A quantization-coding scheme with variable data rates for cyber-physical systems under DoS attacks

Abstract

This paper considers Cyber-Physical Systems (CPSs) in the presence of Denial-of-Service (DoS) attacks, and investigates the networked control problem under data rate limitations. We develop a framework for CPSs in the presence of DoS attacks, and present a state-estimate control scheme for CPSs under DoS attacks. For the different communication status, the information on the plant state is transmitted to the controller by employing a different data rate in order to mitigate the influence of DoS attacks. Sufficient conditions on the data rate for mean square observability and stabilization of CPSs are derived by applying information theoretic tools of source coding. Our result is less conservative than some existing results. An illustrative example is given to demonstrate the effectiveness of the proposed quantization-coding scheme.

Keywords:

• Cyber-Physical systems (CPSs)
• denial-of-service (DoS)
• variable data rates
• quantization-coding schemes

1. Introduction

In recent years, Cyber-Physical Systems (CPSs) have been an active research area (Fioravanti et al., 2023; Wang & Shen, 2023). As fundamental issues in control theory, the observability and stabilization problem for CPSs has attracted much attention because of their applications, such as intelligent transportation, smart grids and process automation systems (Weng et al., 2023). In CPSs, the information on the plant state is transmitted to the controller over a communication channel, which can be threatened by malicious attacks, such as denial-of-service (DoS) 4attacks. In remote state estimation, DoS attacks can block communication channels. This leads to degraded Quality of Service (QoS) of the communication network. In very severe cases, very few information on the plant state is transmitted to the controller due to DoS attacks such that the close-loop feedback system becomes instable. The networked control problem for CPSs in the presence of DoS attacks has received considerable attention (Liu et al., 2023).

A framework on DoS attacks was proposed in Persis and Tesi (2015). In Qin et al. (2017), the authors proposed an optimal DoS attack scheduling with energy constraint over packet-dropping networks, and in Ding et al. (2017), the authors proposed a multi-channel transmission schedule for remote state estimation under DoS attacks. Li et al. (2017) considered SINR-based DoS attacks on remote state estimation. Kato et al. (2022) employed a deterministic DoS attack model constrained in terms of attacks' frequency and duration, and studied the stabilization problem of networked control systems under DoS attacks. Liu et al. (2022) was concerned with the problem of stabilizing continuous-time linear time-invariant systems subject to quantization and DoS attacks, and proposed a novel structure based on a deadbeat controller as well as a delicate transmission protocol for the input and output channels, and codesigned a deadbeat controller and transmission protocol.

Networked state estimation over a shared communication medium was introduced in Xia et al. (2017). An LMI approach could be used to design the state estimator for networked control systems (Muehlebach & Trimpe, 2018). A comprehensive solution to the plant state estimation problem for networked control systems in a distributed fashion over communication networks was provided in Li et al. (2018). A remote state estimator over a wireless fading channel was designed (Ren et al., 2018). Furthermore, the problem of estimating the state of weakly coupled linear systems was considered, and an optimal allocation of the channel capacity was proposed (Chen et al., 2017). Kircher et al. (2022) gave an optimization framework for resilient batch estimation in CPSs. Liu et al. (2023) addressed remote estimation for energy harvesting systems under multiplicative noises. Xu et al. (2022) addressed state estimation under joint false data injection attacks.

A lot of excellent results on data rate limitation have been reported in the literature. A lower bound on the data rate required to ensure observability and stabilizability of linear control systems was provided by Tatikonda and Mitter (2004). Liu and Jin (2017) further developed the result, and gave a smaller lower bound by employing the quantization and coding scheme where all plant states are quantized, coded and converted together into a codeword. Liu and Rui (2018) constructed a binary code where the codeword length is determined by the real state prediction error, and further gave a less conservative result. A data rate condition to stabilize a continuous-time scalar linear system was derived on the basis of event triggering (Ling, 2017). Furthermore, event-triggering control under bandwidth limitation was also studied by Li et al. (2017). Kostina et al. (2022) gave sufficient conditions on the exact minimum number of bits to stabilize a linear system. Based on the spherical polar coordinate quantizer, a quantization method with finite data rate was proposed for handling the nested quantization and achieving the convergence to the origin of the system with input and output quantizations (Wang, 2023).

In this paper, we will consider networked control systems under data-rate constraints and DoS attacks. Namely, communication between sensors and controllers may take place over data-rate-limitation channels in the presence of DoS attacks. There exists the trade-offs between system resilience and data rate, which affects the stability of control systems (Feng et al., 2021). Feng et al. (2022) further developed a variable bit rate (VBR) encoding–decoding protocol and quantized controller to stabilize the control system in the presence of DoS attacks. Ran et al. (2023) were concerned with the quantized consensus problem for uncertain nonlinear multiagent systems under data-rate constraints and DoS attacks, and proposed a novel dynamic quantization with zooming-in and holding capabilities. The main differences between this paper and Feng et al. (2022) include the following points. (1) Feng et al. (2022) considered a system without the disturbance or noise, but we consider a system with the disturbance or noise. Clearly, our results are more meaningful. (2) (Feng et al., 2022) gave conditions for exponentially stability. However, we give ones for mean square observability and stabilization. (3) We gave a lower bound of data rates for mean square observability and stabilization, which is smaller than one given by Feng et al. (2022). Thus, our result is less conservative than the results given by Feng et al., (2021, 2022). (4) Our proof of technical methods is different from the proof given by Feng et al. (2022).

The remainder of this paper is organized as follows: Section 2 introduces problem formulation. Section 3 proposes a state-estimate control scheme under DoS attacks, and presents sufficient conditions for observability and stabilization. The result of a numerical example is presented in Section 4. Conclusions are stated in Section 5.

2. Problem formulation

2.1. Process model

In this paper, we consider the CPS in Figure 1, where the process is a linear time-invariant control system as given below: (1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) Here, $X(t)\in {\mathbb{R}}^n$ denotes the plant state, $U(t)\in {\mathbb{R}}^p$ denotes the control input, and $W(t)\in {\mathbb{R}}^q$ denotes the disturbance or noise. A, B and F are known real matrices.

Figure 1. Networked control systems.

Here, we considered a networked control system with a state-estimate feedback controller, which is different from other existing frameworks. Notice that, the sensor is able to connect to the controller via a stationary memoryless digital communication channel. Thus, the plant state $X(t)$ needs to be quantized, encoded and transmitted to the estimator. Then, the estimator will compute the estimation on the basis of data packets sent by the encoder. However, it is possible that some data packets are intercepted by attackers in communication networks. DoS attacks may cause failure of some transmission attempts. In remote state estimation, packet losses have important effects on observability and stabilization of the system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ).

Here, we impose the following assumptions (Feng et al., 2022) on system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ).

Proof

• Assumption 2.1

The initial state $X(0)$ may be considered as a zero mean, Gaussian random variable, satisfying (2) $\begin{array}{r}E[\Vert X(0){\Vert }^2]\le {\phi }_0<\mathrm{\infty }\end{array}$(2) and the disturbance $W(t)$ may be considered as a stationary stochastic process, satisfying (3) $\begin{array}{r}E[\Vert W(t){\Vert }^2]\le {\phi }_w<\mathrm{\infty }\end{array}$(3)

Proof

• Assumption 2.2

Our model of the communication channel neglects the effect of networked-induced delay.

Proof

• Assumption 2.3

System (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is a TCP-like system, which employs the packet acknowledgment (ACK) scheme.

Proof

• Assumption 2.4

The network communication between the sensor and controller takes place over a data-rate limited communication channel subject to DoS attacks.

2.2. Attacker model

Defending against DoS attacks has become important in CPSs. DoS attacks block the communication channel to prevent information exchange between the sensor and controller. Namely, transmission attempts will fail in DoS status. Let T denote the sampling interval. We define the set of the state sampling instants by $\begin{array}{r}{\varphi }_T=\{t_0,t_1,t_2,\dots ,t_k,\dots \},k\in {\mathbb{Z}}^{\mathbb{+}}\end{array}$where we set $t_k=\mathrm{kT}$ and $t_0=0$.

As stated in Feng et al. (2021), the sequence of DoS off/on transitions is defined by $\{h_n{\}}_{n\in {\mathbb{Z}}^{\mathbb{+}}}$ with $h_0>0$. Let $(\tau ,t)$ denote the finite time interval where t and τ is the end or beginning, respectively. Then, given $\tau ,t\in \mathbb{R}$ with $t>\tau \ge 0$, we define $f(\tau ,t)$ as the number of DoS off/on transitions in time $[\tau ,t]$, and define $s(\tau ,t)$ as the time set where the communication channel lies in the DoS status. Here, we impose the following assumptions on DoS attacks.

Proof

• Assumption 2.5 (Feng et al., 2021)

There exist constant $\eta \in {\mathbb{R}}^{\mathbb{+}}$ and constant $\omega \in {\mathbb{R}}^{\mathbb{+}}$ such that $\begin{array}{r}f(\tau ,t)\le \eta +\frac{t-\tau }{\omega }\end{array}$holds for any $\tau ,t\in \mathbb{R}$ with $t>\tau \ge 0$.

Proof

• Assumption 2.5 (Feng et al., 2021)

There exist constant $\theta \in {\mathbb{R}}^{\mathbb{+}}$ and constant $\varpi \in {\mathbb{R}}^{\mathbb{+}}$ such that $\begin{array}{r}s(\tau ,t)\le \theta +\frac{t-\tau }{\varpi }\end{array}$holds for any $\tau ,t\in \mathbb{R}$ with $t>\tau \ge 0$.

We define the subset ${\varphi }_z=\{z_0,z_1,z_2,\dots \}\in {\varphi }_T$ as the sequence of the time instants where transmissions are successful. Then, we give Lemma 2.1 following from Feng et al. (2021).

Lemma 2.1

Feng et al., 2021

Consider the system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) over a communication channel subject to DoS attacks characterized by Assumptions 2.1–2.6. Let $N(z_0,t)$ denote the number of successful transmissions in the time interval $[z_0,t]$. If $0<\frac{1}{\varpi }+\frac{T}{\omega }<1$ hold, it follows that $\begin{array}{r}N(z_0,t)\ge (1-\frac{1}{\varpi }-\frac{T}{\omega })\frac{t-z_0}{T}-\frac{\theta +\mathrm{\eta T}}{T}\end{array}$

2.3. Problem formulation and contribution

Let $\hat{X}(k)$ and $V(k)$ denote the state estimate and estimation error, respectively. Namely, we have (4) $\begin{array}{r}V(t)=X(t)-\hat{X}(t)\end{array}$(4) Here, we implement a state-estimate feedback control law of the form (5) $\begin{array}{r}U(t)=K\hat{X}(t)\end{array}$(5) where we select K such that A + BK is a Hurwitz matrix.

The value of the plant state can be quantized and encoded into a finite number of bits. We define $r(t_k)$ as the number of bits transmitted at the kth sampling interval. The data rate $R(t)$ at the kth sampling interval is defined as $\begin{array}{r}R(t)=\frac{r(t_k)}{T}(\mathrm{bits}/\mathrm{s})\end{array}$

Definition 2.1

System (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) with the initial state $X(0)$ and disturbance $W(t)$ satisfying conditions (2(2) $\begin{array}{r}E[\Vert X(0){\Vert }^2]\le {\phi }_0<\mathrm{\infty }\end{array}$(2) ) and (3(3) $\begin{array}{r}E[\Vert W(t){\Vert }^2]\le {\phi }_w<\mathrm{\infty }\end{array}$(3) ) is said to be mean square observable if there exists a quantization and coding scheme such that $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim sup}E\Vert V(t)\Vert \le {\phi }_v<\mathrm{\infty }\end{array}$holds, where ${\phi }_v$ is a know constant.

Definition 2.2

System (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) with the initial state $X(0)$ and disturbance $W(t)$ satisfying conditions (2(2) $\begin{array}{r}E[\Vert X(0){\Vert }^2]\le {\phi }_0<\mathrm{\infty }\end{array}$(2) ) and (3(3) $\begin{array}{r}E[\Vert W(t){\Vert }^2]\le {\phi }_w<\mathrm{\infty }\end{array}$(3) ) is said to be mean square stabilizable if there exists a state-estimate feedback control law of the form (5(5) $\begin{array}{r}U(t)=K\hat{X}(t)\end{array}$(5) ) such that $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim sup}E\Vert X(t)\Vert \le {\phi }_x<\mathrm{\infty }\end{array}$holds, where ${\phi }_x$ is a know constant.

In this paper, we will derive a lower bound of the data rate for mean square observability and stabilization of the system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ), and give a state-estimate feedback control scheme for system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) in the presence of DoS attacks.

The main contributions of this paper are as follows.

• We develop a framework for CPSs in the presence of DoS attacks, and present a state-estimate control scheme for CPSs under DoS attacks.

• We discuss the effect of the disturbance, and derive sufficient conditions for mean square observability and stabilization of CPSs.

• We present a lower bound on the data rate for observability and stabilization of CPSs, which is less conservative than some existing results (Feng et al., 2021, 2022).

3. State-estimate control under doS attacks

In order to achieve a tight result on the data rate for observability and stabilization of system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) in the presence of DoS attacks, we first carry out the transformation on $X(t)$. Let $H\in {\mathbb{R}}^{n\times n}$ denote a real orthogonal matrix that can diagonalize $\begin{array}{r}A=H^{-1}\mathrm{\Lambda H}\end{array}$where we define the ith eigenvalue by ${\lambda }_i$ and further define $\begin{array}{r}\mathrm{\Lambda }=\mathrm{diag}[{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n]\end{array}$Here, we only consider this case where matrix A has real eigenvalues each with geometric multiplicity one. For other cases, there exist the same result (Feng et al., 2022). Next, we define (6) $\begin{array}{rl}& \overline{X}(t)=\mathrm{HX}(t)=[{\overline{x}}_1(t){\overline{x}}_2(t)\dots {\overline{x}}_n(t){]}^T\\ & \tilde{X}(t)=H\hat{X}(t)=[{\tilde{x}}_1(t){\tilde{x}}_2(t)\dots {\tilde{x}}_n(t){]}^T\\ & \overline{V}(t)=\mathrm{HV}(t)=[{\overline{v}}_1(t){\overline{v}}_2(t)\dots {\overline{v}}_n(t){]}^T\end{array}$(6) Clearly, we have (7) $\begin{array}{r}{\overline{v}}_i(t)={\overline{x}}_i(t)-{\tilde{x}}_i(t),i=1,2,\dots ,n.\end{array}$(7) Thus, system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) can be rewritten as $\begin{array}{r}\dot{\overline{X}}(t)=\mathrm{\Lambda }\overline{X}(t)+\mathrm{HBU}(t)+\mathrm{HFW}(t)\end{array}$For brevity, we define $\begin{array}{rl}& \mathrm{HBU}(t)=[{\overline{u}}_1(t){\overline{u}}_2(t)\dots {\overline{u}}_n(t){]}^T\\ & \mathrm{HFW}(t)=[{\overline{w}}_1(t){\overline{w}}_2(t)\dots {\overline{w}}_n(t){]}^T\end{array}$In particular, system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is also rewritten as (8) $\begin{array}{r}{\dot{\overline{x}}}_i(t)={\lambda }_i{\overline{x}}_i(t)+{\overline{u}}_i(t)+{\overline{w}}_i(t),i=1,2,\dots ,n\end{array}$(8) We employ a predictive coding method for plant states. Namely, at time $t_k^{-}$, ${\overline{v}}_i(t_k^{-})$, not ${\overline{x}}_i(t_k^{-})$, is quantized and encoded,$i=1,2,\dots ,n$. By doing so, our result will be less conservative than the existing results (Feng et al., 2021, 2022). Let $q_i(t_k^{-})$ and $e_i(t_k^{-})$ denote the quantization value and quantization error at time $t_k^{-}$, respectively. Then, $q_i(t_k^{-})$ can be encoded into a finite number of bits. We define $r_i(t_k)$ as the number of bits transmitted at the kth sampling interval. Then, a data packet of $r_i(t_k)$ bits will be sent to the decoder. Namely, we can obtain the data rate $R(t)$ given by $\begin{array}{r}R(t)=\frac{1}{T}\sum _{i=1}^n{r}_i(t_k)\mathrm{bits}/\mathrm{s}\end{array}$Then, we give Lemma 3.1 following from Cover and Thomas (2006).

Lemma 3.1

Cover & Thomas, 2006

Consider the system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) over a communication channel. Let $q_i(t_k^{-})$ denote the quantization value of ${\overline{v}}_i(t_k^{-})$. For a given $D_i(t_k^{-})$, find $q_i(t_k^{-})$ such that (9) $\begin{array}{r}E[({\overline{v}}_i(t_k^{-})-q_i(t_k^{-}){)}^2]\le D_i(t_k^{-})\mathrm{for}\mathrm{all}{t}_k^{-}\end{array}$(9) If set $r_i(t_k)\ge \frac{1}{2}{\log }_2\frac{E[{\overline{v}}_i^2(t_k^{-})]}{D_i(t_k^{-})}$ (bits), there exists a quantization and coding scheme such that condition (9(9) $\begin{array}{r}E[({\overline{v}}_i(t_k^{-})-q_i(t_k^{-}){)}^2]\le D_i(t_k^{-})\mathrm{for}\mathrm{all}{t}_k^{-}\end{array}$(9) ) holds. Thus, the data rate $R(t)$ is given by $\begin{array}{r}R(t)\ge \frac{1}{2T}\sum _{i=1}^{n}max\{{\log }_2\frac{E[{\overline{v}}_i^2(t_k^{-})]}{D_i(t_k^{-})},0\}(\mathrm{bits}/\mathrm{s})\end{array}$

The information of the plant state needs to be transmitted over the communication channel under DoS attacks. Here, we divide time into three time zones. The first one is the DoS attack time zone defined as $T_a$, where the communication channel is in the DoS attack status. The second one is the defense time zone defined as $T_d$, where the communication channel is in the defense status. The last one is the peace time zone defined as $T_p$, which is the time zone behind the defense time zone and in front of the next DoS attack time zone. Thus, we divide the set ${\varphi }_T$ of the state sampling instants into three types of sets: DoS attack instant set ${\varphi }_a$, defense instant set ${\varphi }_d$ and peace instant set ${\varphi }_p$. We define $\begin{array}{rl}& {\varphi }_a=\{t_k:t_k=\mathrm{kT},t_k\in T_a,k\in {\mathbb{Z}}^{+}\}\\ & {\varphi }_d=\{t_k:t_k=\mathrm{kT},t_k\in T_d,k\in {\mathbb{Z}}^{+}\}\\ & {\varphi }_p=\{t_k:t_k=\mathrm{kT},t_k\in T_p,k\in {\mathbb{Z}}^{+}\}\end{array}$Here, we employ a variable data rate $R(t)$ in order to mitigate the influence of DoS attacks. Especially, the data rate $R(t)$ changes with the communication status, and is determined by (10) $\begin{array}{r}R(t)=\{\begin{array}{ll}0(\mathrm{bits}/\mathrm{s})& \mathrm{when}t\in {\varphi }_a\\ {\displaystyle \sum _{i=1}^n\lceil max\{{\lambda }_i{\log }_{2}e,0\}\rceil (\mathrm{bits}/\mathrm{s})}& \mathrm{when}t\in {\varphi }_p\\ \begin{array}{l}{\displaystyle \sum _{i=1}^n\lceil max\{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}{\lambda }_i}\\ {\log }_{2}e,0\phantom{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}}\}\rceil (\mathrm{bits}/\mathrm{s})\end{array}& \mathrm{when}t\in {\varphi }_d\end{array}\end{array}$(10) where e denotes the natural constant.

Notice that, the communication between the sensor and controller is subject to data rate limitations. We want to give a tight result on the data rate for observability and stabilization of system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) in the presence of DoS attacks. First, we give the main result on observability of system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ).

Theorem 3.1

Consider system (3.1) in the presence of DoS attacks in Assumptions 2.1–2.6. Let ${\lambda }_1,\dots ,{\lambda }_n$ denote n positive real eigenvalues of matrix A. Assume that $0<\frac{1}{\varpi }+\frac{T}{\omega }<1$ hold. If the information of the plant state is transmitted by employing the data rate given by (10(10) $\begin{array}{r}R(t)=\{\begin{array}{ll}0(\mathrm{bits}/\mathrm{s})& \mathrm{when}t\in {\varphi }_a\\ {\displaystyle \sum _{i=1}^n\lceil max\{{\lambda }_i{\log }_{2}e,0\}\rceil (\mathrm{bits}/\mathrm{s})}& \mathrm{when}t\in {\varphi }_p\\ \begin{array}{l}{\displaystyle \sum _{i=1}^n\lceil max\{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}{\lambda }_i}\\ {\log }_{2}e,0\phantom{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}}\}\rceil (\mathrm{bits}/\mathrm{s})\end{array}& \mathrm{when}t\in {\varphi }_d\end{array}\end{array}$(10) ), the system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is mean square observable.

Proof.

First, we consider the case where the communication channel is in DoS attack status.

In TCP/IP network communication, the encoder may receive a data packet acknowledgment (ACK) from the decoder if one data packet is successfully transmitted to the decoder. However, for the DoS attack status, no data packet can be received at the decoder, and no ACK can be received at the encoder too. Thus, both the encoder and the decoder know that the communication channel is in the DoS attack status. For this case, there is no need to transmit any data packet. Clearly, the data rate $R(t)$ is equal to zero in DoS attack status. Thus, we have $\begin{array}{r}R(t)=0(\mathrm{bit}/\mathrm{s})\end{array}$when the system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is in the DoS attack status.

In the DoS attack status, both the encoder and the decoder compute the state estimate using the same method. First, we compute ${\tilde{x}}_i(t)$ on the basis of (11) $\begin{array}{r}{\dot{\tilde{x}}}_i(t)={\lambda }_i{\tilde{x}}_i(t)+{\overline{u}}_i(t),t>z_l\end{array}$(11) where we define $\begin{array}{r}{z}_l:=\{t_k:t_k=\mathrm{kT},t_k\in {\varphi }_d\cup {\varphi }_p,t_{k+1}\in {\varphi }_a,k\in {\mathbb{Z}}^{+}\}\end{array}$According to Equations (7(7) $\begin{array}{r}{\overline{v}}_i(t)={\overline{x}}_i(t)-{\tilde{x}}_i(t),i=1,2,\dots ,n.\end{array}$(7) ), (8(8) $\begin{array}{r}{\dot{\overline{x}}}_i(t)={\lambda }_i{\overline{x}}_i(t)+{\overline{u}}_i(t)+{\overline{w}}_i(t),i=1,2,\dots ,n\end{array}$(8) ), and (11(11) $\begin{array}{r}{\dot{\tilde{x}}}_i(t)={\lambda }_i{\tilde{x}}_i(t)+{\overline{u}}_i(t),t>z_l\end{array}$(11) ), we can obtain $\begin{array}{r}{\dot{\overline{v}}}_i(t)={\lambda }_i{\overline{v}}_i(t)+{\overline{w}}_i(t),i=1,2,\dots ,n,t>z_l\end{array}$By solving the differential equation above, we may give (12) $\begin{array}{rl}{\overline{v}}_i(t)& =e^{{\lambda }_i(t-z_l)}{\overline{v}}_i(z_l)+{\int }_{z_l}^t{e}^{{\lambda }_i(\tau -z_l)}{\overline{w}}_i(\tau )\mathrm{d}\tau ,\\ & i=1,2,\dots ,n,t>z_l\end{array}$(12) If ${\lambda }_i\le 0$, ${\overline{v}}_i(t)$ will not increase. For this case, it is unnecessary for the encoder to encode ${\overline{v}}_i(t)$. Namely, we do not transmit the information on ${\overline{v}}_i(t)$, and set $r_i(t)=0$. On the contrary, ${\lambda }_i>0$, ${\overline{v}}_i(t)$ will be enlarged more than $e^{{\lambda }_i(\tau -z_l)}$ times.

It follows from Equation (12(12) $\begin{array}{rl}{\overline{v}}_i(t)& =e^{{\lambda }_i(t-z_l)}{\overline{v}}_i(z_l)+{\int }_{z_l}^t{e}^{{\lambda }_i(\tau -z_l)}{\overline{w}}_i(\tau )\mathrm{d}\tau ,\\ & i=1,2,\dots ,n,t>z_l\end{array}$(12) ) that (13) $\begin{array}{rl}{\overline{v}}_i(t_k)& =e^{{\lambda }_{i}T}{\overline{v}}_i(t_{k-1})+{\int }_{t_{k-1}}^{t_k}{e}^{{\lambda }_i(\tau -t_{k-1})}{\overline{w}}_i(\tau )\mathrm{d}\tau ,\\ & i=1,2,\dots ,n,t_k\in {\varphi }_a\end{array}$(13) Furthermore, notice that the disturbance $W(t)$ may be considered as a stationary stochastic process, satisfying (3(3) $\begin{array}{r}E[\Vert W(t){\Vert }^2]\le {\phi }_w<\mathrm{\infty }\end{array}$(3) ). Then, it follows from Assumptions 1 that we can define (14) $\begin{array}{rl}{\phi }_{{\overline{w}}_i}& :={\int }_{t_{k-1}}^{t_k}{e}^{{\lambda }_i(\tau -t_{k-1})}{\overline{w}}_i(\tau )\mathrm{d}\tau ={\int }_0^T{e}^{{\lambda }_i\tau }{\overline{w}}_i(\tau )\mathrm{d}\tau <\mathrm{\infty },\\ & i=1,2,\dots ,n\end{array}$(14) Substituting Equation (14(14) $\begin{array}{rl}{\phi }_{{\overline{w}}_i}& :={\int }_{t_{k-1}}^{t_k}{e}^{{\lambda }_i(\tau -t_{k-1})}{\overline{w}}_i(\tau )\mathrm{d}\tau ={\int }_0^T{e}^{{\lambda }_i\tau }{\overline{w}}_i(\tau )\mathrm{d}\tau <\mathrm{\infty },\\ & i=1,2,\dots ,n\end{array}$(14) ) into Equation (13(13) $\begin{array}{rl}{\overline{v}}_i(t_k)& =e^{{\lambda }_{i}T}{\overline{v}}_i(t_{k-1})+{\int }_{t_{k-1}}^{t_k}{e}^{{\lambda }_i(\tau -t_{k-1})}{\overline{w}}_i(\tau )\mathrm{d}\tau ,\\ & i=1,2,\dots ,n,t_k\in {\varphi }_a\end{array}$(13) ), we obtain (15) $\begin{array}{r}{\overline{v}}_i(t_k)=e^{{\lambda }_{i}T}{\overline{v}}_i(t_{k-1})+{\phi }_{{\overline{w}}_i},i=1,2,\dots ,n,t_k\in {\varphi }_a\end{array}$(15) According to the recurrence formula (15(15) $\begin{array}{r}{\overline{v}}_i(t_k)=e^{{\lambda }_{i}T}{\overline{v}}_i(t_{k-1})+{\phi }_{{\overline{w}}_i},i=1,2,\dots ,n,t_k\in {\varphi }_a\end{array}$(15) ) above, we can further obtain $\begin{array}{rl}{\overline{v}}_i(t_k)& =e^{{\lambda }_i(t_k-z_l)}{\overline{v}}_i(z_l)+\frac{1-e^{{\lambda }_i(t_k-z_l)}}{1-e^{{\lambda }_{i}T}}{\phi }_{{\overline{w}}_i},t_k>z_l,\\ & i=1,2,\dots ,n,t_k\in {\varphi }_a\end{array}$Thus, it follows that (16) $\begin{array}{rl}E[{\overline{v}}_i^2(t_k)]& =e^{2{\lambda }_{i}T{n}_a}E[{\overline{v}}_i^2(z_l)]+{\left(\frac{1-e^{{\lambda }_{i}T{n}_a}}{1-e^{{\lambda }_{i}T}}\right)}^2{\phi }_{{\overline{w}}_i}^2,\\ & t_k>z_l,i=1,2,\dots ,n,t_k\in {\varphi }_a\end{array}$(16) where we define $n_a:=\frac{t_k-z_l}{T}$. Clearly, the uncertainty of the state estimation error will increase in the DoS attack status because the information on the plant state cannot be sent to the estimator due to DoS attacks.

In defense status, both the encoder and the decoder compute the state estimate using the same method. First, we compute ${\tilde{x}}_i(t)$ on the basis of (17) $\begin{array}{rl}{\dot{\tilde{x}}}_i(t)& ={\lambda }_i{\tilde{x}}_i(t)+{\overline{u}}_i(t),\mathrm{when}t>z_d\\ {\tilde{x}}_i(t)& ={\tilde{x}}_i(t^{-})+q_i(t^{-}),\mathrm{when}t=z_d\end{array}$(17) where we define $\begin{array}{r}{z}_d:=\{t_k:t_k=\mathrm{kT},t_k\in {\varphi }_d,t_{k-1}\in {\varphi }_a,k\in {\mathbb{Z}}^{+}\}\end{array}$According to Equations (7(7) $\begin{array}{r}{\overline{v}}_i(t)={\overline{x}}_i(t)-{\tilde{x}}_i(t),i=1,2,\dots ,n.\end{array}$(7) ), (8(8) $\begin{array}{r}{\dot{\overline{x}}}_i(t)={\lambda }_i{\overline{x}}_i(t)+{\overline{u}}_i(t)+{\overline{w}}_i(t),i=1,2,\dots ,n\end{array}$(8) ), and (17(17) $\begin{array}{rl}{\dot{\tilde{x}}}_i(t)& ={\lambda }_i{\tilde{x}}_i(t)+{\overline{u}}_i(t),\mathrm{when}t>z_d\\ {\tilde{x}}_i(t)& ={\tilde{x}}_i(t^{-})+q_i(t^{-}),\mathrm{when}t=z_d\end{array}$(17) ), we can obtain $\begin{array}{rl}{\dot{\overline{v}}}_i(t)={\lambda }_i{\overline{v}}_i(t)+{\overline{w}}_i(t),& \mathrm{when}{z}_d<t<z_d+1\\ {\overline{v}}_i(t)=e_i(t^{-}),& \mathrm{when}t=z_d\end{array}$By solving the differential equation above, we may give (18) $\begin{array}{rl}{\overline{v}}_i(t)& =e^{{\lambda }_i(t-z_d)}{\overline{v}}_i(z_d)+{\int }_{z_d}^t{e}^{{\lambda }_i(\tau -z_d)}{\overline{w}}_i(\tau )\mathrm{d}\tau ,\\ & i=1,2,\dots ,n,z_d<t<z_d+1\end{array}$(18) with ${\overline{v}}_i(z_d)=e_i(z_d^{-})$.

Then, it follows from Equation (18(18) $\begin{array}{rl}{\overline{v}}_i(t)& =e^{{\lambda }_i(t-z_d)}{\overline{v}}_i(z_d)+{\int }_{z_d}^t{e}^{{\lambda }_i(\tau -z_d)}{\overline{w}}_i(\tau )\mathrm{d}\tau ,\\ & i=1,2,\dots ,n,z_d<t<z_d+1\end{array}$(18) ) that (19) $\begin{array}{rl}{\overline{v}}_i(t_{k+1}^{-})& =e^{{\lambda }_{i}T}{\overline{v}}_i(t_k)+{\int }_{t_k}^{t_{k+1}}{e}^{{\lambda }_i(\tau -t_k)}{\overline{w}}_i(\tau )\mathrm{d}\tau ,\\ & i=1,2,\dots ,n,t_k\in {\varphi }_d\end{array}$(19) where we have $\begin{array}{r}{\overline{v}}_i(t_k)=e_i(t_k^{-})\end{array}$Clearly, it means that ${\overline{v}}_i(t_{k+1})=e_i(t_{k+1}^{-})$ holds too. Substituting Equation (14(14) $\begin{array}{rl}{\phi }_{{\overline{w}}_i}& :={\int }_{t_{k-1}}^{t_k}{e}^{{\lambda }_i(\tau -t_{k-1})}{\overline{w}}_i(\tau )\mathrm{d}\tau ={\int }_0^T{e}^{{\lambda }_i\tau }{\overline{w}}_i(\tau )\mathrm{d}\tau <\mathrm{\infty },\\ & i=1,2,\dots ,n\end{array}$(14) ) into Equation (19(19) $\begin{array}{rl}{\overline{v}}_i(t_{k+1}^{-})& =e^{{\lambda }_{i}T}{\overline{v}}_i(t_k)+{\int }_{t_k}^{t_{k+1}}{e}^{{\lambda }_i(\tau -t_k)}{\overline{w}}_i(\tau )\mathrm{d}\tau ,\\ & i=1,2,\dots ,n,t_k\in {\varphi }_d\end{array}$(19) ), we obtain $\begin{array}{r}{\overline{v}}_i(t_{k+1}^{-})=e^{{\lambda }_{i}T}{\overline{v}}_i(t_k)+{\phi }_{{\overline{w}}_i},i=1,2,\dots ,n,t_k\in {\varphi }_d\end{array}$with ${\overline{v}}_i(t_{k+1})=e_i(t_{k+1}^{-})$. Then, it follows that (20) $\begin{array}{rl}E[{\overline{v}}_i^2(t_{k+1}^{-})]& =e^{2{\lambda }_{i}T}E[{\overline{v}}_i^2(t_k)]+{\phi }_{{\overline{w}}_i}^2,\\ & i=1,2,\dots ,n,t_k\in {\varphi }_d\end{array}$(20) where we have (21) $\begin{array}{r}E[{\overline{v}}_i^2(t_{k+1})]=E[e_i^2(t_{k+1}^{-})]\end{array}$(21) Furthermore, notice that $\begin{array}{r}{e}_i(t_{k+1}^{-})={\overline{v}}_i(t_{k+1})-q_i(t_{k+1})\end{array}$Then, it follows from Lemma 3.1 that we have (22) $\begin{array}{r}E[e_i^2(t_{k+1}^{-})]=E[({\overline{v}}_i(t_{k+1}^{-})-q_i(t_{k+1}^{-}){)}^2]\le D_i(t_{k+1}^{-})\end{array}$(22) and (23) $\begin{array}{r}{r}_i(t_{k+1})\ge \frac{1}{2}{\log }_2\frac{E[{\overline{v}}_i^2(t_{k+1}^{-})]}{D_i(t_{k+1}^{-})}(\mathrm{bits})\end{array}$(23) According to (22(22) $\begin{array}{r}E[e_i^2(t_{k+1}^{-})]=E[({\overline{v}}_i(t_{k+1}^{-})-q_i(t_{k+1}^{-}){)}^2]\le D_i(t_{k+1}^{-})\end{array}$(22) ) and (23(23) $\begin{array}{r}{r}_i(t_{k+1})\ge \frac{1}{2}{\log }_2\frac{E[{\overline{v}}_i^2(t_{k+1}^{-})]}{D_i(t_{k+1}^{-})}(\mathrm{bits})\end{array}$(23) ), we quantize, encode ${\overline{v}}_i(t_{k+1}^{-})$ and have (24) $\begin{array}{r}E[e_i^2(t_{k+1}^{-})]=\frac{E[{\overline{v}}_i^2(t_{k+1}^{-})]}{2^{2r_i(t_{k+1})}}\end{array}$(24) Substituting Equations (20(20) $\begin{array}{rl}E[{\overline{v}}_i^2(t_{k+1}^{-})]& =e^{2{\lambda }_{i}T}E[{\overline{v}}_i^2(t_k)]+{\phi }_{{\overline{w}}_i}^2,\\ & i=1,2,\dots ,n,t_k\in {\varphi }_d\end{array}$(20) ) and (21(21) $\begin{array}{r}E[{\overline{v}}_i^2(t_{k+1})]=E[e_i^2(t_{k+1}^{-})]\end{array}$(21) ) into Equation (24(24) $\begin{array}{r}E[e_i^2(t_{k+1}^{-})]=\frac{E[{\overline{v}}_i^2(t_{k+1}^{-})]}{2^{2r_i(t_{k+1})}}\end{array}$(24) ), we obtain $\begin{array}{rl}E[{\overline{v}}_i^2(t_{k+1})]& =\frac{e^{2{\lambda }_{i}T}}{2^{2r_i(t_{k+1})}}E[{\overline{v}}_i^2(t_k)]+\frac{{\phi }_{{\overline{w}}_i}^2}{2^{2r_i(t_{k+1})}},\\ & i=1,2,\dots ,n,t_{k+1}\in {\varphi }_d\end{array}$According to the recurrence formula above, we can further obtain (25) $\begin{array}{rl}E[{\overline{v}}_i^2(t_k)]& =\frac{e^{2{\lambda }_{i}T{n}_d}}{2^{2n_d{r}_i(t_k)}}E[{\overline{v}}_i^2(z_d)]+\frac{1-\frac{e^{2{\lambda }_{i}T{n}_d}}{2^{2n_d{r}_i(t_k)}}}{1-\frac{e^{2{\lambda }_{i}T}}{2^{2r_i(t_{k+1})}}}\frac{{\phi }_{{\overline{w}}_i}^2}{2^{2r_i(t_{k+1})}},\\ & t_k>z_d,i=1,2,\dots ,n,t_k\in {\varphi }_d\end{array}$(25) where we define $n_d:=\frac{t_k-z_d}{T}$.

Notice that, the uncertainty of the state estimation error will increase in the DoS attack status because the information on the plant state cannot be sent to the estimator due to DoS attacks. In order to decrease the state estimation error, we increase the data rate $R(t)$ and send more information on the plant state to the estimator. Let $z_m$ denote the last state sampling instant in defense status. Here, we define $\begin{array}{r}{z}_m:=\{t_k:t_k=\mathrm{kT},t_k\in {\varphi }_d,t_{k+1}\in {\varphi }_p,k\in {\mathbb{Z}}^{+}\}\end{array}$According to Equations (16(16) $\begin{array}{rl}E[{\overline{v}}_i^2(t_k)]& =e^{2{\lambda }_{i}T{n}_a}E[{\overline{v}}_i^2(z_l)]+{\left(\frac{1-e^{{\lambda }_{i}T{n}_a}}{1-e^{{\lambda }_{i}T}}\right)}^2{\phi }_{{\overline{w}}_i}^2,\\ & t_k>z_l,i=1,2,\dots ,n,t_k\in {\varphi }_a\end{array}$(16) ) and (25(25) $\begin{array}{rl}E[{\overline{v}}_i^2(t_k)]& =\frac{e^{2{\lambda }_{i}T{n}_d}}{2^{2n_d{r}_i(t_k)}}E[{\overline{v}}_i^2(z_d)]+\frac{1-\frac{e^{2{\lambda }_{i}T{n}_d}}{2^{2n_d{r}_i(t_k)}}}{1-\frac{e^{2{\lambda }_{i}T}}{2^{2r_i(t_{k+1})}}}\frac{{\phi }_{{\overline{w}}_i}^2}{2^{2r_i(t_{k+1})}},\\ & t_k>z_d,i=1,2,\dots ,n,t_k\in {\varphi }_d\end{array}$(25) ), we can obtain $\begin{array}{rl}E[{\overline{v}}_i^2(z_m)]& =\frac{e^{2{\lambda }_{i}T(n_a+n_d)}}{2^{2n_d{r}_i(t)}}E[{\overline{v}}_i^2(z_l)]\\ & +[{\left(\frac{1-e^{{\lambda }_{i}T{n}_a}}{1-e^{{\lambda }_{i}T}}\right)}^2\frac{e^{2{\lambda }_{i}T{n}_d}}{2^{2n_d{r}_i(t)}}\\ & ++\frac{1-\frac{e^{2{\lambda }_{i}T{n}_d}}{2^{2n_d{r}_i(t)}}}{1-\frac{e^{2{\lambda }_{i}T}}{2^{2r_i(t)}}}\frac{1}{2^{2r_i(t)}}]{\phi }_{{\overline{w}}_i}^2\end{array}$There exists a ${\phi }_m\in {\mathbb{R}}^{+}$ such that $\begin{array}{r}[{\left(\frac{1-e^{{\lambda }_{i}T{n}_a}}{1-e^{{\lambda }_{i}T}}\right)}^2\frac{e^{2{\lambda }_{i}T{n}_d}}{2^{2n_d{r}_i(t)}}+\frac{1-\frac{e^{2{\lambda }_{i}T{n}_d}}{2^{2n_d{r}_i(t)}}}{1-\frac{e^{2{\lambda }_{i}T}}{2^{2r_i(t)}}}\frac{1}{2^{2r_i(t)}}]{\phi }_{{\overline{w}}_i}^2<{\phi }_m\end{array}$with $t\to \mathrm{\infty }$. If we choose $r_i(t)$ such that (26) $\begin{array}{r}\frac{e^{2{\lambda }_{i}T(n_a+n_d)}}{2^{2n_d{r}_i(t)}}<1\end{array}$(26) holds, then the mean square value of the state estimation error is bounded and given by $\begin{array}{r}E[{\overline{v}}_i^2(t)]<{\phi }_m\end{array}$with $t\to \mathrm{\infty }$. Thus, there exists a state-estimate feedback control law of the form (5(5) $\begin{array}{r}U(t)=K\hat{X}(t)\end{array}$(5) ) such that $\begin{array}{r}E\Vert V(t)\Vert \le {\phi }_v<\mathrm{\infty }\end{array}$holds for any $t\in T_d$.

Next, we set $t=z_m$, and have (27) $\begin{array}{r}T(n_a+n_d)=t-z_l\end{array}$(27) Clearly, $n_d$ denotes the number of successful transmissions in the time interval $[z_l,t]$. If assume that $0<\frac{1}{\varpi }+\frac{T}{\omega }<1$ holds, according to Lemma 1, we may give (28) $\begin{array}{r}{n}_d=N(z_l,t)\ge (1-\frac{1}{\varpi }-\frac{T}{\omega })\frac{t-z_l}{T}-\frac{\theta +\mathrm{\eta T}}{T}\end{array}$(28) in the time interval $[z_l,t]$. Substituting (27(27) $\begin{array}{r}T(n_a+n_d)=t-z_l\end{array}$(27) ) and (28(28) $\begin{array}{r}{n}_d=N(z_l,t)\ge (1-\frac{1}{\varpi }-\frac{T}{\omega })\frac{t-z_l}{T}-\frac{\theta +\mathrm{\eta T}}{T}\end{array}$(28) ) into (26(26) $\begin{array}{r}\frac{e^{2{\lambda }_{i}T(n_a+n_d)}}{2^{2n_d{r}_i(t)}}<1\end{array}$(26) ), we have (29) $\begin{array}{r}{r}_i(t)>{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}{\lambda }_{i}T{\log }_{2}e(\mathrm{bits})\end{array}$(29) with $t\to \mathrm{\infty }$. Namely, we set the number of bits given by (29(29) $\begin{array}{r}{r}_i(t)>{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}{\lambda }_{i}T{\log }_{2}e(\mathrm{bits})\end{array}$(29) ) for encoding when the ith eigenvalue of the system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is greater than or equal to zero. On the contrary, we set $r_i(t)=0$ when the ith eigenvalue of the system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is less than zero. Furthermore, notice that the number of bits $r_i(t)$ must be an integer. Thus, we obtain $\begin{array}{r}{r}_i(t)>\lceil max\{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}{\lambda }_{i}T{\log }_{2}e,0\}\rceil (\mathrm{bits})\end{array}$The data rate $R(t)$ in the defense status is given by $\begin{array}{rl}& R(t)>\sum _{i=1}^n\lceil max\{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}{\lambda }_i{\log }_{2}e,0\}\rceil \\ & \times (\mathrm{bits}/\mathrm{s})\end{array}$In the peace status, both the encoder and the decoder compute the state estimate using the same method. First, we compute ${\tilde{x}}_i(t)$ on the basis of (30) $\begin{array}{rl}{\dot{\tilde{x}}}_i(t)={\lambda }_i{\tilde{x}}_i(t)+{\overline{u}}_i(t),& \mathrm{when}t>z_p\\ {\tilde{x}}_i(t)={\tilde{x}}_i(t^{-})+q_i(t^{-}),& \mathrm{when}t=z_p\end{array}$(30) where we define $\begin{array}{r}{z}_p:=\{t_k:t_k=\mathrm{kT},t_k\in {\varphi }_p,t_{k-1}\in {\varphi }_d,k\in {\mathbb{Z}}^{+}\}\end{array}$According to Equations (7(7) $\begin{array}{r}{\overline{v}}_i(t)={\overline{x}}_i(t)-{\tilde{x}}_i(t),i=1,2,\dots ,n.\end{array}$(7) ), (8(8) $\begin{array}{r}{\dot{\overline{x}}}_i(t)={\lambda }_i{\overline{x}}_i(t)+{\overline{u}}_i(t)+{\overline{w}}_i(t),i=1,2,\dots ,n\end{array}$(8) ), and (30(30) $\begin{array}{rl}{\dot{\tilde{x}}}_i(t)={\lambda }_i{\tilde{x}}_i(t)+{\overline{u}}_i(t),& \mathrm{when}t>z_p\\ {\tilde{x}}_i(t)={\tilde{x}}_i(t^{-})+q_i(t^{-}),& \mathrm{when}t=z_p\end{array}$(30) ), we can obtain $\begin{array}{rl}& {\dot{\overline{v}}}_i(t)={\lambda }_i{\overline{v}}_i(t)+{\overline{w}}_i(t),\mathrm{when}{z}_p<t<z_p+1\\ & {\overline{v}}_i(t)=e_i(t^{-}),\mathrm{when}t=z_p\end{array}$By solving the differential equation above, we may give (31) $\begin{array}{rl}{\overline{v}}_i(t)& =e^{{\lambda }_i(t-z_p)}{\overline{v}}_i(z_p)+{\int }_{z_p}^t{e}^{{\lambda }_i(\tau -z_p)}{\overline{w}}_i(\tau )\mathrm{d}\tau ,\\ & i=1,2,\dots ,n,z_p<t<z_p+1\end{array}$(31) with ${\overline{v}}_i(z_p)=e_i(z_p^{-})$.

Then, it follows from Equation (31(31) $\begin{array}{rl}{\overline{v}}_i(t)& =e^{{\lambda }_i(t-z_p)}{\overline{v}}_i(z_p)+{\int }_{z_p}^t{e}^{{\lambda }_i(\tau -z_p)}{\overline{w}}_i(\tau )\mathrm{d}\tau ,\\ & i=1,2,\dots ,n,z_p<t<z_p+1\end{array}$(31) ) that (32) $\begin{array}{rl}{\overline{v}}_i(t_{k+1}^{-})& =e^{{\lambda }_{i}T}{\overline{v}}_i(t_k)+{\int }_{t_k}^{t_{k+1}}{e}^{{\lambda }_i(\tau -t_k)}{\overline{w}}_i(\tau )\mathrm{d}\tau ,\\ & i=1,2,\dots ,n,t_k\in {\varphi }_p\end{array}$(32) where we have $\begin{array}{r}{\overline{v}}_i(t_k)=e_i(t_k^{-})\end{array}$Clearly, it means that ${\overline{v}}_i(t_{k+1})=e_i(t_{k+1}^{-})$ holds too. Substituting Equation (14(14) $\begin{array}{rl}{\phi }_{{\overline{w}}_i}& :={\int }_{t_{k-1}}^{t_k}{e}^{{\lambda }_i(\tau -t_{k-1})}{\overline{w}}_i(\tau )\mathrm{d}\tau ={\int }_0^T{e}^{{\lambda }_i\tau }{\overline{w}}_i(\tau )\mathrm{d}\tau <\mathrm{\infty },\\ & i=1,2,\dots ,n\end{array}$(14) ) into Equation (32(32) $\begin{array}{rl}{\overline{v}}_i(t_{k+1}^{-})& =e^{{\lambda }_{i}T}{\overline{v}}_i(t_k)+{\int }_{t_k}^{t_{k+1}}{e}^{{\lambda }_i(\tau -t_k)}{\overline{w}}_i(\tau )\mathrm{d}\tau ,\\ & i=1,2,\dots ,n,t_k\in {\varphi }_p\end{array}$(32) ), we obtain $\begin{array}{r}{\overline{v}}_i(t_{k+1}^{-})=e^{{\lambda }_{i}T}{\overline{v}}_i(t_k)+{\phi }_{{\overline{w}}_i},i=1,2,\dots ,n,t_k\in {\varphi }_p\end{array}$with ${\overline{v}}_i(t_{k+1})=e_i(t_{k+1}^{-})$. Then, it follows that (33) $\begin{array}{rl}E[{\overline{v}}_i^2(t_{k+1}^{-})]& =e^{2{\lambda }_{i}T}E[{\overline{v}}_i^2(t_k)]+{\phi }_{{\overline{w}}_i}^2,\\ & i=1,2,\dots ,n,t_k\in {\varphi }_p\end{array}$(33) where we have (34) $\begin{array}{r}E[{\overline{v}}_i^2(t_{k+1})]=E[e_i^2(t_{k+1}^{-})]\end{array}$(34) Furthermore, we notice that $\begin{array}{r}{e}_i(t_{k+1}^{-})={\overline{v}}_i(t_{k+1})-q_i(t_{k+1})\end{array}$Then, it follows from Lemma 2 that we have (35) $\begin{array}{r}E[e_i^2(t_{k+1}^{-})]=E[({\overline{v}}_i(t_{k+1}^{-})-q_i(t_{k+1}^{-}){)}^2]\le D_i(t_{k+1}^{-})\end{array}$(35) and (36) $\begin{array}{r}{r}_i(t_{k+1})\ge \frac{1}{2}{\log }_2\frac{E[{\overline{v}}_i^2(t_{k+1}^{-})]}{D_i(t_{k+1}^{-})}(\mathrm{bits})\end{array}$(36) According to (35(35) $\begin{array}{r}E[e_i^2(t_{k+1}^{-})]=E[({\overline{v}}_i(t_{k+1}^{-})-q_i(t_{k+1}^{-}){)}^2]\le D_i(t_{k+1}^{-})\end{array}$(35) ) and (36(36) $\begin{array}{r}{r}_i(t_{k+1})\ge \frac{1}{2}{\log }_2\frac{E[{\overline{v}}_i^2(t_{k+1}^{-})]}{D_i(t_{k+1}^{-})}(\mathrm{bits})\end{array}$(36) ), we quantize, encode ${\overline{v}}_i(t_{k+1}^{-})$ and have (37) $\begin{array}{r}E[e_i^2(t_{k+1}^{-})]=\frac{E[{\overline{v}}_i^2(t_{k+1}^{-})]}{2^{2r_i(t_{k+1})}}\end{array}$(37) Substituting Equations (33(33) $\begin{array}{rl}E[{\overline{v}}_i^2(t_{k+1}^{-})]& =e^{2{\lambda }_{i}T}E[{\overline{v}}_i^2(t_k)]+{\phi }_{{\overline{w}}_i}^2,\\ & i=1,2,\dots ,n,t_k\in {\varphi }_p\end{array}$(33) ) and (34(34) $\begin{array}{r}E[{\overline{v}}_i^2(t_{k+1})]=E[e_i^2(t_{k+1}^{-})]\end{array}$(34) ) into Equation (37(37) $\begin{array}{r}E[e_i^2(t_{k+1}^{-})]=\frac{E[{\overline{v}}_i^2(t_{k+1}^{-})]}{2^{2r_i(t_{k+1})}}\end{array}$(37) ), we obtain (38) $\begin{array}{rl}E[{\overline{v}}_i^2(t_{k+1})]& =\frac{e^{2{\lambda }_{i}T}}{2^{2r_i(t_{k+1})}}E[{\overline{v}}_i^2(t_k)]+\frac{{\phi }_{{\overline{w}}_i}^2}{2^{2r_i(t_{k+1})}},\\ & i=1,2,\dots ,n,t_{k+1}\in {\varphi }_p\end{array}$(38) According to the recurrence formula (38(38) $\begin{array}{rl}E[{\overline{v}}_i^2(t_{k+1})]& =\frac{e^{2{\lambda }_{i}T}}{2^{2r_i(t_{k+1})}}E[{\overline{v}}_i^2(t_k)]+\frac{{\phi }_{{\overline{w}}_i}^2}{2^{2r_i(t_{k+1})}},\\ & i=1,2,\dots ,n,t_{k+1}\in {\varphi }_p\end{array}$(38) ) above, we can further obtain $\begin{array}{rl}E[{\overline{v}}_i^2(t_k)]& =\frac{e^{2{\lambda }_{i}T{n}_p}}{2^{2n_p{r}_i(t_k)}}E[{\overline{v}}_i^2(z_p)]+\frac{1-\frac{e^{2{\lambda }_{i}T{n}_p}}{2^{2n_p{r}_i(t_k)}}}{1-\frac{e^{2{\lambda }_{i}T}}{2^{2r_i(t_{k+1})}}}\frac{{\phi }_{{\overline{w}}_i}^2}{2^{2r_i(t_{k+1})}},\\ & t_k>z_p,i=1,2,\dots ,n,t_k\in {\varphi }_p\end{array}$where we define $n_p:=\frac{t_k-z_p}{T}$.

Notice that the uncertainty of the state estimation error has decreased to the level that is considered the allowable error range of system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) in peace status. Thus, we only transmit the information on the plant state such that the uncertainty of the state estimation error will no longer increase. Clearly, the number of bits in peace status is less than one in defense status.

There exists a ${\phi }_s\in {\mathbb{R}}^{+}$ such that $\begin{array}{r}\frac{1-\frac{e^{2{\lambda }_{i}T{n}_p}}{2^{2n_p{r}_i(t_k)}}}{1-\frac{e^{2{\lambda }_{i}T}}{2^{2r_i(t_{k+1})}}}\frac{{\phi }_{{\overline{w}}_i}^2}{2^{2r_i(t_{k+1})}}<{\phi }_s\end{array}$with $t\to \mathrm{\infty }$. If we choose $r_i(t)$ such that (39) $\begin{array}{r}\frac{e^{2{\lambda }_{i}T{n}_p}}{2^{2n_p{r}_i(t)}}<1\end{array}$(39) holds, then the mean square value of the state estimation error is bounded and given by $\begin{array}{r}E[{\overline{v}}_i^2(t)]<{\phi }_s\end{array}$with $t\to \mathrm{\infty }$. Thus, there exists a state-estimate feedback control law of the form (5(5) $\begin{array}{r}U(t)=K\hat{X}(t)\end{array}$(5) ) such that $\begin{array}{r}E\Vert V(t)\Vert \le {\phi }_v<\mathrm{\infty }\end{array}$holds for any $t\in T_p$.

Then, it follows from (39(39) $\begin{array}{r}\frac{e^{2{\lambda }_{i}T{n}_p}}{2^{2n_p{r}_i(t)}}<1\end{array}$(39) ) that the number of bits $r_i(t)$ needs to satisfy (40) $\begin{array}{r}{r}_i(t)>{\lambda }_{i}T{\log }_{2}e(\mathrm{bits})\end{array}$(40) such that the system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is mean square observable. Namely, we set the number of bits given by (40(40) $\begin{array}{r}{r}_i(t)>{\lambda }_{i}T{\log }_{2}e(\mathrm{bits})\end{array}$(40) ) for encoding when the ith eigenvalue of the system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is greater than or equal to zero. On the contrary, we set $r_i(t)=0$ when the ith eigenvalue of the system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is less than zero. Furthermore, notice that the number of bits $r_i(t)$ must be an integer. Thus, we obtain $\begin{array}{r}{r}_i(t)>\lceil max\{{\lambda }_{i}T{\log }_{2}e,0\}\rceil (\mathrm{bits})\end{array}$The data rate $R(t)$ in peace status is given by $\begin{array}{r}R(t)>\sum _{i=1}^n\lceil max\{{\lambda }_i{\log }_{2}e,0\}\rceil (\mathrm{bits}/\mathrm{s})\end{array}$Combined with the arguments above, we can show that there exists a constant ${\phi }_v\in {\mathbb{R}}^{\mathbb{+}}$ such that $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim sup}E\Vert V(t)\Vert \le {\phi }_v<\mathrm{\infty }\end{array}$holds. Clearly, system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is mean square observable if data packets on the information of the plant state are transmitted by employing the data rate given by (10(10) $\begin{array}{r}R(t)=\{\begin{array}{ll}0(\mathrm{bits}/\mathrm{s})& \mathrm{when}t\in {\varphi }_a\\ {\displaystyle \sum _{i=1}^n\lceil max\{{\lambda }_i{\log }_{2}e,0\}\rceil (\mathrm{bits}/\mathrm{s})}& \mathrm{when}t\in {\varphi }_p\\ \begin{array}{l}{\displaystyle \sum _{i=1}^n\lceil max\{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}{\lambda }_i}\\ {\log }_{2}e,0\phantom{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}}\}\rceil (\mathrm{bits}/\mathrm{s})\end{array}& \mathrm{when}t\in {\varphi }_d\end{array}\end{array}$(10) ).

Remark 3.1

• Notice that, we consider system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) with random disturbance. Thus, system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is mean square stable, not exponentially stable.

• The chief difference in our case is that we consider the communication channel which has three states: DoS status, defense status, and peace status. While the communication status is changing, the data rate is being changed. Thus, our result is less conservative than some existing results (Feng et al., 2021, 2022).

• For the proof technique in Theorem 3.1, we employ not only control theoretic tools but also information theoretic tools.

Next, we derive the sufficient condition on the data rate for mean square stabilization of system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) under DoS attacks, and give the following result.

Theorem 3.2

Consider system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) in the presence of DoS attacks in Assumptions 2.1–2.6. Let ${\lambda }_1,\dots ,{\lambda }_n$ denote n positive real eigenvalues of matrix A. Assume that $0<\frac{1}{\varpi }+\frac{T}{\omega }<1$ hold. If the information of the plant state is transmitted by employing the data rate given by (10(10) $\begin{array}{r}R(t)=\{\begin{array}{ll}0(\mathrm{bits}/\mathrm{s})& \mathrm{when}t\in {\varphi }_a\\ {\displaystyle \sum _{i=1}^n\lceil max\{{\lambda }_i{\log }_{2}e,0\}\rceil (\mathrm{bits}/\mathrm{s})}& \mathrm{when}t\in {\varphi }_p\\ \begin{array}{l}{\displaystyle \sum _{i=1}^n\lceil max\{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}{\lambda }_i}\\ {\log }_{2}e,0\phantom{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}}\}\rceil (\mathrm{bits}/\mathrm{s})\end{array}& \mathrm{when}t\in {\varphi }_d\end{array}\end{array}$(10) ), system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is mean square stabilizable.

Proof.

According to (4(4) $\begin{array}{r}V(t)=X(t)-\hat{X}(t)\end{array}$(4) ) and (6(6) $\begin{array}{rl}& \overline{X}(t)=\mathrm{HX}(t)=[{\overline{x}}_1(t){\overline{x}}_2(t)\dots {\overline{x}}_n(t){]}^T\\ & \tilde{X}(t)=H\hat{X}(t)=[{\tilde{x}}_1(t){\tilde{x}}_2(t)\dots {\tilde{x}}_n(t){]}^T\\ & \overline{V}(t)=\mathrm{HV}(t)=[{\overline{v}}_1(t){\overline{v}}_2(t)\dots {\overline{v}}_n(t){]}^T\end{array}$(6) ), we obtain $\begin{array}{r}X(t)=H^{-1}(\tilde{X}(t)+\overline{V}(t))\end{array}$Notice that both the encoder and decoder have a knowledge of $\tilde{X}(t)$ at any time t. However, $\overline{V}(t)$ is a random variable for the encoder and decoder. Then, we have (41) $\begin{array}{r}E\Vert X(t)\Vert <E\Vert \tilde{X}(t)\Vert +E\Vert \overline{V}(t)\Vert \end{array}$(41) As stated in Theorem 3.1, we have (42) $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim sup}E\Vert \overline{V}(t)\Vert \le {\phi }_v<\mathrm{\infty }\end{array}$(42) If there exists a constant ${\phi }_a\in {\mathbb{R}}^{\mathbb{+}}$ such that (43) $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim sup}E\Vert \tilde{X}(t)\Vert \le {\phi }_a<\mathrm{\infty }\end{array}$(43) holds, we may obtain $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim sup}E\Vert X(t)\Vert \le {\phi }_x<\mathrm{\infty }\end{array}$Now, we will drive Inequality (43(43) $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim sup}E\Vert \tilde{X}(t)\Vert \le {\phi }_a<\mathrm{\infty }\end{array}$(43) ) in three different situations. First, we consider the case where the communication channel is in DoS attack status. For this case, there is no need to transmit any data packet. Thus, we have $\begin{array}{r}R(t)=0(\mathrm{bit}/\mathrm{s})\end{array}$when the system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is in DoS attack status.

According to (5(5) $\begin{array}{r}U(t)=K\hat{X}(t)\end{array}$(5) ) and (6(6) $\begin{array}{rl}& \overline{X}(t)=\mathrm{HX}(t)=[{\overline{x}}_1(t){\overline{x}}_2(t)\dots {\overline{x}}_n(t){]}^T\\ & \tilde{X}(t)=H\hat{X}(t)=[{\tilde{x}}_1(t){\tilde{x}}_2(t)\dots {\tilde{x}}_n(t){]}^T\\ & \overline{V}(t)=\mathrm{HV}(t)=[{\overline{v}}_1(t){\overline{v}}_2(t)\dots {\overline{v}}_n(t){]}^T\end{array}$(6) ), we obtain $\begin{array}{r}U(t)=K{H}^{-1}\tilde{X}(t)\end{array}$Then, we have $\begin{array}{r}\tilde{X}(t)=e^{(\mathrm{\Lambda }+\mathrm{HBK}{H}^{-1})(t-z_l)}\tilde{X}(z_l)=e^{H(A+\mathrm{BK})H^{-1}(t-z_l)}\tilde{X}(z_l)\end{array}$with $t>z_l$. Notice that, we select K such that A + BK is a Hurwitz matrix. Thus, it follows that there exists a constant ${\phi }_a\in {\mathbb{R}}^{\mathbb{+}}$ such that $\begin{array}{r}E\Vert \tilde{X}(t)\Vert \le {\phi }_a<\mathrm{\infty }\end{array}$holds in DoS attack status.

Secondly, we consider the case where the communication channel is in defense status. For this case, we increase the data rate $R(t)$ and send more information on the plant state to the estimator in order to decrease the state estimation error. Namely, the data rate $R(t)$ in defense status is given by $\begin{array}{rl}& R(t)>\sum _{i=1}^n\lceil max\{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}{\lambda }_i{\log }_{2}e,0\}\rceil \\ & \times (\mathrm{bits}/\mathrm{s})\end{array}$We define $\begin{array}{r}Q(t^{-}):=[q_1(t^{-})q_2(t^{-})\dots q_n(t^{-}){]}^T\end{array}$For any $t_k\in {\varphi }_d$, we have $\begin{array}{r}\tilde{X}(t)=\{\begin{array}{ll}{e}^{H(A+\mathrm{BK})H^{-1}(t-t_k)}\tilde{X}(t_k),& \mathrm{when}{t}_k<t<t_{k+1}\\ \tilde{X}(t^{-})+Q(t^{-}),& \mathrm{when}t=t_k\end{array}\end{array}$Notice that, $Q(t^{-})$ is the quantization value of $\overline{V}(t)$, and is norm-bounded. At the same time, A + BK is a Hurwitz matrix. Thus, it follows that there exists a constant ${\phi }_a\in {\mathbb{R}}^{\mathbb{+}}$ such that $\begin{array}{r}E\Vert \tilde{X}(t)\Vert \le {\phi }_a<\mathrm{\infty }\end{array}$holds in defense status.

Finally, we consider the case where the communication channel is in peace status. For this case, the uncertainty of the state estimation error has decreased to the level that is considered the allowable error range of system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ). Thus, we only transmit the information on the plant state such that the uncertainty of the state estimation error will no longer increase. The data rate $R(t)$ in peace status is given by (44) $\begin{array}{r}R(t)>\sum _{i=1}^n\lceil max\{{\lambda }_i{\log }_{2}e,0\}\rceil (\mathrm{bits}/\mathrm{s})\end{array}$(44) For any $t_k\in {\varphi }_p$, we have $\begin{array}{r}\tilde{X}(t)=\{\begin{array}{ll}{e}^{H(A+\mathrm{BK})H^{-1}(t-t_k)}\tilde{X}(t_k),& \mathrm{when}{t}_k<t<t_{k+1}\\ \tilde{X}(t^{-})+Q(t^{-}),& \mathrm{when}t=t_k\end{array}\end{array}$Arguing as before, we can show that there exists a constant ${\phi }_a\in {\mathbb{R}}^{\mathbb{+}}$ such that $\begin{array}{r}E\Vert \tilde{X}(t)\Vert \le {\phi }_a<\mathrm{\infty }\end{array}$holds in peace status.

Combined with the arguments above, there exists a constant ${\phi }_a\in {\mathbb{R}}^{\mathbb{+}}$ such that (45) $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim sup}E\Vert \tilde{X}(t)\Vert \le {\phi }_a<\mathrm{\infty }\end{array}$(45) holds. Thus, according to (41(41) $\begin{array}{r}E\Vert X(t)\Vert <E\Vert \tilde{X}(t)\Vert +E\Vert \overline{V}(t)\Vert \end{array}$(41) ), (42(42) $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim sup}E\Vert \overline{V}(t)\Vert \le {\phi }_v<\mathrm{\infty }\end{array}$(42) ), and (45(45) $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim sup}E\Vert \tilde{X}(t)\Vert \le {\phi }_a<\mathrm{\infty }\end{array}$(45) ), we can show that there exists a constant ${\phi }_x\in {\mathbb{R}}^{\mathbb{+}}$ such that $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim sup}E\Vert X(t)\Vert \le {\phi }_x<\mathrm{\infty }\end{array}$holds. Clearly, if the information of the plant state is transmitted by employing the data rate given by (10(10) $\begin{array}{r}R(t)=\{\begin{array}{ll}0(\mathrm{bits}/\mathrm{s})& \mathrm{when}t\in {\varphi }_a\\ {\displaystyle \sum _{i=1}^n\lceil max\{{\lambda }_i{\log }_{2}e,0\}\rceil (\mathrm{bits}/\mathrm{s})}& \mathrm{when}t\in {\varphi }_p\\ \begin{array}{l}{\displaystyle \sum _{i=1}^n\lceil max\{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}{\lambda }_i}\\ {\log }_{2}e,0\phantom{{(1-\frac{1}{\varpi }-\frac{T}{\omega })}^{-1}}\}\rceil (\mathrm{bits}/\mathrm{s})\end{array}& \mathrm{when}t\in {\varphi }_d\end{array}\end{array}$(10) ), system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is mean square stabilizable.

Remark 3.2

• It is shown in Theorem 3.2 that there exists a lower bound on the data rate above which one can find a state-estimate feedback control scheme to stabilize system (1) under DoS attacks.

• No information on the plant state can be sent to the controller over the communication channel due to DoS attacks. Thus, one needs to transmit more information in defense status by employing higher data rate.

• If assume that the disturbance is equal to 0, it follows from Theorem 2 that system (1(1) $\begin{array}{r}\dot{X}(t)=\mathrm{AX}(t)+\mathrm{BU}(k)+\mathrm{FW}(t),t\in {\mathbb{R}}^{+}.\end{array}$(1) ) is exponentially stable.

4. Numerical examples and simulations

In order to better understand our results, we give a simple numerical example. Consider a networked control system which is open-loop unstable and is given by $\begin{array}{rl}\left[\begin{array}{c}{\dot{x}}_1(t)\\ {\dot{x}}_2(t)\\ {\dot{x}}_3(t)\end{array}\right]& =\left[\begin{array}{ccc}0.6427& 0.019& -0.0535\\ 0.0362& 0.6537& -0.089\\ 0.0247& 0.0188& 0.5636\end{array}\right]\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\end{array}\right]\\ & +\left[\begin{array}{c}{u}_1(t)\\ u_2(t)\\ u_3(t)\end{array}\right]+10\left[\begin{array}{c}{w}_1(t)\\ w_2(t)\\ w_3(t)\end{array}\right]\end{array}$Here, we implement a state-estimate feedback control law of the form $\begin{array}{r}U(t)=K\hat{X}(t)\end{array}$with $\begin{array}{r}K=\left[\begin{array}{ccc}2.1& 3.74& 2.31\\ 4.025& 2.72& 3.465\\ 1.925& 2.55& 2.64\end{array}\right]\end{array}$Let the sampling period T = 1 s. As stated before, we consider the communication channel in the presence of DoS attacks, and set $\varpi =4.13$ and $\omega =4.02$. Clearly, we have $\frac{1}{\varpi }+\frac{T}{\omega }=0.4909<1$. Over a simulation horizon of 0–20 s, the communication channel is in peace status. Over a simulation horizon of 21–70 s, the communication channel is in DoS attack status. Over a simulation horizon of 71–100 s, the communication channel is in defense status. Over a simulation horizon of 101–150 s, the communication channel is in peace status again.

First, we select the data rate $R_1(t)$ given by $\begin{array}{r}{R}_1(t)=\{\begin{array}{ll}0(\mathrm{bits}/\mathrm{s})& \mathrm{when}t\in {\varphi }_a\\ 3(\mathrm{bits}/\mathrm{s})& \mathrm{when}t\in {\varphi }_p\\ 6(\mathrm{bits}/\mathrm{s})& \mathrm{when}t\in {\varphi }_d\end{array}\end{array}$which satisfies the condition in Theorems 3.1 and 3.2. The data rate $R_2(t)$ is given by Feng et al. (2022). The curves of the data rate response are shown in Figure 2. Clearly, our result is less conservative than the one in Feng et al. (2022). The simulation results of $V(t)$ are shown in Figure 3. Clearly, $V(t)$ is mean square convergent and the system is mean square observable. The simulation results of $X(t)$ are shown in Figure 4. Clearly, $X(t)$ is mean square convergent and the system is mean square stabilizable too.

Figure 2. The curve of the data rate response.

Figure 3. The curve of the estimation error response.

Figure 4. The curve of the plant state response.

5. Conclusion

In this paper, we addressed the networked control problem for CPSs in the presence of DoS attacks, and derived the sufficient conditions on the data rate for mean square observability and stabilization. There exists a lower bound of the data rate, above which one can find a quantization, coding, and control scheme to stabilize the unstable plant. It is shown in our result that the information on the plant state can be transmitted to the controller by employing the different data rate for the different communication status. The numerical simulation was also given to illustrate the validity of our result.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by Basic scientific research project of colleges and universities of Department of Education, Liaoning Provincial (general project) (Project Number: LJKMZ20220608) and 14th Five Year National Key R&D Program Project (Project Number: 2023YFB3211001).