenting the Fundamental Lemma, the standard
definition of persistency of excitation is introduced first.

######  Definition 2 (_Persistency of excitation_).

A sequence $u_{N}:=\\{u_{t}\in\mathbb{R}^{n_{u}}\\}_{t=0}^{N-1}$ is said to be
persistently exciting of order $L$ if ${\rm{rank}}(H_{L}(u_{N}))=n_{u}L$.

Based on Definition 2, it has been shown in [21] that any input-output
trajectory can be expressed as a linear combination of pre-collected
persistently exciting input-output data, which is also known as the
Fundamental Lemma.

######  Lemma 2 (_Fundamental Lemma[21]_).

Consider the noise-free version of the LTI system (3) described by

$\displaystyle x_{t+1}$ $\displaystyle=Ax_{t}+Bu_{t}$ (14a) $\displaystyle
y_{t}$ $\displaystyle=Cx_{t}+Du_{t}.$ (14b)

Suppose that the input-output sequence
$\\{\bar{u}_{N}^{s},\bar{y}_{N}^{s}\\}:=\\{\bar{u}_{t}^{s},\bar{y}_{t}^{s}\\}_{t=0}^{N-1}$
is a trajectory of the system (14) , induced by a persistently exciting input
sequence $\bar{u}^{s}$ of order $L+n_{x}$. Then,
$\\{\bar{u},\bar{y}\\}:=\\{\bar{u}_{t},\bar{y}_{t}\\}_{t=0}^{L-1}$ is a
trajectory of the system (14) if and only if there exists a vector
$g\in\mathbb{R}^{N-L+1}$ such that the following holds

$\left[\begin{matrix}H_{L}\\!\left(\bar{u}^{s}_{N}\right)\\\
H_{L}\\!\left(\bar{y}^{s}_{N}\right)\end{matrix}\right]g=\left[\begin{matrix}\bar{u}\\\
\bar{y}\end{matrix}\right].$ (15)

Regarding the Fundamental Lemma, two remarks come ready.

######  Remark 4 (_Requirements of Fundamental Lemma_).

Lemma 2 indicates that the length $L$ of the constructed trajectory
$\\{\bar{u},\bar{y}\\}$ depends on the persistently exciting order of the pre-
collected data $\\{\bar{u}_{t}^{s},\bar{y}_{t}^{s}\\}_{t=0}^{N-1}$. More
importantly, it offers a feasible way to design a data-dependent model for the
noise-free LTI system (14). When dealing with the noisy system (3), some
modifications should be made.

######  Remark 5 (_Different ways of pre-collecting trajectories_).

The sequence $\\{\bar{u}_{N}^{s},\bar{y}_{N}^{s}\\}$ can be obtained from
either a single long enough trajectory or formed by multiple short
trajectories obtained by simulating the system using different initial
conditions and input sequences. In fact, both ways of collecting data are
equivalent; please refer to [40] for details.

###  II-D Model-based observer-based controller

When the system matrices of (3) are perfectly known, model-based control
strategies for achieving stability under DoS attacks were designed in, e.g.,
[14, 42, 35]. In [42, 35], although MPC scheme were used for resilient
control, neither the system model nor the attack model resemble that in the
present work. On the other hand, the setting in [14] is the same as this
paper. Moreover, system adopting the model-based observer-based control
strategy in [14] achieves maximum resilient against DoS attacks. The key idea
behind this strategy is to equip an observer at the sensor side to estimate
the state $x_{t}$ at every instant, and adopt a predictor-based state-feedback
controller using the predicted state $\hat{x}_{t}$-based control law
$u_{t}=K\hat{x}_{t}$. When there is no DoS attack (i.e., $t=s_{r}$), the
predictor receives new estimated state $\bar{x}_{t}$ successfully from the
observer, and updates the predicted state $\hat{x}_{t}$; during DoS attacks
(i.e., $t\neq s_{r}$) however, the predictor simply updates the predicted
state following a prediction step based on the system model. Mathematically,
the observer is given by

$\displaystyle\bar{x}_{t+1}$
$\displaystyle=A\bar{x}_{t}+L(y_{t}-C\bar{x}_{t})+Bu_{t}$ (16a) $\displaystyle
u_{t}$ $\displaystyle=K\hat{x}_{t}$ (16b)

and the predictor-based controller is given by

$\displaystyle\hat{x}_{t+1}$ $\displaystyle=A\hat{x}_{t}+Bu_{t},$ (17a)
$\displaystyle\hat{x}_{t}$ $\displaystyle=\bar{x}_{t},~{}~{}$ $\displaystyle
t=s_{r}$ (17b) $\displaystyle u_{t}$ $\displaystyle=K\hat{x}_{t}$ (17c)

where matrix $L$ is a deadbeat observer gain matrix such that
$(A-LC)^{\eta}=0$, and $\eta$ is the observability index of the system [43];
and $K$ is any stabilizing feedback gain matrix, i.e., $A+BK$ is Schur stable.

It has been shown in [14] that, for DoS attacks satisfying (12), the error
between $\hat{x}_{t}$ and ${x}_{t}$ is reset to a constant depending on the
noise bound and the system matrices at least once every $T$ time instants.
Indeed, this resetting is guaranteed because there exists at least one
successful transmission, in any sequence of $T$ time instants, as guaranteed
by Lemma 1.

Without loss of generality, consider the interval $[0,T-1]$, with $s_{0}=T-1$
being the first successful transmission. Define the error
$\bar{e}_{t}:=\bar{x}_{t}-x_{t}$, and the error
$\hat{e}_{t}:=\hat{x}_{t}-x_{t}$. It can be obtained from (3), (16), and (17)
that

$\displaystyle\bar{e}_{t+1}$ $\displaystyle=(A-LC)\bar{e}_{t}+Ln_{t}-w_{t}$
(18a) $\displaystyle\hat{e}_{t+1}$
$\displaystyle=A\hat{e}_{t}-w_{t},~{}~{}~{}$ $\displaystyle t\neq s_{r}$ (18b)
$\displaystyle\hat{e}_{t}$ $\displaystyle=\bar{e}_{t},~{}~{}~{}$
$\displaystyle t=s_{r}.$ (18c)

For $t=s_{0}$, it follows from the recursion in (18) that

$\displaystyle\hat{e}_{s_{0}}=\bar{e}_{s_{0}}$
$\displaystyle=(A-LC)^{\eta}\bar{e}_{s_{0}-\eta}$
$\displaystyle~{}~{}~{}~{}-\sum_{i=0}^{\eta-1}(A-LC)^{i}(Ln_{s_{0}-i-1}-w_{s_{0}-i-1})$
(19a)
$\displaystyle=-\sum_{i=0}^{\eta-1}(A-LC)^{i}(Ln_{s_{0}-i-1}-w_{s_{0}-i-1})$
(19b)

where (19b) follows because $(A-LC)^{\eta}=0$. Moreover, since matrices $A$,
$L$, $C$, and noise $w_{t}$, $n_{t}$ are all finite and bounded, then the
summation of products of finite moments of these terms is also bounded. This
implies that when there is a successful transmission, the error is reset to a
bounded value; that is, the estimated state matches the actual state well. For
DoS attacks obeying (12), Lemma 1 guarantees that this _“predict-then-reset”_
cycle of the estimation error, happens frequently (in fact, at least once
every $T$ time instants). In conclusion, under our working assumptions on the
DoS attacks, the estimation error of the predictor in (17) with the selected
parameters is always bounded, and the system can be stabilized. Obviously,
successful application and implementation of the predictor-based controller in
(17) to defend against DoS attacks hinges on perfect knowledge of the system
model. When this knowledge is not available, although standard system
identification approaches can be used to obtain an estimate of the model, this
step can be time-consuming and computationally expensive as the size of the
system increases. For this reason, data-driven methods, deriving control
strategies directly from data without performing explicit system
identification procedures, have become prevalent recently.

##  III Data-driven Resilient Control under DoS

In this section, we address the challenging stabilization problem of unknown
LTI systems in the presence of DoS attacks. Our proposal is to redesign the
robust predictor-based controller in (17) leveraging the Fundamental Lemma
(15), to develop a novel resilient control scheme that capitalizes purely on
data. Specifically, a data-dependent controller is first constructed to follow
the predict-then-reset circle, which resembles the data-driven MPC developed
in [32]. Capitalizing on this controller, a DoS-resilient MPC algorithm is
proposed and subsequently corroborated by a stability analysis.

###  III-A Data-driven resilient predictive control

According to Section II-D, a known LTI system under DoS attacks can be
stabilized by the predictor-based controller (17). Without knowing the system
matrices $A$ and $B$, MPCs are preferred instead of state-feedback controllers
in which closed-loop stabilizing feedback gain matrices $K$ are in general
hard to find. Traditional MPC schemes (e.g., [29, 27]) can predict future
trajectories based on the system model, initial conditions, and using some
terminal constraints. The optimal future trajectory is generated by minimizing
a cost function. Choosing a control input from the optimal future trajectory,
the system can be stabilized. Thanks to Lemma 2, any trajectory of an LTI
system can be exactly characterized by a (single) input-output trajectory that
is persistently exciting of enough order. This naturally inspires one to
construct a data-driven MPC scheme, that is predicting future trajectories
using some input-output collections to replace a system model. To sum up, our
idea is to replace the model-based controller in (17) with a data-driven MPC,
which builds solely on data consisting of a pre-collected input-output
trajectory, initial conditions, as well as terminal constraints.

Before formally introducing our data-driven MPC, a prerequisite assumption on
the pre-collected data is posed. Let $L$ denote the prediction horizon of the
MPC scheme. At the time instant when no DoS attack occurs (i.e., $t=s_{r}$),
the MPC scheme predicts a trajectory of $L$ steps in the future, i.e., from
$t$ to $t+L-1$, and only the input obtained for time $t$ will be used to
control the system. According to Lemma 2, to be able to generate a predicted
trajectory of length $L$, the pre-collected data should be persistently
exciting of at least order $L$. To validate this requirement, we make the
following assumption on the pre-collected data.

######  Assumption 6 (_Pre-collected data_).

Let sequence
$\\{u^{s}_{N},\\!y^{s}_{N}\\}:=\\{u^{s}_{t},y^{s}_{t}\\}_{i=0}^{N-1}$ denote
an input-output trajectory generated by the LTI system (3) from initial
condition $x_{0}^{s}$, in which the input sequence $\\{u^{s}_{N}\\}$ is
persistently exciting of order $L+n_{x}+\eta$, and the output sequence
$\\{y^{s}_{N}\\}$ is collected offline without network-induced noise $n_{t}$.

Based on the assumption above, we build on the work of [32] and advocate the
following data-driven MPC for control of unknown LTI systems without DoS
attacks, which is solved at every time $t\in\mathbb{N}_{0}$.

$\displaystyle J^{*}_{L}(u_{[t-\eta,t-1]},$
$\displaystyle\zeta_{[t-\eta,t-1]}):=$
$\displaystyle\underset{g(t),h(t)\atop\bar{u}_{i}(t),\bar{y}_{i}(t)}{\min}~{}$
$\displaystyle\sum_{i=0}^{L-1}\ell(\bar{u}_{i}(t),\bar{y}_{i}(t))\\!+\\!\lambda_{g}\bar{v}\|g(t)\|^{2}\\!+\\!\frac{\lambda_{h}}{\bar{v}}\|h(t)\|^{2}$
$\displaystyle{\rm s.t.}\quad\;$
$\displaystyle\left[\begin{matrix}\bar{u}(t)\\\
\bar{y}(t)+h(t)\end{matrix}\right]=\left[\begin{matrix}{H}_{L+\eta}(u^{s})\\\
{H}_{L+\eta}(y^{s})\end{matrix}\right]g(t),$ (20a)
$\displaystyle\left[\begin{matrix}\bar{u}_{[-\eta,-1]}(t)\\\
\bar{y}_{[-\eta,-1]}(t)\end{matrix}\right]=\left[\begin{matrix}u_{[t-\eta,t-1]}\\\
\zeta_{[t-\eta,t-1]}\end{matrix}\right],$ (20b)
$\displaystyle\left[\begin{matrix}\bar{u}_{[L-\eta,L-1]}(t)\\\
\bar{y}_{[L-\eta,L-1]}(t)\end{matrix}\right]=\left[\begin{matrix}0\\\
0\end{matrix}\right],$ (20c)
$\displaystyle~{}\bar{u}_{i}\in\mathbb{U},~{}~{}i\in[0,L-1].$ (20d)

where

  * 1)

Constraint (20a) is reminiscent of (15) in Lemma 2, where
$\bar{u}(t)=[\bar{u}_{-\eta}^{\top}(t),\cdots\\!,\bar{u}^{\top}_{L}(t)]^{\top}\in\mathbb{R}^{n_{u}(L+\eta)}$
and
$\bar{y}(t)=[\bar{y}_{-\eta}^{\top}(t),\cdots\\!,\bar{y}^{\top}_{L}(t)]^{\top}\in\mathbb{R}^{n_{y}(L+\eta)}$.
To sustain feasibility of the optimization problem, a slack vector
$h(t)=[h_{-\eta}^{\top}(t),\cdots\\!,h^{\top}_{L}(t)]^{\top}\\!\in\\!\mathbb{R}^{n_{y}(L+\eta)}$
is introduced to compensate for the network-induced noise $n_{t}$;

  * 2)

Constraints (20b) and (20c) use $\eta$ input and output pairs to restrict the
state $x$ at time instants $t$ and $t+L-1$, which can be implemented thanks to
the observability of the system. To be specific, Constraint (20b) is imposed
to maintain the continuity of the true trajectory at time $t$, and Constraint
(20c) ensures convergence of the predicted trajectories;

  * 3)

Constraint (20d) indicates that for $i\in[0,L-1]$, the control inputs
$\bar{u}_{i}$ are chosen from a given convex set $\mathbb{U}$ with
$0\in\mathbb{U}$, e.g., $\mathbb{U}=[-u_{\max},u_{\max}]$ for some $u_{\max}$;

  * 4)

The cost function $\ell(\bar{u},\bar{y})$ is a quadratic stage cost given by
$\ell(\bar{u},\bar{y})=\|\bar{u}\|^{2}_{R_{1}}+\|\bar{y}\|^{2}_{R_{2}}$, where
$R_{1}\succ 0$, and $R_{2}\succ 0$; the penalty term $\|h(t)\|^{2}$ enforces
sparing use of slacks to produce solutions of minimal constraint violation;
and the coefficient $\lambda_{h}>0$ balances between minimizing the cost and
penalizing the constraint violation; and $\|g(t)\|^{2}$ with coefficient
$\lambda_{g}>0$ is introduced to restrain the effect of process noise $w_{t}$.
Notice that the regularization coefficients of $\|h(t)\|^{2}$ and
$\|g(t)\|^{2}$ also depend on the upper bound on the noise. This indicates
that i) the slack variable $h(t)$ decreases with the noise level $\bar{v}$;
and ii) $\|h(t)\|^{2}$ is small enough compared with the
$\lambda_{g}\bar{v}^{2}\|g(t)\|^{2}/\lambda_{h}$, which is required to prove
the system stability.

It is easy to see that all the constraints and the objective function are
convex, so Problem (20) is convex and can be solved efficiently by means of
off-the-shelf convex programming solvers.

######  Remark 6 (_Comparison with the work[32]_).

Difference between our data-driven MPC scheme in (20) and that of [32] lies in
two aspects. First, instead of an ideal system, we consider LTI systems
containing process noise $w_{t}$. In this setting, the Hankel matrix is a
noisy one that does not contain exact system trajectories. Therefore, to
invoke the Fundamental Lemma, inspired by [44], the error between the Hankel
matrix constructed by the ideal input-output trajectory and the actual
trajectory is characterized in the following proof. Moreover, the data-driven
MPC problem in [32] is non-convex, which is NP-hard in general and
computationally challenging. Inspired by [45], the cost function $J^{*}_{L}$
in the present work is modified by adding noise bound as the coefficients of
$\|h(t)\|^{2}$ and $\|g(t)\|^{2}$. In this manner, the non-convex constraint
$\|h_{i}(t)\|_{\infty}\leq\bar{n}(1+\|g(t)\|_{1})$ (where $\|\cdot\|_{1}$, and
$\|\cdot\|_{\infty}$ are the $\ell_{1}$-, and $\ell_{\infty}$-norm of vectors,
respectively) employed in [32] to guarantee closed-loop stability for their
data-driven MPC, is not needed. Furthermore, the data-driven MPC scheme in
(20) can also guarantee the system stability if the pre-collected output data
is corrupted by noise, i.e., $\tilde{y}^{s}=y^{s}+\tilde{n}$. In this setting,
a similar stability analysis can be conducted yet involving more complicated
noise terms. For ease of exposition, the present work assumes that the pre-
collected output is free from noise.

Upon solving (20), a future trajectory of length $L$ is obtained. In addition,
due to constraints (20b) and (20c), length $L$ should be long enough.
Therefore, the following assumption is made.

######  Assumption 7 (_Prediction horizon_).

The problem horizon satisfies $L\geq\eta+n_{x}$.

It is worth stressing that the proposed MPC in (20) requires no system
matrices but only some input-output trajectories collected from the system (3)
by means of off-line experiments. However, when the linear system is subject
to DoS attacks, one may not be able to solve (20) at every time instant, since
packets containing the last $\eta$ outputs may not successfully arrive at the
controller. To address the challenge emerged with missing packets due to the
DoS attacks, the proposed data-driven MPC scheme is modified by virtue of the
idea behind the predictor-based resilient controller in Section II-D.
Specifically, when there is no DoS attack (i.e., $t=s_{r}$), the controller
receives new output packets successfully from the plant, solves Problem (20),
and uses the input obtained for time $t$ as the control input. However, when a
DoS attack occurs (i.e., $t\neq s_{r}$), if $t-s_{r}\leq L$, it simply sends
the input for time $t$ from the most recent solution of Problem (20); that is,
for $t\in[s_{r},s_{r}+L]$, the first $t-s_{r}$ computed inputs
$\bar{u}_{i}(s_{r})\in[0,t-s_{r}]$ are to be used sequentially when there is a
DoS attack. On the other hand, if $t>s_{r}+L$, i.e., no control input for time
$t$ has been computed from previous solving of (20), then zero inputs will be
used until the next successful transmission comes and Problem (20) is solved
again for new control inputs.

For reference, our proposed resilient data-driven control scheme is summarized
in Algorithm 1.

Algorithm 1 Data-driven resilient control under DoS attacks.

1: Input: Predict horizon $L\geq\eta+{n_{x}}$; parameters of the cost function
$R_{1}\succ 0$, $R_{2}\succ 0$, $\lambda_{g}>0$ and $\lambda_{h}>0$; noise
bound $\bar{v}$; input-output trajectories
$\\{(u^{s}_{0,N-1},y^{s}_{0,N-1})\\}$ of system (3) from initial condition
$x_{0}^{s}$, where input sequence $\\{u^{s}\\}$ is persistently exciting of
order $L+2\eta$.

2: Construct Hankel matrix for the input-output trajectory, i.e.,
$H\\!=\\![H^{\top}_{L+\eta}(u^{s}),H^{\top}_{L+\eta}(y^{s})]^{\top}$.

3: If $t=s_{r}$, do

4: Use the past $\eta$ measurements, i.e., $u_{[t-\eta,t-1]}$ and
$\zeta_{[t-\eta,t-1]}$, to solve Problem (20). Set $u_{t}=\bar{u}_{0}(t)$.

5: Else if $t\neq s_{r}$

6: if $t-s_{r}\leq L$

7: Set $u_{t}=\bar{u}_{t-s_{r}}(s_{r})$.

8: else if $t-s_{r}>L$

9: Set $u_{t}=0$.

10: End if

11: End if

12: Set $t=t+1$ and go back to 3.

###  III-B Stability analysis

In this following, performance of the proposed data-driven resilient
controller is analyzed. The stability analysis follows the same line as that
of model-based resilient controller, in e.g., [14]. It proceeds in two steps.
Similar to the model-based controller in Section II-D, we first show that the
error between the predicted output by solving Problem (20) and the actual
output is always bounded. Next, we construct a Lyapunov function, and
establish the stability of system (3) with the devised data-driven resilient
controller under conditions on the DoS attacks and level of noise.

Before proceeding, we define some variables that will be frequently used
throughout the proof. According to (20b), at each time instant $t$, the most
recent $\eta$ inputs and outputs, i.e.,
$[u^{\top}_{[t-\eta,t-1]},\zeta^{\top}_{[t-\eta,t-1]}]^{\top}$, are used to
initialize (the trajectory of) Problem (20), so its solution depends on
$[u^{\top}_{[t-\eta,t-1]},\zeta^{\top}_{[t-\eta,t-1]}]^{\top}$. Moreover, it
has been proved in [46] that the state $x_{t-\eta}$ of system (3) can be
derived from $u_{[t-\eta,t-1]}$ and $y_{[t-\eta,t-1]}$ when system matrices
$(A,B,C)$ are known, and pair $(C,A)$ is observable. are known. To this end,
let us define the following augmented state vectors of the system

$z_{t}:=\left[\begin{matrix}u_{[t-\eta,t-1]}\\\
y_{[t-\eta,t-1]}\end{matrix}\right],\qquad\tilde{z}_{t}:=\left[\begin{matrix}u_{[t-\eta,t-1]}\\\
\zeta_{[t-\eta,t-1]}\end{matrix}\right].$ (21)

In addition, let $\Gamma_{z}$ denote the linear transformation from $z$ to an
arbitrary but fixed state $x$ that obeys a similar transformation expressed
analytically using $(A,B,C)$, i.e., $x_{t}=\Gamma_{z}z_{t}$, and we have that
$\|x_{t}\|\leq\gamma_{z}\|z_{t}\|$, where $\gamma_{z}:=\|\Gamma_{z}\|$.

The following lemma provides an upper bound on the error between the predicted
output $\bar{y}$ and the actual output $y$.

######  Lemma 3 (_Bounded output prediction error_).

Consider the system (3) with the controller in Algorithm 1. Suppose that
Assumptions 1–7 hold. If i) DoS attacks satisfy Assumptions 4 and 5 with
parameters $\nu_{d}$ and $\nu_{f}$ obey (12) and, ii) Problem (20) is feasible
at $s_{r}$, and $J^{*}_{L}(\tilde{z}_{s_{r}})\leq\bar{J}$, then the error
$e_{y,t}:=y_{t}-\bar{y}^{*}_{t-s_{r}}(s_{r})$ between the predicted output
contained in the solution of (20) and the actual output satisfies

$\|e_{y,s_{r}+q}\|^{2}\leq\beta_{1}(\bar{v},q),\quad q\in[0,s_{r+1}-s_{r})$
(22)

with a $\mathcal{KL}$-function $\beta_{1}(\bar{v},q)$ defined by

$\displaystyle\beta_{1}(\bar{v},q):=b_{3}\xi^{q+\eta}\sqrt{\eta}\bar{v}+b_{2}\xi^{q+\eta}\Bigg{[}\sqrt{\eta}\bar{v}+\sqrt{\bar{J}\bar{v}/\lambda_{h}}$
(23)
$\displaystyle+\|\Upsilon_{\eta}(I)\|\sqrt{\frac{\eta(N-L-\eta+1)\bar{J}\bar{v}}{\lambda_{g}}}\Bigg{]}+b_{1}\\!\\!\\!\\!\\!\sum^{q+\eta-1}_{j=0}\\!\\!\\!\\!\xi^{j}\bar{v}+\sqrt{\frac{\bar{J}\bar{v}}{\lambda_{h}}}$
$\displaystyle+\\!\big{[}2b_{1}\\!\\!\sum^{N-1}_{j=\eta}\xi^{j}\\!+\\!\|\Upsilon_{L+\eta}(I)\|\big{]}\\!\sqrt{\frac{(L+\eta)(N\\!-\\!L\\!-\\!\eta)\bar{J}\bar{v}}{\lambda_{g}}}$

where $b_{1},~{}b_{2},~{}b_{3},~{}\xi>0$ are constants such that
$\|CA^{j}\|\leq b_{1}\xi^{j}$, $\|CA^{j+\eta}\Theta^{{\dagger}}\|\leq
b_{2}\xi^{j}$, and
$\|CA^{j+\eta}\Theta_{\eta}^{{\dagger}}\Upsilon_{\eta}(I)\|\leq b_{3}\xi^{j}$.
For $n=1,2,\cdots$ matrices $\Theta_{n}$ and $\Upsilon_{n}(I)$ are defined by

$\displaystyle\Theta_{n}\\!$ $\displaystyle\,:=\\!\left[\begin{matrix}C\\\
CA\\\ \vdots\\\ CA^{n-1}\end{matrix}\right]\\!$ (24a)
$\displaystyle\Upsilon_{n}(I)\\!$
$\displaystyle\,:=\\!\left[\begin{matrix}\\!0\\!&0&\cdots&0\\\
\\!C\\!&0&\cdots&0\\\ \\!\vdots\\!&\\!\vdots\\!&\\!\ddots\\!&\\!\vdots\\!\\\
\\!CA^{n-2}\\!&CA^{n-3}&\cdots&0\end{matrix}\right]\\!.$ (24b)

###### Proof.

It follows from (3) that the inputs and outputs of system can be rewritten as

$\displaystyle\quad\,\left[\begin{matrix}u_{[s_{r}-\eta,s_{r}+L-1]}\\\
y_{[s_{r}-\eta,s_{r}+L-1]}\end{matrix}\right]$ (25)
$\displaystyle=\Psi_{L+\eta}\left[\begin{matrix}u_{[s_{r}-\eta,s_{r}+L-1]}\\\
x_{s_{r}-\eta}\end{matrix}\right]+\left[\begin{matrix}0\\\
\Upsilon_{L+\eta}(I)\end{matrix}\right]w_{[s_{r}-\eta,s_{r}+L-1]}$ (26)

where

$\Psi_{L+\eta}:=\left[\begin{matrix}I&0\\\
\Upsilon_{L+\eta}(B)&\Theta_{L+\eta}\end{matrix}\right]$ (27)

and symbols $I$ and $0$ denote respectively identity and zero matrices with
suitable dimensions. For $n=1,2,\cdots$, matrix $\Upsilon_{n}(B)$ is given by

$\Upsilon_{n}(B)\\!\,:=\\!\left[\begin{matrix}\\!0\\!&0&\cdots&0\\\
\\!CB\\!&0&\cdots&0\\\ \\!\vdots\\!&\\!\vdots\\!&\\!\ddots\\!&\\!\vdots\\!\\\
\\!CA^{n-2}B\\!&CA^{n-3}B&\cdots&0\end{matrix}\right]\\!.\\\ $ (28)

Let $x^{s}_{N}:=\\{x^{s}_{t}\\}_{t=0}^{N-1}$ and
$w^{s}_{N}:=\\{w^{s}_{t}\\}_{t=0}^{N-1}$ denote the state trajectory and noise
corresponding to the collected trajectory $(u^{s}_{N},y^{s}_{N})$,
respectively, and one gets from (25) that

$\displaystyle\left[\begin{matrix}H_{L+\eta}(u^{s}_{N})\\\
H_{L+\eta}(y^{s}_{N})\end{matrix}\right]$
$\displaystyle=\Psi_{L+\eta}\left[\begin{matrix}H_{L+\eta}(u^{s}_{N})\\\
H_{1}(x^{s}_{N-L-\eta+1})\end{matrix}\right]$ (29a)
$\displaystyle~{}~{}~{}+\left[\begin{matrix}0\\\
\Upsilon_{L+\eta}(I)\end{matrix}\right]H_{L+\eta}(w^{s}_{N}).$ (29b)

Similarly, let $(\hat{x}^{s}_{N},u^{s}_{N},\hat{y}^{s}_{N})$ denote a
trajectory of the system $\eqref{eq:idealsys}$, it can be derived that

$\displaystyle\left[\begin{matrix}H_{L+\eta}(u^{s}_{N})\\\
H_{L+\eta}(\hat{y}^{s}_{N})\end{matrix}\right]=\Psi_{L+\eta}\left[\begin{matrix}H_{L+\eta}(u^{s}_{N})\\\
H_{1}(\hat{x}^{s}_{N-L-\eta+1})\end{matrix}\right].$ (30a)

Choosing

$\displaystyle x^{s}_{0}=\hat{x}^{s}_{0}$ (31a)

and substituting (30) into (29), we arrive at

$\displaystyle\quad\,\left[\begin{matrix}H_{L+\eta}(u^{s}_{N})\\\
H_{L+\eta}(y^{s}_{N})\end{matrix}\right]$
$\displaystyle=\Psi_{L+\eta}\left[\begin{matrix}H_{L+\eta}(u^{s}_{N})\\\
H_{1}(\hat{x}^{s}_{N-L-\eta+1})\end{matrix}\right]+\Psi_{L+\eta}\left[\begin{matrix}0\\\
W^{s}\end{matrix}\right]$ $\displaystyle~{}~{}~{}+\left[\begin{matrix}0\\\
\Upsilon_{L+\eta}(I)\end{matrix}\right]H_{L+\eta}(w^{s}_{N})$ (32)
$\displaystyle=\left[\\!\begin{matrix}H_{L+\eta}(u^{s}_{N})\\\
H_{L+\eta}(\hat{y}^{s}_{N})\end{matrix}\\!\right]\\!\\!+\\!\Psi_{L+\eta}\\!\left[\begin{matrix}0\\\
W^{s}\end{matrix}\right]\\!\\!+\\!\\!\left[\begin{matrix}0\\\
\Upsilon_{L+\eta}(I)\end{matrix}\right]\\!H_{L+\eta}(w^{s}_{N})$ (33)

where
$W^{s}:=[0,w_{0}^{s},\cdots\\!,\sum_{i=0}^{N-L-\eta-1}A^{i}w^{s}_{N-L-\eta-1-i}]$.
Let
$\left(\bar{y}^{*}(s_{r}),\bar{u}^{*}(s_{r}),g^{*}(s_{r}),h^{*}(s_{r})\right)$
denote an optimal solution of Problem (20) at time $s_{r}$. It follows from
(20a) that

$\displaystyle\bar{y}^{*}(s_{r})$
$\displaystyle=H_{L+\eta}(\hat{y}^{s}_{N})g^{*}(s_{r})-h^{*}(s_{r})+\Theta_{L+\eta}W^{s}g^{*}(s_{r})$
$\displaystyle~{}~{}~{}+\Upsilon_{L+\eta}(I)H_{L+\eta}(w^{s}_{N})g^{*}(s_{r}).$
(34)

Since $\hat{y}^{s}_{N}$ is an output trajectory of system (14), it follows
from Lemma 2 that $H_{L+\eta}(\hat{y}^{s}_{N})g^{*}(s_{r})$ is a trajectory of
system (14).

If we let $\tilde{y}=y-H_{L+\eta}(\hat{y}^{s}_{N})g^{*}(s_{r})$, and
$\tilde{x}_{t}$ denote its corresponding state, then dynamics of
$\tilde{x}_{t}$ is given by

$\displaystyle\tilde{x}_{t+1}$ $\displaystyle=A\tilde{x}_{t}+w_{t}$ (35a)
$\displaystyle\tilde{y}_{t}$ $\displaystyle=C\tilde{x}_{t}$ (35b)

with initial output condition

$\displaystyle\tilde{y}_{[s_{r}-\eta,s_{r}-1]}$ (36)
$\displaystyle=y_{[s_{r}-\eta,s_{r}-1]}-[H_{\eta}(y^{s}_{N-L+1})g^{*}(s_{r})$
$\displaystyle~{}~{}~{}-\Upsilon_{\eta}(I)H_{\eta}(w^{s}_{N-L+1})g^{*}(s_{r})-\Theta_{\eta}W^{s}g^{*}(s_{r})]$
$\displaystyle=y_{[s_{r}-\eta,s_{r}-1]}-[y_{[s_{r}-\eta,s_{r}-1]}+n_{[s_{r}-\eta,s_{r}-1]}$
$\displaystyle~{}~{}~{}+h^{*}_{[-\eta,-1]}(s_{r})]+\Theta_{\eta}W^{s}g^{*}(s_{r})$
$\displaystyle~{}~{}~{}+\Upsilon_{\eta}(I)H_{\eta}(w^{s}_{N-L+1})g^{*}(s_{r})$
$\displaystyle=-n_{[s_{r}-\eta,s_{r}-1]}-h^{*}_{[-\eta,-1]}(s_{r})+\Theta_{\eta}W^{s}g^{*}(s_{r})$
$\displaystyle~{}~{}~{}+\Upsilon_{\eta}(I)H_{\eta}(w^{s}_{N-L+1})g^{*}(s_{r}).$

Recursively, it can be obtained from (35) that

$\tilde{y}_{[s_{r}-\eta,s_{r}-1]}=\Theta_{\eta}\tilde{x}_{s_{r}-\eta}-\Upsilon_{\eta}(I)w_{[s_{r}-\eta,s_{r}-1]}$
(37)

yielding

$\tilde{x}_{s_{r}-\eta}\\!=\\!\Theta_{\eta}^{{\dagger}}\tilde{y}_{[s_{r}-\eta,s_{r}-1]}\\!-\\!\Theta_{\eta}^{{\dagger}}\Upsilon_{\eta}(I)w_{[s_{r}-\eta,s_{r}-1]}.$
(38)

Based on (35) and (38), it follows recursively that

$\displaystyle\|\tilde{y}_{s_{r}+q}\|$
$\displaystyle=\Big{\|}CA^{q+\eta}\tilde{x}_{s_{r}-\eta}+\sum_{j=0}^{q+\eta-1}CA^{j}w_{s_{r}-\eta+j}\Big{\|}$
$\displaystyle=\Big{\|}CA^{q+\eta}\Theta_{\eta}^{{\dagger}}\tilde{y}_{[s_{r}-\eta,s_{r}-1]}$
$\displaystyle~{}~{}~{}-CA^{q+\eta}\Theta_{\eta}^{{\dagger}}\Upsilon_{\eta}(I)w_{[s_{r}-\eta,s_{r}-1]}$
$\displaystyle~{}~{}~{}+\sum_{j=0}^{q+\eta-1}CA^{j}w_{s_{r}+q-j-1}\Big{\|}.$
(39)

Substituting (36) into (39), one has that

$\displaystyle\|\tilde{y}_{s_{r}+q}\|$
$\displaystyle=\Big{\|}CA^{q+\eta}\Theta_{\eta}^{{\dagger}}\big{[}\\!-\\!n_{[s_{r}-\eta,s_{r}-1]}\\!-\\!h^{*}_{[-\eta,-1]}(s_{r})$
(40)
$\displaystyle~{}~{}~{}+\Upsilon_{\eta}(I)H_{\eta}(w^{s}_{N-\eta+1})g^{*}(s_{r})+\Theta_{\eta}W^{s}g^{*}(s_{r})\big{]}$
$\displaystyle~{}~{}~{}-CA^{q+\eta}\Theta_{\eta}^{{\dagger}}\Upsilon_{\eta}(I)w_{[s_{r}-\eta,s_{r}-1]}$
$\displaystyle~{}~{}~{}+\sum_{j=0}^{q+\eta-1}CA^{j}w_{s_{r}+q-j-1}\Big{\|}$
$\displaystyle\leq
b_{3}\xi^{q+\eta}\sqrt{\eta}\bar{v}+b_{2}\xi^{q+\eta}\big{[}\sqrt{\eta}\bar{v}+\|h^{*}_{[-\eta,-1]}(s_{r})\|$
$\displaystyle~{}~{}~{}+\|\Upsilon_{\eta}(I)\|\|g^{*}(s_{r})\|\sqrt{\eta(N-L-\eta+1)}\bar{v}\big{]}$
$\displaystyle~{}~{}~{}+b_{1}\sqrt{\eta(N\\!-\\!L\\!-\\!\eta)}\\!\\!\\!\\!\sum^{N-L-1}_{j=\eta}\\!\\!\\!\\!\xi^{j}\bar{v}\|g^{*}(s_{r})\|\\!+\\!b_{1}\\!\\!\\!\\!\sum^{q+\eta-1}_{j=0}\\!\\!\\!\\!\xi^{j}\bar{v}$

where constants $b_{1},b_{2},b_{3},\xi>0$ are chosen such that for all
$j\in\mathbb{N}_{0}$, inequalities $\|CA^{j}\|\leq b_{1}\xi^{j}$,
$\|CA^{j+\eta}\Theta^{{\dagger}}\|\leq b_{2}\xi^{j}$, and
$\|CA^{j+\eta}\Theta_{\eta}^{{\dagger}}\Upsilon_{\eta}(I)\|\leq b_{3}\xi^{j}$
hold true.

Next, we derive bounds for $\|g^{*}(s_{r})\|$ and
$\|h^{*}_{[-\eta,-1]}(s_{r})\|$. Noticing that
$J^{*}(\tilde{z}_{s_{r}})\leq\bar{J}$, it can be deduced from (20) that

$\displaystyle\frac{\lambda_{h}}{\bar{v}}\|h^{*}(s_{r})\|^{2}\leq\bar{J}\quad$
$\displaystyle\rightarrow\quad\|h^{*}(s_{r})\|\leq\bigg{(}\frac{\bar{J}\bar{v}}{\lambda_{h}}\bigg{)}^{1/2}$
(41) $\displaystyle\lambda_{g}\bar{v}\|g^{*}(s_{r})\|^{2}\leq\bar{J}\quad$
$\displaystyle\rightarrow\quad\|g^{*}(s_{r})\|\leq\bigg{(}\frac{\bar{J}}{\bar{v}\lambda_{g}}\bigg{)}^{1/2}.$
(42)

Plugging these bounds into (40), leads to

$\displaystyle\|\tilde{y}_{s_{r}+q}\|\leq
b_{3}\xi^{q+\eta}\sqrt{\eta}\bar{v}+b_{2}\xi^{q+\eta}\Big{[}\sqrt{\eta}\bar{v}+\sqrt{\bar{J}\bar{v}/\lambda_{h}}$
$\displaystyle~{}~{}~{}+\|\Upsilon_{\eta}(I)\|\sqrt{\eta(N-L-\eta+1)\bar{J}\bar{v}/\lambda_{g}}\Big{]}$
$\displaystyle~{}~{}~{}+b_{1}\\!\\!\\!\\!\sum^{N-L-1}_{j=\eta}\\!\\!\\!\\!\xi^{j}\sqrt{\eta(N-L-\eta)\bar{J}\bar{v}/\lambda_{g}}+b_{1}\\!\\!\\!\\!\sum^{q+\eta-1}_{j=0}\\!\\!\\!\\!\xi^{j}\bar{v}$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\beta_{2}(\bar{v},q)$
(43)

where the function $\beta_{2}(\bar{v},q)$ is a $\mathcal{KL}$-function.

Finally, combining (34) and (43), we deduce that

$\displaystyle\|e_{y,s_{r}+q}\|=\|y_{s_{r}+q}-\bar{y}^{*}_{q}(s_{r})\big{\|}$
$\displaystyle\leq\big{\|}y_{s_{r}+q}-I_{[q,q+n_{y}-1]}H_{L+\eta}(\hat{y}^{s}_{N})g^{*}(s_{r})\big{\|}$
$\displaystyle~{}~{}+\big{\|}I_{[q,q+n_{y}-1]}H_{L+\eta}(\hat{y}^{s}_{N})g^{*}(s_{r})-\bar{y}^{*}_{q}(s_{r})\big{\|}$
$\displaystyle\leq\beta_{2}(\bar{v},q)+\big{\|}I_{[q,q+n_{y}-1]}\big{[}-\Upsilon_{L+\eta}(I)H_{L+\eta}(w^{s}_{N})g^{*}(s_{r})$
$\displaystyle~{}~{}+\\!h^{*}\\!(s_{r})\\!-\\!\Theta_{L+\eta}W^{s}g^{*}\\!(s_{r})\\!\big{]}\\!\big{\|}$
$\displaystyle\leq\beta_{2}(\bar{v},q)\\!+\\!\\!\sqrt{\frac{\bar{J}\bar{v}}{\lambda_{h}}}+b_{1}\\!\\!\\!\\!\sum^{N-1}_{j=\eta+L}\\!\\!\\!\\!\xi^{j}\sqrt{\frac{(N\\!-\\!L\\!-\eta)\bar{J}\bar{v}}{\lambda_{g}}}$
$\displaystyle~{}~{}+\|\Upsilon_{L+\eta}(I)\|\sqrt{\\!{(L\\!+\\!\eta)(N\\!-\\!L\\!-\eta+1)\bar{J}\bar{v}}/{\lambda_{g}}}$
$\displaystyle\leq\beta_{1}(\bar{v},q)$

holds for all $q\\!\in\\!\\{0,\cdots\\!,L-1\\}$, where the
$\mathcal{KL}$-function $\beta_{1}(\bar{v},q)$ is defined in (23).

So far, we have bounded the error between the predicted output by using the
controller in Algorithm 1 and the actual output for the case of $L\geq T$.
When $L<T$, it follows from Algorithm 1 that zero inputs will be used once the
$L$ predicted control inputs are run out. Notice from (20c) that the last
$\eta$ predicted inputs are all zeros too. Hence, the established bound in
(22) still holds for all $t\in[s_{r}+L,s_{r+1})$. Details of the proof for the
case of $L<T$ are thus omitted. ∎

######  Remark 7 (_Trade-off between level of noise and DoS_).

According to (23), the error between the estimated output
$\bar{y}^{*}_{q}(s_{r})$ and the actual output $y_{s_{r}+q}$ increases as $q$
grows, and reset to a constant related to the noise bound at every successful
transmission (i.e., $q=0$). In addition, Lemma 1 indicates stronger DoS
attacks lead to longer $T\geq s_{r+1}-s_{r}$, and hence larger $q$. Therefore,
it suggests that, to stabilize systems under stronger DoS attacks, the level
of noise $w_{t}$ and $n_{t}$ should be smaller. Therefore, there exists a
trade-off between robustness against noise as well as resilience against DoS
attacks.

Having bounded the prediction error, we analyze the closed-loop stability of
system (3) along with the proposed controller in Algorithm 1. To this end, let
us begin by constructing a Lyapunov function. Based on the observability of
state $x$ and Definition 1, there exists a positive definite matrix
$P=P^{\top}\succ 0$ such that $W(z)=z^{\top}Pz$ is a UIOSS-Lyapunov function.
Consider the following Lyapunov function accounting for both the data-driven
MPC solution and the augmented system state

$V_{t}:=J^{*}_{L}(\tilde{z}_{t})+\gamma W(z_{t})$ (44)

where $\gamma>0$ is any positive constant. The next lemma provides lower and
upper bounds for this Lyapunov function in terms of the augmented state
$z_{t}$.

######  Lemma 4 (_Bounds on the Lyapunov function_).

Consider the system (3) with the controller in Algorithm (1). Suppose that
Assumptions 1–7 are met. If the parameters of DoS attacks satisfying (12),
then there exist constant $\delta>0$ such that for all
$z_{s_{r}}\in\mathbb{B}_{\delta}$, Problem (20) is feasible, and the Lyapunov
function $V_{s_{r}}$ is bounded for $r\in\mathbb{N}_{0}$ as follows

$\gamma\underline{\lambda}_{P}\|z_{s_{r}}\|^{2}\leq
V_{s_{r}}\leq\gamma_{3}\|z_{s_{r}}\|^{2}+\alpha_{3}(\bar{v})$ (45)

where $\gamma_{3}$ is a constant, and $\alpha_{3}(\bar{v})$ is a
$\mathcal{K}_{\infty}$-function.

###### Proof.

Consider an arbitrary $s_{r}\in\mathbb{N}_{0}$ with $r\in\mathbb{N}_{0}$.
Since $J^{*}_{L}(\tilde{z}_{s_{r}})>0$, it follows from (44) that
$V_{s_{r}}\geq\gamma
W(z_{s_{r}})\geq\gamma\underline{\lambda}_{P}\|z_{s_{r}}\|^{2}$. It remains to
establish the upper bound in (45).

To this end, let us start by constructing a candidate solution
$(\bar{g}(s_{r}),\bar{h}(s_{r}),\bar{u}(s_{r}),\bar{y}(s_{r}))$ of Problem
(20), in which, the input sequence is expected to bring the state $x$ (and its
corresponding output $y$) of system (3) to a ball around the origin whose size
depends on the magnitude of noise $\bar{v}$ in $L$ steps. In the following, we
begin by choosing $\bar{u}(s_{r})$ and $\bar{y}(s_{r})$, which is followed by
proving the existence of $\bar{g}(s_{r})$ and $\bar{h}(s_{r})$ such that
constraints in (20) are satisfied.

According to (20b), it holds that
$\bar{u}_{[-\eta,-1]}(s_{r})=u_{[s_{r}-\eta,s_{r}-1]}$ and
$\bar{y}_{[-\eta,-1]}(s_{r})={\zeta}_{[s_{r}-\eta,s_{r}-1]}$. Capitalizing on
the system stabilizability, for the ideal system (14), there exist a constant
$\delta>0$ and an input sequence $u_{[s_{r},s_{r}+L-1]}\in\mathbb{U}^{L}$,
such that for all $x_{s_{r}}$ satisfying
$\|x_{s_{r}}\|/\gamma_{z}\leq\|z_{s_{r}}\|\leq\delta$, this input sequence can
bring the state $x_{[s_{r},s_{r}+L-1]}$ and associated output
$\hat{y}_{[s_{r},s_{r}+L-1]}$ to the origin in $L-\eta$ steps (due to
constraint (20c), there are actually at most $L-\eta$ non-zero inputs in the
solution $\bar{u}_{[s_{r},s_{r}+L-1]}$), while obeying

$\left\|\left[\begin{matrix}u_{[s_{r},s_{r}+L-1]}\\\
\hat{y}_{[s_{r},s_{r}+L-1]}\end{matrix}\right]\right\|^{2}\leq\gamma_{uy}^{2}\|x_{s_{r}}\|^{2}.$
(46)

Hence, we choose $\bar{u}_{[0,L-1]}(s_{r})=u_{[s_{r},s_{r}+L-1]}$ and
$\bar{y}_{[0,L-1]}(s_{r})=\hat{y}_{[s_{r},s_{r}+L-1]}$. Next, we prove that
there exists a vector $g(s_{r})$ such that (20a) is satisfied.

Set

$\bar{g}(s_{r})=H^{{\ddagger}}_{ux}\left[\begin{matrix}u_{[s_{r}-\eta,s_{r}+L-1]}\\\
x_{s_{r}-\eta}\end{matrix}\right]$

with matrix $H_{ux}$ defined by

$\displaystyle H_{ux}:=\left[\begin{matrix}H_{L+\eta}(u^{s}_{N})\\\
H_{1}(\hat{x}^{s}_{N-L-\eta+1})\end{matrix}\right].$

Based on (30), equation (20a) can be rewritten as

$\displaystyle\left[\\!\begin{matrix}{H}_{L+\eta}(u^{s}_{N})\\\
{H}_{L+\eta}(y^{s}_{N})\end{matrix}\\!\right]\\!\bar{g}(s_{r})=\Psi_{L+\eta}H_{ux}H^{{\ddagger}}_{ux}\left[\\!\begin{matrix}u_{[s_{r}-\eta,s_{r}+L-1]}\\\
x_{s_{r}-\eta}\end{matrix}\\!\right]$ (47a)
$\displaystyle~{}~{}~{}+\bigg{(}\\!\\!\Psi_{L+\eta}\\!\left[\\!\begin{matrix}0\\\
W^{s}\end{matrix}\\!\right]\\!+\\!\left[\\!\begin{matrix}0\\\
\Upsilon_{L+\eta}\end{matrix}\\!\right]\\!H_{L+\eta}(w^{s})\\!\\!\bigg{)}\bar{g}(s_{r})$
$\displaystyle=\left[\\!\begin{matrix}u_{[s_{r}-\eta,s_{r}+L-1]}\\\
\hat{y}_{[s_{r}-\eta,s_{r}+L-1]}\end{matrix}\\!\right]\\!\\!+\\!\bigg{(}\\!\\!\Psi_{L+\eta}\\!\left[\\!\begin{matrix}0\\\
W^{s}\end{matrix}\\!\right]\\!+\\!\left[\\!\begin{matrix}0\\\
\Upsilon_{L+\eta}\end{matrix}\\!\right]\\!H_{L+\eta}(w^{s})\\!\\!\bigg{)}\bar{g}(s_{r})$
$\displaystyle=\left[\begin{matrix}u_{[s_{r}-\eta,s_{r}+L-1]}\\\
\zeta_{[s_{r}-\eta,s_{r}-1]}\\\
\hat{y}_{[s_{r},s_{r}+L-1]}\end{matrix}\right]\\!+\\!\left[\begin{matrix}0\\\
\bar{h}_{[-\eta,-1]}(s_{r})\\\ \bar{h}_{[0,L-1]}(s_{r})\end{matrix}\right].$
(47b)

Combining (3) and (14) yields

$\displaystyle\zeta_{[s_{r}-\eta,s_{r}-1]}$
$\displaystyle=\hat{y}_{[s_{r}-\eta,s_{r}-1]}$
$\displaystyle~{}~{}~{}+\Upsilon_{\eta}(I)w_{[s_{r}-\eta,s_{r}-1]}+n_{[s_{r}-\eta,s_{r}-1]}.$

Therefore, we choose the candidate $\bar{h}(s_{r})$ as follows

$\displaystyle\bar{h}_{[-\eta,-1]}(s_{r})$
$\displaystyle\\!=\\![I,0]\\!\big{(}\Theta_{L+\eta}W^{s}-\Upsilon_{\eta}(I)w_{s_{r}-\eta,s_{r}-1}$
(48a)
$\displaystyle~{}~{}+\Upsilon_{L+\eta}(I)H_{L+\eta}(w^{s})\big{)}\bar{g}(s_{r})-n_{[s_{r}-\eta,s_{r}-1]}$
$\displaystyle\bar{h}_{[0,L-1]}(s_{r})$
$\displaystyle\\!=\\![0,I]\\!\big{(}\Theta_{L+\eta}W^{s}+\Upsilon_{L+\eta}(I)H_{L+\eta}(w^{s})\big{)}\bar{g}(s_{r}).$
(48b)

In addition, (48) indicates that

$\displaystyle\|\bar{h}(s_{r})\|$
$\displaystyle\leq\bar{v}\Big{[}\Big{(}b_{1}\\!\\!\\!\\!\sum_{j=L+\eta}^{N-1}\\!\\!\\!\\!\xi^{j}\sqrt{(L+\eta)(N-L-\eta)}$
$\displaystyle~{}~{}~{}+\|\Upsilon_{L+\eta}(I)\|\sqrt{(L+\eta)(N-L-\eta+1)}\Big{)}\|\bar{g}(s_{r})\|$
$\displaystyle~{}~{}~{}+(1+\|\Upsilon_{\eta}(I)\|)\sqrt{\eta}\Big{]}.$ (49)

Noticing from (20) that the coefficients for terms $\|\bar{h}(s_{r})\|$ and
$\|\bar{g}(s_{r})\|$ are $\lambda_{h}/\bar{v}$ and $\lambda_{g}\bar{v}$,
respectively. Therefore, to minimize the cost function $J^{*}_{L}$, the
inequality (III-B) is naturally guaranteed, and a candidate solution of
Problem (20) is constructed.

Now, we are ready to upper bound $V_{s_{r}}$ in terms of $z_{s_{r}}$ and
$\bar{v}$. Since

$\left[\begin{matrix}u_{[s_{r}-\eta,s_{r}-1]}\\\
y_{[s_{r}-\eta,s_{r}-1]}\end{matrix}\right]=\Psi_{\eta}\left[\begin{matrix}u_{[s_{r}-\eta,s_{r}-1]}\\\
x_{s_{r}-\eta}\end{matrix}\right]+\left[\begin{matrix}0\\\
\Upsilon_{\eta}(I)\end{matrix}\right]w_{[s_{r}-\eta,s_{r}-1]}$

from the observability of system (3) we have that

$\left[\begin{matrix}u_{[s_{r}-\eta,s_{r}-1]}\\\
x_{s_{r}-\eta}\end{matrix}\right]=\Psi^{{\dagger}}_{\eta}z_{s_{r}}-\Psi^{{\dagger}}_{\eta}\left[\begin{matrix}0\\\
\Upsilon_{\eta}(I)\end{matrix}\right]w_{[s_{r}-\eta,s_{r}-1]}.$ (50)

Therefore, $\|\bar{g}(s_{r})\|$ satisfies

$\displaystyle\|\bar{g}(s_{r})\|$
$\displaystyle\leq\|H_{ux}^{{\ddagger}}\|^{2}\bigg{(}\\!\|u_{[s_{r},s_{r}+L-1]}\|^{2}+\left\|\left[\begin{matrix}u_{[s_{r}-\eta,s_{r}-1]}\\\
x_{s_{r}-\eta}\end{matrix}\right]\right\|^{2}\\!\bigg{)}$
$\displaystyle\leq\|H_{ux}^{{\ddagger}}\|^{2}(\gamma_{uy}^{2}\|x_{s_{r}}\|^{2}+2\|\Psi^{{\dagger}}_{\eta}\|^{2}\|z_{s_{r}}\|^{2}$
$\displaystyle~{}~{}~{}+2\|\Theta^{\dagger}_{\eta}\Upsilon_{\eta}(I)\|^{2}\eta\bar{v}^{2})$
$\displaystyle\leq\gamma_{1}\|z_{s_{r}}\|^{2}+\gamma_{2}\bar{v}^{2}$

where
$\gamma_{1}:=\|H_{ux}^{\ddagger}\|^{2}(\gamma_{uy}^{2}\gamma_{z}^{2}+2\|\Psi^{{\dagger}}_{\eta}\|^{2})$
and
$\gamma_{2}:=2\|H_{ux}^{\ddagger}\|^{2}\|\Theta^{\dagger}_{\eta}\Upsilon_{\eta}(I)\|^{2}\eta$.
The first inequality is derived using the definition of $2$-norm. Substituting
(46) and (50) into the first inequality, we arrive at the second inequality.
It follows from (III-B) that

$\displaystyle\|\bar{h}(s_{r})\|^{2}$ $\displaystyle\leq
2\Bigg{[}2\Bigg{(}b_{1}\\!\\!\\!\\!\sum_{j=L+\eta}^{N-1}\\!\\!\\!\\!\xi^{j}\Bigg{)}^{2}(L+\eta)(N-L-\eta)$
$\displaystyle~{}~{}~{}+\\!2\|\Upsilon_{L+\eta}(I)\|^{2}(L\\!+\\!\eta)\\!(N\\!-\\!L\\!-\\!\eta\\!+\\!1)\\!\Bigg{]}\\!\bar{v}^{2}\|\bar{g}(s_{r})\|^{2}$
$\displaystyle~{}~{}~{}+\\!2\|\Upsilon_{\eta}(I)\|^{2}\eta\bar{v}^{2}+2\eta\bar{v}^{2}$
$\displaystyle\leq\alpha_{1}(\bar{v}^{2})\gamma_{1}\|z_{s_{r}}\|^{2}+\alpha_{2}(\bar{v}^{2})$

where $\alpha_{1}(\bar{v}^{2})$ and $\alpha_{2}(\bar{v}^{2})$ are
$\mathcal{K}_{\infty}$-functions defined by
$\alpha_{1}(\bar{v}^{2}):=2[2(b_{1}\sum_{j=L+\eta}^{N-1}\xi^{j})^{2}(L+\eta)(N-L-\eta)+2\|\Upsilon_{L+\eta}(I)\|^{2}(L+\eta)(N-L-\eta-1)]\bar{v}^{2}$
and
$\alpha_{2}(\bar{v}^{2}):=\alpha_{1}(\bar{v}^{2})\bar{v}^{2}\gamma_{2}+2\|\Upsilon_{\eta}(I)\|^{2}\eta\bar{v}^{2}+2\eta\bar{v}^{2}$.
Recalling $V_{t}:=J^{*}_{L}(\tilde{z}_{t})+\gamma W(z_{t})$, it can be deduced
that

$\displaystyle V_{s_{r}}$
$\displaystyle\leq\overline{\lambda}_{\\{R_{1},R_{2}\\}}\gamma_{uy}^{2}\gamma_{z}^{2}\|z_{s_{r}}\|^{2}+\gamma\overline{\lambda}_{P}\|z_{s_{r}}\|^{2}+\lambda_{g}\bar{v}(\gamma_{1}\|z_{s_{r}}\|^{2}$
$\displaystyle~{}~{}~{}+\gamma_{2}\bar{v}^{2})+(\lambda_{h}/\bar{v})(\alpha_{1}(\bar{v}^{2})\gamma_{1}\|z_{s_{r}}\|^{2}+\alpha_{2}(\bar{v}^{2}))$
$\displaystyle\leq\gamma_{3}\|z_{s_{r}}\|^{2}+\alpha_{3}(\bar{v})$

where
$\gamma_{3}:=\overline{\lambda}_{\\{R_{1},R_{2}\\}}\gamma_{uy}^{2}\gamma_{z}^{2}+\gamma\overline{\lambda}_{P}+\lambda_{g}\gamma_{1}\bar{v}+\lambda_{h}\gamma_{1}\alpha_{1}(\bar{v}^{2})/\bar{v}$,
and
$\alpha_{3}(\bar{v}):=\lambda_{g}\gamma_{2}\bar{v}^{3}+(\lambda_{h}/\bar{v})\alpha_{2}(\bar{v}^{2})$
is a $\mathcal{K}_{\infty}$-function. ∎

Now, based on Lemmas 3 and 4, our stability result is presented as follows.

######  Theorem 1.

Consider the system (3) adopting the controller in Algorithm 1, and there
exist a constant $\delta>0$ such that the initial condition
$z_{0}\\!\in\\!\mathbb{B}_{\delta}$. Let Assumptions 1–7 hold. For any
$V_{in}>0$, there exist positive constants $\overline{\lambda}_{g}$,
$\underline{\lambda}_{g}$, $\overline{\lambda}_{h}$,
$\underline{\lambda}_{h}$, and $\bar{v}_{0}>0$ such that for all
$\underline{\lambda}_{g}\\!\leq\\!\lambda_{g}\\!\leq\\!\overline{\lambda}_{g}$,
$\underline{\lambda}_{h}\\!\leq\\!\lambda_{h}\\!\leq\\!\overline{\lambda}_{h}$,
and $\bar{v}\leq\bar{v}_{0}$, if i) the DoS attacks obey (12) in Lemma 1, ii)
Problem (20) is feasible at $s_{0}$, and the Lyapunov function satisfies
$V_{s_{0}}\leq V_{in}$, then the following statements hold true

  1. 1.

Problem (20) is feasible at any successful transmission instant $s_{r}$ with
$r\in\mathbb{N}_{0}$; and,

  2. 2.

System (3) is ISS under Algorithm 1.

###### Proof.

We begin by proving 1), which is also known as _Recursive feasibility_.

The proof consists of two steps. In the first step, we show that if Problem
(20) is feasible at $s_{0}$, it is feasible at $s_{1}$. In the second step, we
prove that the Lyapunov function decreases at every successful transmission
time instant, which indicates the recursive feasibility of Problem (20).

Step 1: First, we prove that Problem (20) is feasible at $s_{1}$, by means of
carefully constructing a candidate solution. Denote an optimal solution of the
Problem (20) at time instant $s_{0}$ by
$(g^{*}(s_{0}),h^{*}(s_{0}),\bar{u}^{*}(s_{0}),\bar{y}^{*}(s_{0}))$. The true
trajectory of system (3) within the time interval $[s_{0},s_{1})$ with initial
condition $(u_{[s_{0}-\eta,s_{0}-1]},\zeta_{[s_{0}-\eta,s_{0}-1]})$ and the
input sequence $\bar{u}^{*}(s_{0})$ is represented by $y_{[s_{0},s_{1}]}$.
Denote a candidate solution at $s_{1}$ by
$(\bar{g}(s_{1}),\bar{h}(s_{1}),\bar{u}(s_{1}),\bar{y}_{s_{1}})$. The
following proof is divided into three cases with respect to the relationship
between $s_{1}-s_{0}$ and $n_{x}$.

Case 1: $s_{1}-s_{0}<n_{x}$.

Similar to the proof of Lemma 4, we first construct $\bar{u}$, $\bar{y}$, and
then $\bar{g}$ and $\bar{h}$ can be derived using the constraints in Problem
(20). For $q\in[-\eta,L-\eta-n_{x}-1]$, let
$\bar{u}_{q}(s_{1})=\bar{u}^{*}_{q+s_{1}-s_{0}}(s_{0})$. For $q\in[-\eta,-1]$,
let $\bar{y}_{q}(s_{1})=\zeta_{s_{1}+q}$, and for $q\in[0,L-\eta-n_{x}-1]$,
let $\bar{y}_{q}(s_{1})=y_{s_{1}+q}$. Recalling from Lemma 3 that for
$q\in[L-\eta-n_{x},L-\eta-1]$

$\|y_{q}(s_{1})-\bar{y}^{*}_{q+s_{1}-s_{0}}(s_{0})\|\leq\beta_{1}(\bar{v},q).$
(51)

In addition, since $x_{s_{0}}\\!\in\\!\mathbb{B}_{\delta/\gamma_{z}}$, and the
constraint (20c)
$(\bar{y}^{*}_{[L-\eta,L-1]}(s_{0}),\bar{u}^{*}_{[L-\eta,L-1]}(s_{0}))$ equals
to zero, it can be deduced that
$\bar{y}^{*}_{[q+s_{1}-s_{0},q+s_{1}-s_{0}+n_{x}]}(s_{0})$ is close to zero
when $\bar{v}$ approaches to zero (if
$\bar{y}^{*}_{[q+s_{1}-s_{0},q+s_{1}-s_{0}+n_{x}]}(s_{0})$ is large, then
constraint (20c) is violated). Consequently, if $\bar{v}\rightarrow 0$, the
output $y_{[L-\eta-n_{x},L-\eta-1]}(s_{1})$ and its corresponding state
$x_{L-\eta-n_{x}}(s_{1})$ approach to zero. In this setting, since the system
is stabilizable, similar to (46), there exist an input trajectory
$\bar{u}_{[L-\eta-n_{x},L-\eta-1]}(s_{1})$ that brings the state and its
corresponding output of the ideal system (14) $\hat{y}$ to zero while obeying

$\left\|\left[\begin{matrix}\bar{u}_{[L-\eta-n_{x},L-\eta-1]}(s_{1})\\\
\hat{y}_{[s_{1}+L-\eta-
n_{x},s_{1}+L-\eta-1]}\end{matrix}\right]\right\|^{2}\leq\gamma_{uy}^{2}\|x_{s_{1}+L-\eta-
n_{x}}\|^{2}.$ (52)

Let $\bar{y}_{q}(s_{1})=\hat{y}_{s_{1}+q}$ for $q\in[L-\eta-n_{x},L-\eta-1]$.
Moreover, due to constraint (20c), for $q\in[L-\eta,L-1]$, we let
$\bar{y}_{[L-\eta,L-1]}(s_{1})$ and $\bar{u}_{[L-\eta,L-1]}(s_{1})$ equal to
zero.

Based on this input-output pair, $\bar{g}(s_{1})$ and $\bar{h}(s_{1})$ can be
chosen following the same step from that of Lemma (4), i.e.,

$\bar{g}(s_{1})=H^{{\ddagger}}_{ux}\left[\begin{matrix}u_{[s_{1}-\eta,s_{1}+L-1]}\\\
x_{s_{1}-\eta}\end{matrix}\right].$ (53)

and

$\displaystyle\bar{h}_{[-\eta,-1]}(s_{1})\\!=\\!-\Upsilon_{\eta}w_{[s_{1}-\eta,s_{1}-1]}-n_{[s_{1}-\eta,s_{1}-1]}-\Psi^{w}\bar{g}(s_{1})$
$\displaystyle\bar{h}_{[0,L-\eta-n_{x}-1]}(s_{1})=-\Upsilon_{L-\eta-
n_{x}}w_{[s_{1},s_{1}+L-\eta-n_{x}-1]}-\Psi^{w}\bar{g}(s_{1})$
$\displaystyle\bar{h}_{[L-\eta-
n_{x},L-1]}(s_{1})=-\Psi^{w}\bar{g}(s_{1})+\Theta_{n_{x}}\\!\\!\\!\\!\\!\sum_{i=0}^{L-n_{x}-1}\\!\\!\\!\\!A^{i}w_{s_{1}+L-\eta-
n_{x}-i-1}$

where

$\Psi^{w}:=\bigg{(}\\!\\!\Psi_{L+\eta}\\!\left[\\!\begin{matrix}0\\\
W^{s}\end{matrix}\\!\right]\\!+\\!\left[\\!\begin{matrix}0\\\
\Upsilon_{L+\eta}\end{matrix}\\!\right]\\!H_{L+\eta}(w^{s}_{N})\\!\\!\bigg{)}$

Case 2: $n_{x}\leq s_{1}-s_{0}<L$.

For $q\in[-\eta,L-\eta-(s_{1}-s_{0})-1]$, let
$\bar{u}_{q}(s_{1})=\bar{u}^{*}_{q+s_{1}-s_{0}}(s_{0})$. For $q\in[-\eta,-1]$,
let $\bar{y}_{q}(s_{1})=\zeta_{s_{1}+q}$, and for
$q\in[0,L-\eta-(s_{1}-s_{0})-1]$, let $\bar{y}_{q}(s_{1})=y_{s_{1}+q}$. Since
$s_{1}-s_{0}\geq n_{x}$, it follows from (20c) that for
$q\in[L-\eta-(s_{1}-s_{0}),L-1-(s_{1}-s_{0})]$,
$\bar{y}^{*}_{q+s_{1}-s_{0}}(s_{0})=0$. Based on Lemma 3

$\|y_{q}(s_{1})\|\leq\beta_{1}(\bar{v},q)$ (54)

and hence the state $x_{s_{0}+L-\eta}$ is close to zero when $\bar{v}$
approaches to zero. In this setting, similar to that of Case(1), there exist
an input trajectory $\bar{u}_{[L-\eta-(s_{1}-s_{0})],L-\eta-1]}(s_{1})$ that
brings the state and its corresponding output of the ideal system (14)
$\hat{y}$ to zero while obeying

$\left\|\\!\left[\begin{matrix}\bar{u}_{[L-\eta-(s_{1}-s_{0}),L-\eta-1]}(s_{1})\\\
\hat{y}_{[s_{0}+L-\eta,s_{1}+L-\eta-1]}\end{matrix}\right]\\!\right\|^{2}\leq\gamma_{uy}^{2}\|x_{s_{0}+L-\eta}\|^{2}.$
(55)

Let $\bar{y}_{q}(s_{1})=\hat{y}_{s_{1}+q}$ for
$q\in[L-\eta-(s_{1}-s_{0}),L-\eta-1]$. Moreover, due to constraint (20c), for
$q\in[L-\eta,L-1]$, we let $\bar{y}_{[L-\eta,L-1]}(s_{1})$ and
$\bar{y}_{[L-\eta,L-1]}(s_{1})$ equal to zero.

Based on this input-output pair, $\bar{g}(s_{1})$ and $\bar{h}(s_{1})$ can be
chosen as,

$\bar{g}(s_{1})=H^{{\ddagger}}_{ux}\left[\begin{matrix}u_{[s_{1}-\eta,s_{1}+L-1]}\\\
x_{s_{1}-\eta}\end{matrix}\right]$ (56)

and

$\displaystyle\bar{h}_{[-\eta,-1]}(s_{1})$
$\displaystyle~{}~{}~{}=-\Upsilon_{\eta}\\!w_{[s_{1}-\eta,s_{1}-1]}-n_{[s_{1}-\eta,s_{1}-1]}-\Psi^{w}\bar{g}(s_{1})$
(57a) $\displaystyle\bar{h}_{[0,L-\eta-(s_{1}-s_{0})-1]}(s_{1})$
$\displaystyle~{}~{}~{}=-\Psi^{w}\bar{g}(s_{1})-\Upsilon_{L-\eta-(s_{1}-s_{0})}w_{[s_{1},s_{0}+L-\eta-1]}$
(57b) $\displaystyle\bar{h}_{[L-\eta-(s_{1}-s_{0}),L-1]}(s_{1})$
$\displaystyle~{}~{}~{}=-\Psi^{w}\bar{g}(s_{1})+\Theta_{(s_{1}-s_{0})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{i=0}^{L-(s_{1}-s_{0})-1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!A^{i}w_{s_{0}+L-\eta-i-1}.$
(57c)

Case 3: $s_{1}-s_{0}\geq L$.

According to Lemma 3, the error bound between the predicted output and the
actual output still holds true when $s_{1}-s_{0}\geq L$. Therefore, a
candidate solution can be constructed follows the same step as in the Case (2)
and is omitted here.

Step 2: Next, leveraging the candidate solution constructed above, the
recursive feasibility is proved by showing that the Lyapunov function
decreases at every successful transmission instant. For simplicity, we only
prove for Case (2), and the proof for Case (1) and (3) carry through. Suppose
that $\bar{v}_{0}$ is sufficiently small such that candidate solution in Step
1 can be constructed. The optimal cost obeys

$\displaystyle J^{*}_{L}(\tilde{z}(s_{1}))\leq
J_{L}(\tilde{z}(s_{1}),\bar{g}(s_{1}),\bar{h}(s_{1}))$ (58)
$\displaystyle=\bigg{(}J^{*}_{L}(\tilde{z}(s_{0}))-\sum_{q=0}^{L-1}\ell(\bar{u}_{q}^{*}(s_{0}),\bar{y}_{q}^{*}(s_{0}))-\lambda_{g}\bar{v}\|g^{*}(s_{0})\|^{2}$
$\displaystyle~{}~{}~{}-(\lambda_{h}/\bar{v})\|h^{*}(s_{0})\|^{2}\bigg{)}+\lambda_{g}\bar{v}\|\bar{g}(s_{1})\|^{2}+(\lambda_{h}/\bar{v})\|\bar{h}(s_{1})\|^{2}$
$\displaystyle~{}~{}~{}+\sum_{q=0}^{L-1}\ell(\bar{u}_{q}(s_{1}),\bar{y}_{q}(s_{1}))$
(59) $\displaystyle\leq
J^{*}_{L}(\tilde{z}(s_{0}))+(\lambda_{h}/\bar{v})\|\bar{h}(s_{1})\|^{2}+\lambda_{g}\bar{v}\|\bar{g}(s_{1})\|^{2}$
$\displaystyle~{}~{}~{}+\sum_{q=0}^{L-1}\big{(}\ell(\bar{u}_{q}(s_{1}),\bar{y}_{q}(s_{1}))-\ell(\bar{u}_{q}^{*}(s_{0}),\bar{y}_{q}^{*}(s_{0}))\big{)}.$
(60)

In addition, the last term in (60) can be decomposed by

$\displaystyle\sum_{q=0}^{L-1}\ell(\bar{u}_{q}^{*}(s_{1}),\bar{y}_{q}^{*}(s_{1}))-\sum_{q=0}^{L-1}\ell(\bar{u}_{q}(s_{0}),\bar{y}_{q}(s_{0}))$
(61a)
$\displaystyle=-\\!\\!\\!\\!\\!\\!\sum_{q=0}^{s_{1}-s_{0}-1}\\!\\!\\!\\!\ell(\bar{u}_{q}(s_{0}),\bar{y}_{q}(s_{0}))+\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{q=L-\eta-(s_{1}-s_{0})}^{L-\eta-1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\ell(\bar{u}_{q}^{*}(s_{1}),\bar{y}_{q}^{*}(s_{1}))$
(61b)
$\displaystyle~{}~{}~{}+\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{q=0}^{L-\eta-(s_{1}-s_{0})-1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\big{(}\ell(\bar{u}_{q}(s_{1}),\bar{y}_{q}(s_{1}))\\!-\\!\ell(\bar{u}_{q+s_{1}-s_{0}}^{*}\\!(s_{0}),\\!\bar{y}_{q+s_{1}-s_{0}}^{*}\\!(s_{0}))\big{)}.$
(61c)

For the second term in (61b), since $\bar{y}_{i}(s_{1})$ is constructed to be
the output for the ideal system (14), the observability of the system
indicates that

$\|x_{s_{0}+L-\eta}\|^{2}\leq\|\Theta_{\eta}^{{\dagger}}\|^{2}\|\hat{y}_{[s_{0}+L-\eta,s_{0}+L-1]}\|^{2}.$
(62)

In addition, noticing that $\bar{y}^{*}_{[L-\eta,L-1]}(s_{0})=0$, similar to
Lemma 3, there exist a function $\alpha_{4}\in\mathcal{K}_{\infty}$ such that

$\|\hat{y}_{[s_{0}+L-\eta,s_{0}+L-1]}\|\leq\alpha_{4}(\bar{v}).$ (63)

Therefore, leveraging (55), we arrive at

$\displaystyle\sum_{q=L-\eta-(s_{1}-s_{0})}^{L-\eta-1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\ell(\bar{u}_{q}(s_{1}),\bar{y}_{q}(s_{1}))\\!\leq\\!\overline{\lambda}_{\\{R_{1},R_{2}\\}}\gamma_{uy}^{2}\|\Theta_{\eta}^{{\dagger}}\|^{2}\alpha_{4}^{2}(\bar{v})$
(64)

which approaches to zero as $\bar{v}\rightarrow 0$.

Next, we derive an upper bound for the term in (61c). Recalling that
$\bar{u}^{*}_{q+s_{1}-s_{0}}(s_{0})=\bar{u}_{q}(s_{1})$ for
$q\in[0,L-\eta-(s_{1}-s_{0})-1]$, one gets that

$\displaystyle\eqref{eq:ll3}=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{q=0}^{L-\eta-(s_{1}-s_{0})-1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\big{(}\|\bar{y}_{q}(s_{1})\|^{2}_{R_{2}}-\|\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|^{2}_{R_{2}}\big{)}.$
(65)

In addition,

$\displaystyle\|\bar{y}_{q}(s_{1})\|^{2}_{R_{2}}-\|\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|^{2}_{R_{2}}$
$\displaystyle\leq\overline{\lambda}_{R_{2}}\big{(}\|\bar{y}_{q}(s_{1})\|^{2}-\|\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|^{2}\big{)}$
$\displaystyle=\overline{\lambda}_{R_{2}}\big{(}\|\bar{y}_{q}(s_{1})-\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})+\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|^{2}$
$\displaystyle~{}~{}~{}-\|\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|^{2}\big{)}$
$\displaystyle\leq\overline{\lambda}_{R_{2}}\big{(}\|\bar{y}_{q}(s_{1})-\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|^{2}$
$\displaystyle~{}~{}~{}+2\|\bar{y}_{q}(s_{1})-\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|\|\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|\big{)}$
$\displaystyle\leq\overline{\lambda}_{R_{2}}\big{(}\|\bar{y}_{q}(s_{1})-\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|^{2}$
$\displaystyle~{}~{}~{}+2\|\bar{y}_{q}(s_{1})-\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|(1+\|\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|)\big{)}$
$\displaystyle\leq\overline{\lambda}_{R_{2}}\big{(}\|\bar{y}_{q}(s_{1})-\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|^{2}$
$\displaystyle~{}~{}~{}+2\|\bar{y}_{q}(s_{1})-\bar{y}_{q+s_{1}-s_{0}}^{*}(s_{0})\|(1+V_{in})\big{)}$
$\displaystyle\leq\overline{\lambda}_{R_{2}}(\beta_{1}^{2}(\bar{v},q+s_{1}-s_{0})+2\beta_{1}(\bar{v},q+s_{1}-s_{0})(1+V_{in}))$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\beta_{3}(\bar{v},q).$
(66)

Therefore, substituting (64) and (66) into (61), we one has that

$\displaystyle\sum_{q=0}^{L-1}\ell(\bar{u}_{q}(s_{1}),\bar{y}_{q}(s_{1}))-\sum_{q=0}^{L-1}\ell(\bar{u}_{q}^{*}(s_{0}),\bar{y}_{q}^{*}(s_{0}))$
(67)
$\displaystyle\leq-\\!\\!\\!\\!\sum_{q=0}^{s_{1}-s_{0}-1}\\!\\!\\!\\!\ell(\bar{u}_{q}(s_{0}),\bar{y}_{q}(s_{0}))+\overline{\lambda}_{\\{R_{1},R_{2}\\}}\gamma_{uy}^{2}\|\Theta_{\eta}^{{\dagger}}\|^{2}\alpha_{4}(\bar{v})$
$\displaystyle~{}~{}~{}+\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{q=0}^{L-\eta-(s_{1}-s_{0})-1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\beta_{3}(\bar{v},q)$
(68)
$\displaystyle=-\\!\\!\\!\\!\sum_{q=0}^{s_{1}-s_{0}-1}\\!\\!\\!\\!\ell(\bar{u}_{q}(s_{0}),\bar{y}_{q}(s_{0}))+\alpha_{5}(\bar{v})$
(69)

where
$\alpha_{5}(\bar{v}):=\overline{\lambda}_{\\{R_{1},R_{2}\\}}\gamma_{uy}^{2}\|\Theta_{\eta}^{{\dagger}}\|^{2}\alpha_{4}(\bar{v})+\sum_{q=0}^{L-\eta-(s_{1}-s_{0})-1}\beta_{3}(\bar{v},q)$
is a $\mathcal{K}_{\infty}$-function.

Now, it remains to derive upper bounds for $\|\bar{g}(s_{1})\|$ and
$\|\bar{h}(s_{1})\|$. From (56), one gets that

$\displaystyle\|\bar{g}(s_{1})\|^{2}$
$\displaystyle=\|H_{ux}^{{\ddagger}}\|^{2}\big{(}\|x_{s_{1}-\eta}\|^{2}+\|\bar{u}_{[s_{1}-\eta,s_{1}+L-1]}\|^{2}\big{)}$
(70)
$\displaystyle=\|H_{ux}^{{\ddagger}}\|^{2}\big{(}\|x_{s_{1}-\eta}\|^{2}\\!+\\!\|\bar{u}^{*}_{[s_{1}-s_{0}-\eta,L-\eta-1](s_{0})}\|^{2}$
$\displaystyle~{}~{}~{}+\|\bar{u}_{[L-\eta-(s_{1}-s_{0}),L-1]}(s_{1})\|^{2}\big{)}$
(71)
$\displaystyle\leq\|H_{ux}^{{\ddagger}}\|^{2}\big{(}\|x_{s_{1}-\eta}\|^{2}\\!+\\!\|\bar{u}^{*}_{[s_{1}-s_{0}-\eta,L-\eta-1](s_{0})}\|^{2}\big{)}$
$\displaystyle~{}~{}~{}+\|H_{ux}^{{\ddagger}}\|^{2}\gamma_{uy}^{2}\|x_{s_{0}+L-\eta}\|^{2}$
(72)
$\displaystyle\leq\|H_{ux}^{{\ddagger}}\|^{2}\big{(}\|x_{s_{1}-\eta}\|^{2}\\!+\\!\|\bar{u}^{*}_{[s_{1}-s_{0}-\eta,L-\eta-1]}(s_{0})\|^{2}\big{)}$
$\displaystyle~{}~{}~{}+\|H_{ux}^{{\ddagger}}\|^{2}\gamma_{uy}^{2}\|\Theta_{\eta}\|^{2}\alpha_{4}^{2}(\bar{v})$
(73)

where (72) holds true because of (55), and (73) is obtained by substituting
(62) and (63) into (72). Based on (57a)–(57c), and noticing that
$s_{1}-s_{0}\leq T$ from Lemma 1, one gets that

$\displaystyle(\lambda_{h}/\bar{v})\|\bar{h}(s_{1})\|^{2}\leq(\lambda_{h}/\bar{v})\Bigg{[}2\|\Upsilon_{\eta}\|^{2}\eta\bar{v}^{2}+2\eta\bar{v}^{2}$
$\displaystyle+2\Bigg{(}\\!2\Bigg{(}b_{1}\\!\\!\\!\\!\sum_{j=L+\eta}^{N-1}\\!\\!\\!\\!\xi^{j}\Bigg{)}^{2}(L+\eta)(N-L-\eta)\bar{v}^{2}$
$\displaystyle+2\|\Upsilon_{L+\eta}\|^{2}(L+\eta)(N-L-\eta+1)\bar{v}^{2}\Bigg{)}\|\bar{g}(s_{1})\|^{2}$
$\displaystyle+2\bigg{\|}\Theta_{T}\\!\\!\\!\\!\sum_{q=0}^{L-n_{x}-1}\\!\\!\\!\\!A^{q}\bigg{\|}^{2}\bar{v}^{2}+2\|\Upsilon_{L-\eta-
n_{x}}\|^{2}(L-\eta-n_{x})\bar{v}^{2}\Bigg{]},$

together with (73) yielding

$\displaystyle(\lambda_{g}\bar{v})\bar{g}(s_{1})+(\lambda_{h}/\bar{v})\bar{h}(s_{1})\leq\bar{v}(\lambda_{g}+\gamma_{4}\lambda_{h})\big{(}\|x_{s_{1}-\eta}\|^{2}$
$\displaystyle+\|\bar{u}^{*}_{[s_{1}-s_{0}-\eta,L-\eta-1]}(s_{0})\|^{2}\big{)}+\alpha_{6}(\bar{v})$
(74)

where
$\gamma_{4}:=2[2(b_{1}\sum_{j=L+\eta}^{N-1}\xi^{j})^{2}(L+\eta)(N-L-\eta)+2\|\Upsilon_{L+\eta}\|^{2}(L+\eta)(N-L-\eta+1)]\|H_{ux}^{{\ddagger}}\|^{2}$
and $\alpha_{6}$ is a $\mathcal{K}_{\infty}$-function. Combining (69), (73)
and (74), it can be obtained from (60) that

$\displaystyle
J^{*}_{L}(\tilde{z}(s_{1}))-J^{*}_{L}(\tilde{z}(s_{0}))\leq-\\!\\!\\!\\!\\!\sum_{q=0}^{s_{1}-s_{0}-1}\\!\\!\\!\\!\\!\ell(\bar{u}_{q}(s_{0}),\bar{y}_{q}(s_{0}))+\alpha_{5}(\bar{v})$
$\displaystyle\\!+\\!\bar{v}(\lambda_{g}\\!+\\!\gamma_{4}\lambda_{h})\big{(}\|x_{s_{1}-\eta}\|^{2}\\!+\\!\|\bar{u}^{*}_{[s_{1}-s_{0}-\eta,L-\eta-1]}\\!(s_{0})\\!\|^{2}\big{)}\\!+\\!\alpha_{6}(\bar{v}).$

Moreover, it follows from (7) that

$\displaystyle W(z_{s_{1}})-W(z_{s_{0}})$
$\displaystyle=\big{(}W(z_{s_{1}})-W(z_{s_{1}-1})\big{)}+\cdots+\big{(}W(z_{s_{0}+1})-W(z_{s_{0}})\big{)}$
$\displaystyle\leq-\sigma_{1}\|z_{[s_{0},s_{1}-1]}\|^{2}+\sigma_{2}\|u_{[s_{0},s_{1}-1]}\|^{2}+\sigma_{3}\|y_{[s_{0},s_{1}-1]}\|^{2}$
$\displaystyle\leq-\sigma_{1}\|z_{[s_{0},s_{1}-1]}\|^{2}+\sigma_{3}(2\|\bar{y}^{*}_{[0,s_{1}-s_{0}-1]}(s_{0})\|^{2}$
$\displaystyle~{}~{}~{}+2\|y_{[s_{0},s_{1}-1]}-\bar{y}^{*}_{[0,s_{1}-s_{0}-1]}(s_{0})\|^{2})+\sigma_{2}\|u_{[s_{0},s_{1}-1]}\|^{2}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:beta_1}}}{{\leq}}\\!\\!-\sigma_{1}\|z_{[s_{0},s_{1}-1]}\|^{2}+\sigma_{2}\|u_{[s_{0},s_{1}-1]}\|^{2}$
$\displaystyle~{}~{}~{}+2\sigma_{3}\|\bar{y}^{*}_{[0,s_{1}-s_{0}-1]}(s_{0})\|^{2}+2\sigma_{3}\sum_{q=0}^{s_{1}-1}\beta_{1}^{2}(\bar{v},q).$

Therefore, the Lyapunov function satisfies

$\displaystyle
V_{s_{1}}\\!-\\!V_{s_{0}}=J^{*}_{L}(\tilde{z}(s_{1}))-J^{*}_{L}(\tilde{z}(s_{0}))+\gamma(W(z_{s_{1}})-W(z_{s_{0}}))$
$\displaystyle\leq-\\!\\!\\!\\!\\!\sum_{q=0}^{s_{1}-s_{0}-1}\\!\\!\\!\\!\\!\ell(\bar{u}_{q}(s_{0}),\bar{y}_{q}(s_{0}))+\alpha_{5}(\bar{v})\\!+\\!\alpha_{6}(\bar{v})$
$\displaystyle~{}~{}~{}+\bar{v}(\lambda_{g}\\!+\\!\gamma_{4}\lambda_{h})\big{(}\|x_{s_{1}-\eta}\|^{2}\\!+\\!\|\bar{u}^{*}_{[s_{1}-s_{0}-\eta,L-\eta-1]}\\!(s_{0})\\!\|^{2}\big{)}$
$\displaystyle~{}~{}~{}+\gamma\big{(}-\sigma_{1}\|z_{[s_{0},s_{1}-1]}\|^{2}+\sigma_{2}\|u_{[s_{0},s_{1}-1]}\|^{2}$
$\displaystyle~{}~{}~{}+2\sigma_{3}\|\bar{y}^{*}_{[0,s_{1}-s_{0}-1]}(s_{0})\|^{2}+2\sigma_{3}\sum_{q=0}^{s_{1}-1}\beta_{1}^{2}(\bar{v},q)\big{)}$
$\displaystyle\stackrel{{\scriptstyle\gamma=\frac{\overline{\lambda}{\\{R_{1},R_{2}\\}}}{\sigma_{2},2\sigma_{3}}}}{{\leq}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\underbrace{\alpha_{5}(\bar{v})\\!+\\!\alpha_{6}(\bar{v})\\!+\\!2\gamma\sigma_{3}\sum_{q=0}^{s_{1}-1}\beta_{1}^{2}(\bar{v},q)}_{\stackrel{{\scriptstyle\triangle}}{{=}}\alpha_{7}(\bar{v})\in\mathcal{K}_{\infty}}\\!-\Big{(}\frac{\gamma\sigma_{1}}{\gamma_{z}}\Big{)}\|x_{s_{0}}\|^{2}$
$\displaystyle~{}~{}~{}+\big{(}\bar{v}(\lambda_{g}\\!+\\!\gamma_{4}\lambda_{h})-(\gamma\sigma_{1}/\gamma_{z})\big{)}\|x_{s_{1}-\eta}\|^{2}$
$\displaystyle~{}~{}~{}+\bar{v}(\lambda_{g}+\gamma_{4}\lambda_{h})\|\bar{u}^{*}_{[s_{1}-s_{0}-\eta,L-\eta-1]}\\!(s_{0})\\!\|^{2}$
$\displaystyle\leq\alpha_{7}(\bar{v})\\!-\\!(\gamma\sigma_{1}/\gamma_{z})\|x_{s_{0}}\|^{2}\\!+\\!\big{(}\bar{v}(\lambda_{g}\\!+\\!\gamma_{4}\lambda_{h})\\!-\\!(\gamma\sigma_{1}/\gamma_{z})\big{)}$
$\displaystyle~{}~{}~{}\times\bigg{\|}A^{s_{1}-\eta-
s_{0}}x_{s_{0}}+\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{q=0}^{s_{1}-s_{0}-\eta-1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!A^{q}Bu_{s_{1}-\eta-q}+\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{q=0}^{s_{1}-\eta-
s_{0}-q}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!w_{s_{1}-\eta-q}\bigg{\|}^{2}$
$\displaystyle~{}~{}~{}+\bar{v}(\lambda_{g}+\gamma_{4}\lambda_{h})\|\bar{u}^{*}_{[s_{1}-s_{0}-\eta,L-\eta-1]}\\!(s_{0})\\!\|^{2}$
$\displaystyle\stackrel{{\scriptstyle\bar{u}\in\mathbb{U}}}{{\leq}}\\!\\!\\!\\!-\Big{(}\frac{\gamma\sigma_{1}}{\gamma_{z}}\\!-\\!2\big{(}\bar{v}(\lambda_{g}\\!+\\!\gamma_{4}\lambda_{h})\\!-\\!\frac{\gamma\sigma_{1}}{\gamma_{z}}\big{)}\|A^{T-\eta}\|\\!\Big{)}\|x_{s_{0}}\|^{2}\\!+\\!\underbrace{\alpha_{7}(\bar{v})}$
$\displaystyle~{}~{}~{}\underbrace{+\bar{v}\bigg{(}\\!\\!4(\\!\lambda_{g}\\!+\\!\gamma_{4}\lambda_{h}\\!-\\!\frac{\gamma\sigma_{1}}{\gamma_{z}}\\!)\\!\\!\\!\\!\\!\sum_{q=0}^{T-\eta-1}\\!\\!\\!\\!\\!\|A^{q}B\|^{2}\\!+\\!(\lambda_{g}\\!+\\!\gamma_{4}\lambda_{h})\\!\\!\bigg{)}u_{\max}^{2}}_{\stackrel{{\scriptstyle\triangle}}{{=}}\alpha_{8}(\bar{v})\in\mathcal{K}_{\infty}}$
$\displaystyle\leq-\gamma_{5}\|x_{s_{0}}\|^{2}+\alpha_{8}(\bar{v})$

where
$\gamma_{5}\\!:=\\!(\gamma\sigma_{1}/\gamma_{z})\\!-\\!2\big{(}\bar{v}(\lambda_{g}\\!+\\!\gamma_{4}\lambda_{h})-(\gamma\sigma_{1}/\gamma_{z})\big{)}\|A^{T-\eta}\|>0$
holds true if
$\bar{v}_{0}\leq((2\|A^{T-\eta}\|+1)\gamma\sigma_{1}/(2\|A^{T-\eta}\|(\lambda_{g}+\gamma_{4}\lambda_{h})\gamma_{z}))$.
Further, adopting (45) in Lemma 4, and choosing $\gamma_{x}>0$ such that
$\gamma_{x}\|z_{t}\|\leq\|x_{t}\|$, one gets that

$V_{s_{1}}\\!-\\!V_{s_{0}}\leq\\!-(\gamma_{5}\gamma_{x}^{2}/\gamma_{3})V_{s_{0}}\\!+\\!(\gamma_{5}\gamma_{x}^{2}/\gamma_{3})\alpha_{3}(\bar{v})\\!+\\!\alpha_{8}(\bar{v})$
(75)

In conclusion, the Lyapunov function decreases at every successful
transmission time instant, and the Problem (20) is feasible at $s_{r}$ for all
$r\in\mathbb{N}_{0}$.

_s2) ISS_

Based on (75), it can be obtained recursively that

$V_{s_{r}}\\!\leq\\!\bigg{(}\\!\\!1-\frac{\gamma_{5}\gamma_{x}^{2}}{\gamma_{3}}\\!\bigg{)}^{r+1}\\!\\!\\!\\!\\!\\!\\!\\!\\!V_{s_{0}}\\!+\\!\sum_{q=0}^{r}\bigg{(}\\!1\\!-\\!\frac{\gamma_{5}\gamma_{x}^{2}}{\gamma_{3}}\\!\bigg{)}^{q}\\!\\!\bigg{(}\\!\frac{\gamma_{5}\gamma_{x}^{2}}{\gamma_{3}}\alpha_{3}(\bar{v})\\!+\\!\alpha_{8}(\bar{v})\\!\\!\bigg{)}.$

Finally, adopting (45), we arrive at

$\displaystyle\|z_{s_{r}}\|^{2}\leq\frac{\gamma_{3}}{\gamma\underline{\lambda}_{P}}\bigg{(}\\!\\!1-\frac{\gamma_{5}\gamma_{x}^{2}}{\gamma_{3}}\\!\bigg{)}^{r+1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\|z_{s_{0}}\|^{2}+\alpha_{9}(\bar{v})$
(76)

with function $\alpha_{9}(\bar{v})\in\mathcal{K}_{\infty}$, which completes
the proof according to Definition 1. ∎

######  Remark 8 (_Global ISS_).

In Theorem 1, $z_{0}\in\mathbb{B}_{\delta}$ is required to establish the
recursive feasibility. That is, system (3) achieves only local ISS under the
proposed data-driven resilient controller in Algorithm 1. To recover global
ISS, we provide next two solutions, each of which comes at the price of either
reduced resilience against DoS attacks or increased computational complexity.
The first solution is to solve Problem (20) once every $n_{x}$ time instants,
and apply the control inputs in
$u_{[t,t+n_{x}-1]}=\bar{u}^{*}_{[0,n_{x}-1]}(t)$ sequentially to the system
over the ensuing $n_{x}$ steps. Under such a method, global recursive
feasibility of Problem (20) can be shown similarly to that of Theorem 1, and
the closed-loop system achieves global ISS provided that condition
$(1/\nu_{f})+(1/\nu_{d})<1-((n_{x}-1)/\nu_{f})$ on DoS attacks is met. In
addition, a data-driven MPC scheme without the terminal constraint (20c) was
recently proposed in [45], under which global ISS can be ensured if the
prediction horizon (i.e., $L$) is sufficiently large. Condition on $L$ implies
that this scheme also requires sufficiently many pre-collected trajectories
(i.e., large $N$). In this manner, system (3) can achieve maximum resilience
against DoS attacks at the price of increased computational complexity.
Stability analysis for these two schemes can be performed by leveraging the
results in [32] and [45].

######  Remark 9 (_DoS attacks also in controller-to-plant channel_).

When both controller-to-plant and sensor-to-controller channels are subject to
DoS attacks, a packetized transmission policy can be used in the input channel
too. At each successful transmission time instant, the plant receives a packet
consisting of $b$ inputs from the controller (i.e.,
$\bar{u}^{*}_{0,b-1}(s_{r})$), where $b$ is the buffer size at the plant side.
Our proposed data-driven resilient controller along with the stability and
robustness guarantees of this paper can be generalized to this case, and the
trade-off between the buffer size $b$ and DoS attacks can be characterized by
the decrease rate and the increase rate of the Lyapunov function (c.f. [38]).

##  IV Numerical Example

Consider the unstable batch reactor example in [14] (originally studied in
[47]), where model-based resilient control was investigated under DoS attacks.
This model is given by $\dot{x}(t)=Ax(t)+Bu(t)$ and $y=Cx(t)$, with

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

Here, we consider a discrete-time version of the unstable batch reactor with a
sampling period of $0.1s$, and evaluate the resilient performance of the
proposed controller. To this end, a number of input-output trajectories of
length $N=100$ were obtained by means of simulating the open-loop system off-
line using input sequences obeying Assumption 6. Based on Assumption 7,
parameters of Algorithm 1 were set as follows: $L=10\geq n_{x}+2\eta=8$,
$\lambda_{g}=0.1$, $\lambda_{h}=100$, $R_{1}=10^{-4}I_{2}$, and $R_{2}=3{\rm
I_{2}}$. Over a simulation horizon of $200$ time instants, Figs. 2–4 depict
the comparative performance of the data-driven method and the model-based
method under different levels of noise and DoS attacks (signified by the gray
shades). In addition, the identification step uses the same data as the data-
driven method. The red dashed line (output using the proposed controller) in
Figs. 2–4 confirmed that there is a trade-off between system resilience
against DoS attacks and robustness against noise, which aligns with our
observations in Remark 7. Compared with the data-riven method, the model-based
method either using the controller in [14] or standard model predictive
controller yields poorer system performance, which resembles the results in
[44]. This is because the identification step tend to over-fit the noise in
the pre-collected data when system dimension is high (e.g., $n_{x}\geq 4$).

Furthermore, relationship between system performance, prediction horizon $L$,
and length of the pre-collected data $N$ is discussed in Fig. 5. It was shown
in [22] that the length of pre-collected data should obey
$N\geq(\max\\{n_{u},n_{y}\\}+1)L+n_{x}-1=33$. The top panel of Fig. 5
illustrates that when $N$ is small (i.e., $33<N\leq 60$), system performance
improves fast as $N$ increases; and when $N$ becomes large, system performance
remains almost steady as $N$ increases. The bottom panel of Fig. 5 reports
system performance for different $L$ values. According to Assumption 7, the
predicted horizon obeys $2\eta+n_{x}\leq
L\leq(N-n_{x}+1)/(\max\\{n_{u},n_{y}\\}+1)$. Therefore, when $N=40$, one gets
that $8\leq L\leq 12$. It has been shown that, as long as the $L$ and $N$ are
enough for evaluating Problem (20), they have little effect on the system
performance. Finally, the bottom panel of Fig. 5 implies that when the dataset
is small, the data-driven control method achieves better performance than
using the model-based one.

Figure 2: Norm of the closed-loop output $\|y\|$:
$1/\nu_{f}+1/\nu_{d}=0.8841$.
Figure 3: Norm of the closed-loop output $\|y\|$:
$1/\nu_{f}+1/\nu_{d}=0.9142$.
Figure 4: Norm of the closed-loop output $\|y\|$:
$1/\nu_{f}+1/\nu_{d}=0.9317$.
Figure 5: Relationship between system performance, $L$, and $N$:
$1/\nu_{f}+1/\nu_{d}=0.8841$.

##  V Conclusions

This paper considered the stabilization problem of unknown stochastic LTI
systems under DoS attacks. A data-driven MPC scheme was developed, under which
a sequence of inputs and associated predicted outputs can be computed purely
from some pre-collected input-output data by solving convex programs. On top
of this scheme, a data-driven resilient controller was proposed such that the
system can achieve local input-to-state stability under conditions on the DoS
attacks and level of noise, even without any knowledge of system model or a
priori system identification procedure. Moreover, the condition on DoS attacks
is the same as the maximum resilience achieved in existing works under model-
based controller. To achieve global stability, two modifications of the
proposed controller were discussed at the price of sacrificing system
resilience against DoS attacks, or increasing computational complexity.
Finally, a numerical example was provided to demonstrate the effectiveness of
the proposed method as well as the practical merits of the established theory.

## References

  * [1] A. Cárdenas, S. Amin, and S. Sastry, “Secure control: Towords survivable cyber-physical systems,” in _Proc. Int. Conf. Distrib. Comput. Syst. Workshops_ , July 2008, pp. 495–500.
  * [2] E. A. Lee, “Cyber physical systems: Design challenges,” in _Proc. Int. Symp. Obj. Component-Oriented Real-Time Distrib. Comput._ , Orlando, FL, USA, May 5-7 2008.
  * [3] M. Todescato, J. W. Simpson-Porco, F. Dörfler, R. Carli, and F. Bullo, “Online distributed voltage stress minimization by optimal feedback reactive power control,” _IEEE Trans. Control Netw. Syst._ , vol. 5, no. 3, pp. 1467–1478, July 2018.
  * [4] G. Wu, G. Wang, J. Sun, and J. Chen, “Optimal partial feedback attacks in cyber-physical power systems,” _IEEE Trans. Autom. Control_ , vol. 65, no. 9, pp. 3919–3926, Sept. 2020.
  * [5] L. Ren, Z. Meng, X. Wang, R. Lu, and L. T. Yang, “A wide-deep-sequence model-based quality prediction method in industrial process analysis,” _IEEE Trans. Neural Netw. Learn. Syst._ , vol. 31, no. 9, pp. 3721–3731, Sept. 2020.
  * [6] F. Pasqualetti, F. Dörfler, and F. Bullo, “Attack detection and identification in cyber-physical systems,” _IEEE Trans. Autom. Control_ , vol. 58, no. 11, pp. 2715–2729, June 2013.
  * [7] F. Pasqualetti, R. Carli, and F. Bullo, “Distributed estimation via iterative projections with application to power network monitoring,” _Automatica_ , vol. 48, no. 5, pp. 747–758, May 2012.
  * [8] G. Wu and J. Sun, “Optimal switching integrity attacks on sensors in industrial control systems,” _J. Syst. Sci. Complex_ , vol. 32, pp. 1290–1305, Jan. 2019.
  * [9] M. Zhu and S. Martinez, “On the performance analysis of resilient networked control systems under replay attacks,” _IEEE Trans. Autom. Control_ , vol. 59, no. 3, pp. 804–808, Aug. 2014.
  * [10] B. Chen, D. W. C. Ho, G. Hu, and L. Yu, “Secure fusion estimation for bandwidth constrained cyber-physical systems under replay attacks,” _IEEE Trans. Cybern._ , vol. 48, no. 6, pp. 1862–1876, July 2018.
  * [11] A. Cetinkaya, H. Ishii, and T. Hayakawa, “An overview on Denial-of-Service attacks in control systems: Attack models and security analyses,” _Entropy_ , vol. 21, no. 2, pp. 210–238, Feb. 2019.
  * [12] D. Winder, “Powerful cyber attack takes down 25% of Iranian internet,” https://www.forbes.com/sites/daveywinder/2020/02/09/powerful-iran-cyber-attack-takes-down-25-of-national-internet/?sh=2ea15ad20dce.
  * [13] C. De Persis and P. Tesi, “Input-to-state stabilizing control under Denial-of-Service,” _IEEE Trans. Autom. Control_ , vol. 60, no. 11, pp. 2930–2944, Nov. 2015.
  * [14] S. Feng and P. Tesi, “Resilient control under Denial-of-Service: Robust design,” _Automatica_ , vol. 79, pp. 42–51, Mar. 2017.
  * [15] M. Wakaiki, A. Cetinkaya, and H. Ishii, “Stabilization of networked control systems under DoS attacks and output quantization,” _IEEE Trans. Autom. Control_ , vol. 65, no. 8, pp. 3560–3575, 2020.
  * [16] 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_ , 2021, doi: 10.1109/TAC.2021.3107145.
  * [17] A. Y. Lu and G.-H. Yang, “Input-to-state stabilizing control for cyber-physical systems with multiple transmission channels under Denial-of-Service,” _IEEE Trans. Autom. Control_ , vol. 63, no. 6, pp. 1813–1820, June 2018.
  * [18] C. De Persis and P. Tesi, “Networked control of nonlinear systems under Denial-of-Service,” _Syst. Control Lett._ , vol. 96, pp. 124–131, Oct. 2016.
  * [19] L. A. Montestruque and P. J. Antsaklis, “Model-based control of networked systems,” _Automatica_ , vol. 39, no. 10, pp. 1837–1843, Oct. 2003.
  * [20] L. Ren, Z. Meng, X. Wang, L. Zhang, and L. T. Yang, “A data-driven approach of product quality prediction for complex production systems,” _IEEE Trans. Industr. Inform._ , vol. 17, no. 9, pp. 6457–6465, Sept. 2021.
  * [21] J. C. Willems, I. Markovsky, P. Rapisarda, and B. L. M. De Moor, “A note on persistency of excitation,” _Syst. Control Lett._ , vol. 56, no. 4, pp. 325–329, May 2005.
  * [22] C. De Persis and P. Tesi, “Formulas for data-driven control: Stabilization, optimality, and robustness,” _IEEE Trans. Autom. Control_ , vol. 65, no. 3, pp. 909–924, Mar. 2020.
  * [23] J. Berberich, A. Koch, C. W. Scherer, and F. Allgöwer, “Robust data-driven state-feedback design,” in _Proc. Amer. Control Conf._ , Denver, CO, USA, USA, July 1-3 2020, pp. 1532–1538.
  * [24] J. G. Rueda-Escobedo, E. Fridman, and J. Schiffer, “Data-driven control for linear discrete-time delay systems,” _IEEE Trans. Autom. Control_ , 2021, doi = 10.1109/TAC.2021.3096896.
  * [25] M. Zhang, M.-G. Gan, J. Chen, and Z.-P. Jiang, “Data-driven adaptive optimal control of linear uncertain systems with unknown jumping dynamics,” _J. Frankl. Inst._ , vol. 356, no. 12, pp. 6087–6105, Aug. 2019.
  * [26] S. Talebi, S. Alemzadeh, N. Rahimi, and M. Mesbahi, “Online regulation of unstable LTI systems from a single trajectory,” _arXiv:2006.00125_ , 2020\.
  * [27] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: Stability and optimality,” _Automatica_ , vol. 36, no. 6, pp. 789–814, June 2000.
  * [28] D. Limon, I. Alvarado, T. Alamo, and E. F. Camacho, “MPC for tracking of piece-wise constant references for constrained linear systems,” _Automatica_ , vol. 44, no. 9, pp. 2382–2387, May 2008.
  * [29] J. B. Rawlings, D. Q. Mayne, and M. M. Diehl, _Model Predictive Control: Theory and Design_ , 2nd ed. Nob Hill Publishing, 2019.
  * [30] J. Coulson, J. Lygeros, and F. Dörfler, “Data-enabled predictive control: In the shallows of the DeePC,” in _Proc. Eur. Control Conf._ , Naples, Italy, Italy, June, 25-28 2019, pp. 307–312.
  * [31] F. Dörfler, J. Coulson, and I. Markovsky, “Bridging direct & indirect data-driven control formulations via regularizations and relaxations,” _arXiv:2101.01273_ , 2021.
  * [32] J. Berberich, J. Köhler, M. A. Müller, and F. Allgöwer, “Data-driven model predictive control with stability and robustness guarantees,” _IEEE Trans. Autom. Control_ , June 2020, doi: 10.1109/TAC.2020.3000182.
  * [33] Z.-P. Jiang, A. R. Teel, and L. Praly, “Small-gain theorem for ISS systems and applications,” _Math. Control Signals Syst._ , vol. 7, pp. 95–120, 1994.
  * [34] C. Cai and A. R. Teel, “Input–output-to-state stability for discrete-time systems,” _Automatica_ , vol. 44, no. 2, pp. 326–336, Sept. 2008.
  * [35] H. Yang, Y. Li, L. Dai, and Y. Xia, “MPC-based defense strategy for distributed networked control systems under DoS attacks,” _Syst. Control. Lett._ , vol. 128, pp. 9–18, Jun. 2019.
  * [36] G. K. Befekadu, V. Gupta, and P. J. Antsaklis, “Risk-sensitive control under markov modulated Denial-of-Service (DoS) attack strategies,” _IEEE Trans. Autom. Control_ , vol. 60, no. 12, pp. 3299–3304, Dec. 2015.
  * [37] Y. Li, D. E. Quevedo, S. Dey, and L. Shi, “SINR-based DoS attack on remote state estimation: A game-theoretic approach,” _IEEE Trans. Control Netw. Syst._ , vol. 4, no. 3, pp. 632–642, Sept. 2017.
  * [38] S. Feng and P. Tesi, “Networked control systems under Denial-of-Service: Co-located vs. remote architectures,” _Syst. Control Lett._ , vol. 108, pp. 40–47, Sept. 2017.
  * [39] P. Hespanha and A. S. Morse, “Stability of switched systems with average dewell-time,” in _Proc. IEEE Conf. Decis. Control_ , Phoenix, Arizona, USA, Dec. 1999, pp. 2655–2660.
  * [40] H. J. van Waarde, C. De Persis, M. K. Camlibel, and P. Tesi, “Willems’ fundamental lemma for state-space systems and its extension to multiple datasets,” _IEEE Control Syst. Lett._ , vol. 4, no. 3, pp. 602–607, April 2020.
  * [41] Y. Yu, S. Talebi, H. J. van Waarde, U. Topcu, M. Mesbahi, and B. Açıkmeşe, “On controllability and persistency of excitation in data-driven control: Extensions of Willems’ fundamental lemma,” _arXiv:2102.02953_ , 2021.
  * [42] Q. Sun, K. Zhang, and Y. Shi, “Resilient model predictive control of cyber–physical systems under DoS attacks,” _IEEE Trans. Industr. Inform._ , vol. 16, no. 7, pp. 4920–4927, July 2020.
  * [43] J. O’Reilly, _Mathematics in Science and Engineering, Observers for Linear Systems_. London: Academic Press, 1983\.
  * [44] V. Krishnan and F. Pasqualetti, “On direct vs indirect data-driven predictive control,” _arXiv:2103.14936_ , 2021.
  * [45] J. Bongard, J. Berberich, J. Köhler, and F. Allgöwer, “Robust stability analysis of a simple data-driven model predictive control approach,” _arXiv:2103.00851_ , Mar. 2021.
  * [46] F. M. Callier and C. A. Desoer, _Linear System Theory_. Springer New York, 1991.
  * [47] G. C. Walsh and H. Ye, “Scheduling of networked control systems,” _IEEE Control Syst. Mag._ , vol. 21, no. 1, pp. 57–65, Feb. 2002.