The bottleneck and ceiling effects in quantized tracking control of heterogeneous multi-agent systems under DoS attacks

Control of multi-agent systems has attracted substantial attention of researchers. An agent refers to a subsystem in a large-scale system consisting of multiple agents and can be a different subject depending on the context. For example, an agent is a drone in a UAV swarm or a car in a vehicle platoon [1]. Agents are spatially distributed and hence the realization of control significantly depends on the quality of information exchanged among the agents via wireless communication networks. However, the challenges of malicious cyber attacks on information also emerge [2].

The control problems under packet losses have been well studied, e.g. in [3] for stochastic packet losses, and the case of DoS attacks inducing malicious packet losses has been studied in [4, 5, 6]. This paper deals with DoS attacks. The communication failures induced by DoS can exhibit a temporal profile quite different from those caused by genuine packet losses due to network congestion; particularly packet dropouts resulting from malicious DoS need not follow a given class of probability distributions [7], and therefore the analysis techniques relying on probabilistic arguments may not be applicable.

Attention has been focused on centralized systems under DoS attacks for achieving stability[8, 9], attack detection and control [10] and state estimation[11]. In [12], the authors study a stabilization problem under energy-constrained PWM DoS signals, and propose time and event triggering strategies under partially known and unknown attack information, respectively. The paper [13] describes DoS attacks by a Markov modulated model, in which the control packets are jammed stochastically. In [14], the authors model the interplay between a DoS attacker and a defender as a two-player zero-sum game and compute the saddle-point equilibrium. Recently, multi-agent systems under DoS attacks have been increasingly investigated, e.g., consensus, state estimation and real-time attack detection [15, 16, 17]. In [18], the authors propose a novel data-driven based algorithm to estimate the unknown switching matrices of the exosystems, and realize output regulation under DoS for heterogeneous multi-agent systems.

For multi-agent systems, the amount of data can be large especially when the number of agents is large. Consequently, agents can be subject to bandwidth limitation, and hence signals are subject to coarse quantization [19, 20, 21, 22, 23, 24]. Dynamic quantization for centralized systems in the presence of DoS attacks has been recently studied in [8, 25], in which the quantization mechanisms have zooming-out and in capabilities for mitigating the influence of DoS-induced packet losses and achieving asymptotic stability, respectively. Dynamic quantization has also been established for consensus of multi-agent systems without packet losses[26], in which only the zooming-in capability is developed. Recently, the papers [27, 28, 29] study consensus problems under DoS attacks and dynamic quantization. In [27, 28], quantization mechanisms with zooming-in and out capabilities are developed for homogeneous agents. An approach for designing a tight zooming-out parameter is provided, but the condition for consensus is subject to an additional constraint of DoS frequency[28]. As will be shown later, the approach in our paper is free from the additional DoS-frequency constraint.

In practice, a variety of agents may need to cooperate to complete a complicated task, i.e., cooperation of heterogeneous multi-agent systems [30, 31]. Consequently, those results for the control of homogeneous agents may not hold. In [29], heterogeneous nonlinear multi-agent systems under DoS and quantization were studied, while the focus is on the common consensus problem instead of tracking control. In this paper, we are interested in output tracking control of heterogeneous multi-agent systems under quantized communication and DoS attacks, in which the quantizer has a finite quantization range. The quantized controllers to be designed should prevent quantizer saturation all the time. Unlike DoS-free scenarios, the tracking errors can diverge under DoS attacks, and consequently some measurements can overflow the range of the quantizer if the quantized controllers for DoS-free scenarios are implemented.

For quantized tracking control of multi-agent systems, one can classify the overall information flow into the part for leader-follower quantized communication and the part for inter-follower quantized communication. In [32, 27], the leader-follower communication and inter-follower communication adopt an identical quantization mechanism. However, this may not provide optimal performance in communication resource allocation and control. One may also lose the insights about the interactions between the leader-follower and inter-follower quantized communication. For example, in order to save communication resource, one can select a data rate for the leader-follower quantized communication as close as possible to the minimum data rate [33], under which the followers can still estimate the leader’s state asymptotically. As will be shown in this paper, such a data rate (of the leader-follower quantized communication) can influence the inter-follower quantized communication, i.e., the quantizer for quantizing followers’ state can have overflow problem. In this paper, resilience refers to a bound charactering the amount of DoS attacks under which multi-agent systems can achieve quantized tracking control. Under DoS attacks, it is interesting and also important to know whether the leader-follower quantized communication or the inter-follower quantized communication influences more on the resilience. From a practical viewpoint, this can actually lead us to the optimal strategies of limited bandwidth allocation for real applications.

The paper’s contributions are twofold: a) design heterogeneous dynamic quantized controllers for the tracking control of heterogeneous multi-agent systems, b) discover the interactions between the leader-follower quantized communication and inter-follower quantized communication, and the consequent influences on the resilience of tracking control against DoS attacks. We reveal that the leader-follower quantized communication can cause ceiling effect and the inter-follower communication can cause bottleneck effect. Specifically, for a given communication topology and a number of bits for the quantization of the leader’s state, by tuning the followers’ quantized controller, one cannot improve the resilience beyond a level determined by the data rate of leader-follower quantized communication, i.e., the ceiling effect. Otherwise, overflow problems for quantizing followers’ state can occur. For the bottleneck effect, if one selects a “large” data rate for leader-follower quantized communication, further enlarging it cannot improve the resilience. Then the inter-follower quantized communication is the bottleneck of the resilience, which depends on the choices of the zooming-in and out factors for followers’ state quantization. We emphasize that, by our quantized controller design, the tightness for zooming-in factor is the same for DoS-free case [26], and the zooming-out factor is tight, i.e., arbitrarily approaching the spectral radius of the leader’s dynamic matrix.

This paper is organized as follows. In Section 2, we introduce the framework consisting of the control objectives, the heterogeneous leader and followers, and the class of DoS attacks. Section 3 presents the design of quantized controllers. The results of the paper are presented in Section 4. Numerical examples are presented in Section 5, and finally Section 6 ends the paper with conclusions and future research.

Notation. We denote by $\mathbb{R}$ the set of reals. Given $b\in\mathbb{R}$, $\mathbb{R}_{\geq b}$ and $\mathbb{R}_{>b}$ denote the sets of reals no smaller than $b$ and reals greater than $b$, respectively; $\mathbb{R}_{\leq b}$ and $\mathbb{R}_{<b}$ represent the sets of reals no larger than $b$ and reals smaller than $b$, respectively; $\mathbb{Z}$ denotes the set of integers. For any $c\in\mathbb{Z}$, we denote $\mathbb{Z}_{\geq c}:=\{c,c+1,\cdots\}$. Let $\lfloor v\rfloor$ be the floor function such that $\lfloor v\rfloor=\max\{o\in\mathbb{Z}|o\leq v\}$. Given a vector $y$ and a matrix $\Gamma$, let $\|y\|$ and $\|y\|_{\infty}$ denote the $2$- and $\infty$- norms of vector $y$, respectively, and $\|\Gamma\|$ and $\|\Gamma\|_{\infty}$ represent the corresponding induced norms of matrix $\Gamma$. Let $\rho(\Gamma)$ denote the spectral radius of $\Gamma$. Given an interval $\mathcal{I}$, $|\mathcal{I}|$ denotes its length. The Kronecker product is denoted by $\otimes$. We let “ $\genfrac{}{}{1.8pt}{1}{\hphantom{\,\cdot\,}}{\hphantom{\,\cdot\,}}$ $\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{1}{\color[rgb]{0,0,0}\,\cdot\,}{\color[rgb]{0,0,0}\,\cdot\,}$ ” represent element-wise division of vectors, e.g., $\mathchoice{\ooalign{$\genfrac{}{}{1.8pt}{0}{[x_{1}\,\,x_{2}]^{T}}{[y_{1}\,\,y_{2}]^{T}}$\cr$\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{0}{\phantom{[x_{1}\,\,x_{2}]^{T}}}{\phantom{[y_{1}\,\,y_{2}]^{T}}}$}}{\ooalign{$\genfrac{}{}{1.8pt}{1}{[x_{1}\,\,x_{2}]^{T}}{[y_{1}\,\,y_{2}]^{T}}$\cr$\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{1}{\phantom{[x_{1}\,\,x_{2}]^{T}}}{\phantom{[y_{1}\,\,y_{2}]^{T}}}$}}{\ooalign{$\genfrac{}{}{1.8pt}{2}{[x_{1}\,\,x_{2}]^{T}}{[y_{1}\,\,y_{2}]^{T}}$\cr$\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{2}{\phantom{[x_{1}\,\,x_{2}]^{T}}}{\phantom{[y_{1}\,\,y_{2}]^{T}}}$}}{\ooalign{$\genfrac{}{}{1.8pt}{3}{[x_{1}\,\,x_{2}]^{T}}{[y_{1}\,\,y_{2}]^{T}}$\cr$\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{3}{\phantom{[x_{1}\,\,x_{2}]^{T}}}{\phantom{[y_{1}\,\,y_{2}]^{T}}}$}}=[x_{1}/y_{1}\,\,x_{2}/y_{2}]^{T}$. Let “$\circ$” denote the element-wise multiplication of two vectors, e.g., $[a_{1}\,\,a_{2}]^{T}\circ[b_{1}\,\,b_{2}]^{T}=[a_{1}b_{1}\,\,a_{2}b_{2}]^{T}$. In our paper, a switched system refers to a class of systems in the form: $x(k+1)=A_{i}x(k),i\in\{1,2,\cdots,c\},$ in which the value of $i$ depends on the switching rules.

Communication graph: We let graph $\mathcal{G}=(\mathcal{N},\mathcal{E})$ denote the communication topology among the followers, where $\mathcal{N}=\{1,2,\cdots,N\}$ denotes the set of follower agents and $\mathcal{E}\subseteq\mathcal{N}\times\mathcal{N}$ denotes the set of edges. Let $\mathcal{N}_{i}$ denote the set of the neighbors of agent $i$, where $i=1,2,\cdots,N$. We assume that the graph $\mathcal{G}$ is undirected and connected. Let $\mathcal{A}_{\mathcal{G}}=[a_{ij}]\in\mathbb{R}^{N\times N}$ denote the adjacency matrix of $\mathcal{G}$, where $a_{ij}>0$ if and only if $j\in\mathcal{N}_{i}$ and $a_{ii}=0$. Define the Laplacian matrix $\mathcal{L}_{G}=[l_{ij}]\in\mathbb{R}^{N\times N}$, in which $l_{ii}=\sum_{j=1}^{N}a_{ij}$ and $l_{ij}=-a_{ij}$ if $i\neq j$. For the leader-follower communication, we assume that only a fraction of the followers can receive information from the leader. Let $a_{i0}$ represent the leader-follower interaction, i.e., if follower $i$ can directly receive the information from the leader, then $a_{i0}>0$, and otherwise $a_{i0}=0$. Moreover, we let the diagonal matrix be $\mathcal{D}=\text{diag}(a_{10},a_{20},\cdots,a_{N0})\in\mathbb{R}^{N\times N}$. Let $\tilde{\lambda}_{i}$ ($i=1,2,\cdots,N$) denote the eigenvalues of $\mathcal{L}_{G}+\mathcal{D}$.

System description: The follower agents interacting over the network $\mathcal{G}$ are expressed as heterogeneous systems:

	$\displaystyle x_{i}(k+1)$	$\displaystyle=A_{i}x_{i}(k)+B_{i}u_{i}(k)$		(1a)
	$\displaystyle y_{i}(k)$	$\displaystyle=C_{i}x_{i}(k)$		(1b)

where $i\in\mathcal{N}$, $x_{i}(k)\in\mathbb{R}^{n_{i}}$ denotes the state of agent $i$, $u_{i}(k)\in\mathbb{R}^{w_{i}}$ denotes its control input, $y_{i}(k)\in\mathbb{R}^{v}$ denotes its output, and $A_{i}\in\mathbb{R}^{n_{i}\times n_{i}}$, $B_{i}\in\mathbb{R}^{n_{i}\times w_{i}}$ and $C_{i}\in\mathbb{R}^{v\times n_{i}}$. The sampling time between $k$ and $k+1$ is $\Delta$. We assume that $(A_{i},B_{i})$ is stabilizable and thus there exists a feedback gain $K_{i}\in\mathbb{R}^{w_{i}\times n_{i}}$ such that $\rho(A_{i}+B_{i}K_{i})<1$. We assume that an upper bound of the initial condition $x_{i}(0)$ is known, i.e., $\|x_{i}(0)\|_{\infty}\leq C_{x_{0}}\in\mathbb{R}_{>0}$ ($i\in\mathcal{N}$). Note that $C_{x_{0}}$ can be arbitrarily large as long as it satisfies this bound. This is for preventing the overflow of state quantization for the initial condition. The dynamics of the leader can be described by

$\displaystyle v(k+1)=Sv(k)$

(2)

where $v(k)\in\mathbb{R}^{n_{v}}$ is the state and $S\in\mathbb{R}^{n_{v}\times n_{v}}$. We assume that $\rho(S)\geq 1$. Otherwise, as $v(k)\to 0$ when $k\to\infty$, one can simply realize the tracking control by locally stabilizing (1) under which $y_{i}(k)\to 0$. Similarly, we assume that an upper bound of the initial condition $v(0)$ is known, i.e., $\|v(0)\|_{\infty}\leq C_{v_{0}}\in\mathbb{R}_{>0}$.

We assume that $x_{i}(k)$ is available to the local controller of agent $i$. We also assume that only some of the followers can receive information from the leader. Followers can exchange information with their neighbor followers. We assume that transmissions are acknowledgment based and free of delay. Note that in a quantized control problem without packet losses, acknowledgments are not necessary [33, 34]. When packet losses and quantization coexist, acknowledgments are needed to ensure the synchronization of the signals in the encoders and decoders [3, 35]. Due to the data rate limitation, all the information exchanged via communication is encoded by limited numbers of bits. This implies that the transmitted signals are subject to quantization effect. In this paper, we develop two quantization mechanisms: one for followers’ state quantization, and another for leader’s state quantization. The motivations of adopting heterogeneous quantizers will be presented later (see Remark 4.6).

Quantizer for followers’ state: Let $\chi\in\mathbb{R}$ be a scalar before quantization and $q_{R_{f}}(\cdot)$ be the quantization function for scalar input values as

$\displaystyle\!\!q_{R_{f}}(\chi)\!\!=\!\!\left\{\!\!\begin{array}[]{lll}0&-\sigma<\chi<\sigma&\\ 2\psi\sigma&(2\psi-1)\sigma\leq\chi\!<\!(2\psi+1)\sigma\\ 2R_{f}\sigma&\chi\geq(2R_{f}+1)\sigma&\\ -q_{R_{f}}(-\chi)&\chi\leq-\sigma&\end{array}\right.$

(7)

where $R_{f}\in\mathbb{Z}_{>0}$ is to be designed and $\psi=1,2,\cdots,R_{f}$, and $\sigma\in\mathbb{R}_{>0}$. If the quantizer is unsaturated such that $|\chi|\leq(2R_{f}+1)\sigma$, then the quantization error satisfies $|\chi-q_{R_{f}}(\chi)|\leq\sigma.$

Quantizer for leader’s state: Let $\pi_{l}:=e_{l}/\omega_{l}$ be the $l$-th signal in a vector before quantization and $q_{\mathcal{R}_{l}}(\pi_{l})$ represents the quantized signal of $\pi_{l}$ encoded by $\mathcal{R}_{l}$ bits, where $l=1,2,\cdots,v$. The choices of $\mathcal{R}_{l}$, $e_{l}\in\mathbb{R}$ and $\omega_{l}\in\mathbb{R}_{>0}$ will be specified later. We implement a uniform quantizer such that

$\displaystyle q_{\mathcal{R}_{l}}(\pi_{l}):=\left\{\begin{array}[]{ll}\frac{\lfloor 2^{\mathcal{R}_{l}-1}\pi_{l}\rfloor+0.5}{2^{\mathcal{R}_{l}-1}},&\quad\textrm{if }-1\leq\pi_{l}<1\\ 1-\frac{0.5}{2^{\mathcal{R}_{l}-1}},&\quad\textrm{if }\pi_{l}=1\end{array}\right.$

(10)

if $\mathcal{R}_{l}\in\mathbb{Z}_{\geq 1}$, and $q_{\mathcal{R}_{l}}(\pi_{l})=0$ if $\mathcal{R}_{l}=0$. Note that for any $\omega_{l}\in\mathbb{R}_{>0}$ the following holds:

$\displaystyle\left|e_{l}-\omega_{l}q_{\mathcal{R}_{l}}\left(e_{l}/\omega_{l}\right)\right|\leq\omega_{l}/2^{\mathcal{R}_{l}},\quad\textrm{if}\,|e_{l}|/\omega_{l}\leq 1$

(11)

for both cases [3].

The following lemma is necessary to relax the constraints of dwell time of switched systems and obtain a tight data rate (see Remarks 4.4 and 4.6).

If $S$ has only real eigenvalues, the time-varying part in the transformation $\bar{v}(k)=E(k)Tv(k)$ is not needed, i.e., $\bar{v}(k)=Tv(k)$. Then, one only has the Jordan blocks in (17).

Assumption 3   For $i\in\mathcal{N}$, we assume that there exists $(F_{i},V_{i})$ satisfying $F_{i}S=A_{i}F_{i}+B_{i}V_{i}$ and $I_{v}=C_{i}F_{i},$ where $I_{v}$ is the identity matrix of dimension $n_{v}\times n_{v}$, and $V_{i}\in\mathbb{R}^{w_{i}\times n_{v}}$ and $F_{i}\in\mathbb{R}^{n_{i}\times n_{v}}$.  $\blacksquare$

Assumption 3 is a typical assumption for cooperative control of heterogeneous multi-agent systems [30]. We refer the readers to [36] for more details.

Controller design: As shown in Figure 1, for follower agent $i\in\mathcal{N}$, we propose the local control input

$\displaystyle u_{i}(k)=K_{i}x_{i}(k)+(V_{i}-K_{i}F_{i})z_{i}(k)$

(28)

in which

$\displaystyle z_{i}(k):=T^{-1}E^{-1}(k)\bar{z}_{i}(k)\in\mathbb{R}^{n_{v}}.$

(29)

The dynamics of $\bar{z}_{i}(k)$ in (29) follows

		$\displaystyle\bar{z}_{i}(k+1)=$
		$\displaystyle\left\{\begin{array}[]{ll}\bar{S}\bar{z}_{i}(k)+\bar{K}\sum_{j=1}^{N}a_{ij}(\hat{z}_{j}(k)-\hat{z}_{i}(k))\\ \quad\quad\quad+\,\bar{K}a_{i0}(\hat{v}(k)-\hat{z}_{i}(k)),&\text{if}\,k\notin H_{q}\\ \bar{S}\bar{z}_{i}(k),&\text{if}\,k\in H_{q}\end{array}\right.$		(33)

where $\hat{z}_{j}(k)\in\mathbb{R}^{n_{v}}$ and $\hat{v}(k)\in\mathbb{R}^{n_{v}}$ are the estimates of $\bar{z}_{j}(k)$ and $\bar{v}(k)$, respectively, and $\bar{K}\in\mathbb{R}^{n_{v}\times n_{v}}$ is a design parameter to be given later. The computation of $\hat{z}_{j}(k)$ in (3) follows

$\displaystyle\hat{z}_{j}(k)\!\!=\!\!\left\{\!\!\!\begin{array}[]{ll}\bar{S}\hat{z}_{j}(k-1)\!+\!\theta_{k-1}Q\left(\!\frac{\bar{z}_{j}(k)-\bar{S}\hat{z}_{j}(k-1)}{\theta_{k-1}}\!\right)&\text{if $k\notin H_{q}$ }\\ \bar{S}\hat{z}_{j}(k-1)&\text{if $k\in H_{q}$}\end{array}\right.$

(36)

where $j\in\{i\}\bigcup\mathcal{N}_{i}$ and $Q(\cdot)=[q_{R_{f}}(\cdot)\cdots q_{R_{f}}(\cdot)]^{T}$ is the vector version of (7). The scaling parameter $\theta_{k}$ updates as

$\displaystyle\theta_{k}=\left\{\begin{array}[]{ll}\gamma_{1}\theta_{k-1},&\quad\text{if $k\notin H_{q}$}\\ \gamma_{2}\theta_{k-1},&\quad\text{if $k\in H_{q}$ }\end{array}\right.$

(39)

where $\gamma_{1}<1$ and $\gamma_{2}>1$ are zooming in and out parameters, respectively. We assume that the algorithms in (36) and (39) are embedded in the encoders and decoders of $i\in\mathcal{N}$ with identical initial conditions. Under DoS attacks, the variables in $Q(\cdot)$ may diverge. Therefore, the quantizers must zoom out by using $\gamma_{2}$ and increase their ranges so that the states can be measured properly. If the transmissions succeed, the quantizers zoom in and $\theta_{k}\in\mathbb{R}_{>0}$ decreases by using $\gamma_{1}$. By adjusting the scaling parameter $\theta_{k}$ in $Q(\cdot)$ dynamically, the state will be kept within the limited quantization range without saturation and can converge asymptotically. The ranges of $\gamma_{1}$ and $\gamma_{2}$, and $\theta_{0}$ will be specified later. Note that $\gamma_{1},\gamma_{2}$ and $\theta_{0}$ in our paper are homogeneous among the followers. It is interesting to design distributed scaling parameters, e.g., $\gamma_{1}^{i}$ and $\gamma_{2}^{i}$. In this situation, the new scaling parameter $\theta_{k}$ in (39) will be a vector, and zooming-in and out factors composed by $\gamma_{1}^{i}$ and $\gamma_{2}^{i}$ will be matrices. This case will be left for future research.

The update of $\hat{v}(k)$ in (3) is given by

$\displaystyle\!\!\!\hat{v}(k)\!\!=\!\!\left\{\!\!\begin{array}[]{ll}\bar{S}\Big{(}\hat{v}(k-1)\\ \,\,-\left.\omega(k-1)\circ Q_{v}\left(\mathchoice{\ooalign{$\genfrac{}{}{1.8pt}{0}{\bar{S}\hat{v}(k-1)-\bar{v}(k)}{\bar{S}\omega(k-1)}$\cr$\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{0}{\phantom{\bar{S}\hat{v}(k-1)-\bar{v}(k)}}{\phantom{\bar{S}\omega(k-1)}}$}}{\ooalign{$\genfrac{}{}{1.8pt}{1}{\bar{S}\hat{v}(k-1)-\bar{v}(k)}{\bar{S}\omega(k-1)}$\cr$\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{1}{\phantom{\bar{S}\hat{v}(k-1)-\bar{v}(k)}}{\phantom{\bar{S}\omega(k-1)}}$}}{\ooalign{$\genfrac{}{}{1.8pt}{2}{\bar{S}\hat{v}(k-1)-\bar{v}(k)}{\bar{S}\omega(k-1)}$\cr$\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{2}{\phantom{\bar{S}\hat{v}(k-1)-\bar{v}(k)}}{\phantom{\bar{S}\omega(k-1)}}$}}{\ooalign{$\genfrac{}{}{1.8pt}{3}{\bar{S}\hat{v}(k-1)-\bar{v}(k)}{\bar{S}\omega(k-1)}$\cr$\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{3}{\phantom{\bar{S}\hat{v}(k-1)-\bar{v}(k)}}{\phantom{\bar{S}\omega(k-1)}}$}}\right)\!\right),&k\notin\!H_{q}\\ \bar{S}\hat{v}(k-1),&k\in\!H_{q}\\ \end{array}\right.$

(51)

in which $Q_{v}(\cdot)$ is the vector form of (10), $\omega(k)\in\mathbb{R}^{n_{v}}$ is a scaling vector for quantizing $\bar{v}(k)$, “$\circ$” is the element-wise multiplication and “ $\genfrac{}{}{1.8pt}{1}{\hphantom{\,\,\cdot\,\,}}{\hphantom{\cdot}}$ $\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{1}{\color[rgb]{0,0,0}\,\,\cdot\,\,}{\color[rgb]{0,0,0}\cdot}$ ” is the element-wise division. The computation of $\omega(k)$ in (51) is given as follows:

$\displaystyle\!\!\!\!\!\left\{\!\!\begin{array}[]{l}\omega(k)=\left\{\begin{array}[]{ll}\tilde{S}H\omega(k-1),&k\notin H_{q}\\ \tilde{S}\omega(k-1),&k\in H_{q}\end{array}\right.\\ H\!=\!\text{diag}(2^{-R_{1}}I_{1},2^{-R_{2}}I_{2},\cdots,2^{-R_{p}}I_{p})\!\in\!\mathbb{R}^{n_{v}\times n_{v}}\end{array}\right.$

(56)

where the block diagonal matrix $\tilde{S}=\text{diag}(\tilde{S}_{1},\cdots,\tilde{S}_{p})\in\mathbb{R}^{n_{v}\times n_{v}}$. If $r$ corresponds to a real eigenvalue $\lambda_{r}\in\mathbb{R}$, then $\tilde{S}_{r}=\bar{S}_{r}$ in (17). Otherwise, $\tilde{S}_{r}$ takes

$\displaystyle\tilde{S}_{r}\!\!=\!\!\left[\begin{array}[]{cccc}\zeta_{r}I_{2}&O&&\\ &\zeta_{r}I_{2}&O&\\ &&\ddots\\ &&&\zeta_{r}I_{2}\end{array}\right]\!\!\in\!\mathbb{R}^{2n_{r}\times 2n_{r}},O\!=\!\left[\begin{array}[]{cc}1&1\\ 1&1\end{array}\right]\!\!.$

(63)

Similar to [8], the parameter $R_{r}$ ($r=1,\cdots,p$) in $H$ is the number of bits for the quantization process corresponding to $\bar{S}_{r}$. If $R_{r}$ is determined, then $\mathcal{R}_{l}$ in (10) is determined as well. At last, we assume that the initial conditions in the encoding and decoding systems are identical and satisfy

$\displaystyle\hat{v}_{l}(0)=0,\,\,\omega_{l}(0)>|\bar{v}_{l}(0)|,\,\,l={1,2,\cdots,v}$

(64)

where $\hat{v}_{l}(k)\in\mathbb{R}$ and $\omega_{l}(k)\in\mathbb{R}$ denote the $l$-th element in vectors $\hat{v}(k)=[\hat{v}_{1}(k)\cdots\hat{v}_{l}(k)\cdots\hat{v}_{v}(k)]^{T}$ and $\omega(k)=[\omega_{1}(k)\cdots\omega_{l}(k)\cdots\omega_{v}(k)]^{T}$, respectively.

Define vectors $\bar{z}(k):=[\bar{z}_{1}^{T}(k)\cdots\bar{z}_{N}^{T}(k)]^{T}$, $\hat{z}(k):=[\hat{z}_{1}^{T}(k)\cdots\hat{z}_{N}^{T}(k)]^{T}$ and $\delta(k):=\bar{z}(k)-\mathbf{1}_{N}\otimes\bar{v}(k)$. Define errors $e_{z}(k):=\bar{z}(k)-\hat{z}(k)$ (of estimating $\bar{z}(k)$) and $e_{v}(k):=\hat{v}(k)-\bar{v}(k)$ (of estimating $\bar{v}(k)$). Define matrices $G:=\bar{S}_{N}-(\mathcal{L}_{G}+\mathcal{D})\otimes\bar{K},\bar{S}_{N}:=I_{N}\otimes\bar{S},W:=\mathcal{D}\otimes\bar{K},P:=(\mathcal{L}_{G}+\mathcal{D})\otimes\bar{K}$ and $Z:=\bar{S}_{N}+(\mathcal{L}_{G}+\mathcal{D})\otimes\bar{K}.$ We define $\alpha(k):=\delta(k)/\theta_{k}$, $\xi_{z}(k):=e_{z}(k)/\theta_{k}$ and $\xi_{v}(k):=e_{v}(k)/\theta_{k}$. Accordingly, we obtain the dynamics of $\alpha(k)$ and $\xi_{z}(k)$ in four cases as follows. The dynamics of $\xi_{v}(k)$ will be analyzed later.

Case I: $k+1\notin H_{q}$ and $k\notin H_{q}$

	$\displaystyle\!\!\!\!\alpha(k+1)=\frac{G}{\gamma_{1}}\alpha(k)+\frac{P}{\gamma_{1}}\xi_{z}(k)-\frac{W}{\gamma_{1}}(\mathbf{1}_{N}\otimes\xi_{v}(k))$		(65a)
	$\displaystyle\!\!\!\!\xi_{z}(k+1)=\frac{Z}{\gamma_{1}}\xi_{z}(k)-\frac{P}{\gamma_{1}}\alpha(k)-\frac{W}{\gamma_{1}}(\mathbf{1}_{N}\otimes\xi_{v}(k))$
	$\displaystyle\quad-\frac{1}{\gamma_{1}}Q\left(Z\xi_{z}(k)-P\alpha(k)-W(\mathbf{1}_{N}\otimes\xi_{v}(k))\right)$		(65b)

Case II: $k+1\notin H_{q}$ and $k\in H_{q}$

	$\displaystyle\alpha(k+1)=\frac{\bar{S}_{N}}{\gamma_{1}}\alpha(k)$		(66a)
	$\displaystyle\xi_{z}(k+1)=\frac{\bar{S}_{N}}{\gamma_{1}}\xi_{z}(k)-\frac{1}{\gamma_{1}}Q\left(\bar{S}_{N}\xi_{z}(k)\right)$		(66b)

Case III: $k+1\in H_{q}$ and $k\notin H_{q}$

	$\displaystyle\!\!\!\!\alpha(k+1)=\frac{G}{\gamma_{2}}\alpha(k)+\frac{P}{\gamma_{2}}\xi_{z}(k)-\frac{W}{\gamma_{2}}(\mathbf{1}_{N}\otimes\xi_{v}(k))$		(67a)
	$\displaystyle\!\!\!\!\xi_{z}(k+1)=\frac{Z}{\gamma_{2}}\xi_{z}(k)-\frac{P}{\gamma_{2}}\alpha(k)-\frac{W}{\gamma_{2}}(\mathbf{1}_{N}\otimes\xi_{v}(k))$		(67b)

Case IV: $k+1\in H_{q}$ and $k\in H_{q}$

	$\displaystyle\alpha(k+1)=\frac{\bar{S}_{N}}{\gamma_{2}}\alpha(k)$		(68a)
	$\displaystyle\xi_{z}(k+1)=\frac{\bar{S}_{N}}{\gamma_{2}}\xi_{z}(k).$		(68b)

The next proposition specifies the ranges of $\gamma_{1}$, $\gamma_{2}$ and the value of $(2R_{f}+1)\sigma$ preventing the saturation of (7). Its proof with $C_{1}$ and $C_{2}$ is provided in the Appendix.

In Proposition 4.3, by properly selecting $\gamma_{1}$ and $\gamma_{2}$, we have provided the value of $(2R_{f}+1)\sigma$ for the unsaturation of quantizer (7), by supposing $\xi_{v}(k)$ upper bounded. In the following proposition, we provide the conditions to ensure an upper bounded $\xi_{v}(k)$, namely a finite $E_{v}$.

Proof. Let $e_{vl}(k)$ denote the $l$-th element of vector $e_{v}(k)$. To prove a), we will show $|e_{vl}(k)|\leq\omega_{l}(k)$, which implies quantizer unsaturation in view of (10). If $k=1\notin H_{q}$, then by (51) one has

$\displaystyle e_{v}(1)=\bar{S}\left(e_{v}(0)-\omega(0)\circ Q_{v}\left(\mathchoice{\ooalign{$\genfrac{}{}{1.8pt}{0}{\hphantom{e_{v}(0)}}{\hphantom{\omega(0)}}$\cr$\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{0}{\color[rgb]{0,0,0}e_{v}(0)}{\color[rgb]{0,0,0}\omega(0)}$}}{\ooalign{$\genfrac{}{}{1.8pt}{1}{\hphantom{e_{v}(0)}}{\hphantom{\omega(0)}}$\cr$\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{1}{\color[rgb]{0,0,0}e_{v}(0)}{\color[rgb]{0,0,0}\omega(0)}$}}{\ooalign{$\genfrac{}{}{1.8pt}{2}{\hphantom{e_{v}(0)}}{\hphantom{\omega(0)}}$\cr$\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{2}{\color[rgb]{0,0,0}e_{v}(0)}{\color[rgb]{0,0,0}\omega(0)}$}}{\ooalign{$\genfrac{}{}{1.8pt}{3}{\hphantom{e_{v}(0)}}{\hphantom{\omega(0)}}$\cr$\color[rgb]{1,1,1}\genfrac{}{}{0.6pt}{3}{\color[rgb]{0,0,0}e_{v}(0)}{\color[rgb]{0,0,0}\omega(0)}$}}\right)\right).$

(77)

Note that $|e_{vl}(0)|\leq\omega_{l}(0)$ implied by (64). Then according to (11), element-wise inequality of (77) yields

$\displaystyle|e_{vl}(1)|\leq|\bar{S}_{l}|H\omega(0)\leq\tilde{S}_{l}H\omega(0)=\omega_{l}(1)$

(78)

in which $|\bar{S}_{l}|$ is the $l$-th row of $|\bar{S}|$, and $\tilde{S}_{l}$ is the $l$-th row of $\tilde{S}$. Here, $|\bar{S}|$ is a matrix in which the elements are the absolute values of the corresponding elements in $\bar{S}$. If $k=1\in H_{q}$, we have $e_{v}(1)=\bar{S}e_{v}(0)$ by (51). Following a similar element-wise analysis as in (78), one has

$\displaystyle\!\!\!\!|e_{vl}(1)|\!\leq\!|\bar{S}_{l}|[|e_{v1}(0)|\cdots|e_{vv}(0)|]^{T}\!\leq\!\tilde{S}_{l}\omega(0)\!=\!\omega_{l}(1).$

(79)

By the above analysis, we have shown that if overflow does not occur at $k=0$, then no matter $k=1$ is corrupted by DoS or not, overflow cannot occur. By induction, one can confirm that overflow does not occur for all $k$.

To prove b) and c), we first analyze the dynamics of

$\displaystyle\xi_{vl}(k)=\frac{e_{vl}(k)}{\omega_{l}(k)}\frac{\omega_{l}(k)}{\theta(k)}\vspace{-2mm}$

(80)

where $\xi_{vl}(k)$ is the $l$-th element in $\xi_{v}(k)$. We have shown that $|e_{vl}(k)|/\omega_{l}(k)\leq 1$ in a). In order to ensure $\xi_{vl}(k)$ is bounded, one needs to ensure that $w_{l}(k)/\theta(k)$ is upper bounded, namely bounded $w(k)/\theta(k)$ in vector form. Let $\bar{\omega}(k):=\omega(k)/\theta_{k}$. In the absence of DoS, by (39) and (56), one has

$\displaystyle\bar{\omega}(k+1)=\frac{\tilde{S}H}{\gamma_{1}}\bar{\omega}(k).$

(81)

It is clear that if b) or c) holds, then $\rho(\tilde{S}H/\gamma_{1})<1$. In the presence of DoS, we have

$\displaystyle\bar{\omega}(k+1)=\frac{\tilde{S}}{\gamma_{2}}\bar{\omega}(k)$

(82)

where $\rho(\tilde{S}/\gamma_{2})<1$ since $\gamma_{2}>\rho(\bar{S})=\rho(\tilde{S})$ in Proposition 4.3. Because $\tilde{S}H$ and $\tilde{S}$ are upper-triangular matrices, by the stabilization of switched systems, one can confirm that $\bar{\omega}(k)$ is bounded, and hence there exists a positive and finite $E_{v}$ such that $\|\xi_{v}(k)\|_{\infty}\leq E_{v}$ in view of (80).  $\blacksquare$

As a special case, if the dimension of the $\bar{S}_{r}$ corresponding to the spectral radius is 1, the result in Proposition 4.5 b) can be relaxed to $2^{R_{r}}\geq\zeta_{r}/\gamma_{1}$. Note that if the dimension of the $\bar{S}_{r}$ corresponding to the spectral radius is larger than 1, such a relaxation can make (81) unstable. Similarly, if the dimension of the $\bar{S}_{r}$ corresponding to the spectral radius is 1, one can let $\gamma_{2}=\rho(\tilde{S})$.

Note that Proposition 4.5 cannot imply tracking control under DoS. The following main result characterizes the system’s resilience of tracking control under DoS attacks.

Proof. For achieving tracking control, it is necessary that $\delta(k)\to 0$ as $k\to\infty$. This can be obtained by $\theta_{k}\to 0$ in view of $\delta(k)=\theta_{k}\alpha(k)$, in which one can confirm that $\|\alpha(k)\|$ is upper bounded by following a similar analysis in the proof of Proposition 4.3. If DoS attacks satisfy the first inequality in (4.7), then one has $\theta_{k}\to 0$ [27]. By Proposition 4.5, one can infer that $(R_{r},\gamma_{1})$ satisfies $\gamma_{1}>\zeta_{r}/2^{R_{r}}$. Thus, one can obtain the second inequality in (4.7).