Dynamic quantized consensus under DoS attacks:
Towards a tight zooming-out factor

The agents interacting over the network $\mathcal{G}$ are homogeneous linear multi-agent systems with sampling period $\Delta\in\mathbb{R}_{>0}$:

	$\displaystyle x_{i}(k\Delta)$	$\displaystyle=Ax_{i}((k-1)\Delta)+Bu_{i}((k-1)\Delta)$		(1a)
	$\displaystyle y_{i}(k\Delta)$	$\displaystyle=Cx_{i}(k\Delta)$		(1b)

where $i\in\mathcal{N}$, $k\in\mathbb{Z}_{\geq 1}$, $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times w}$ and $C\in\mathbb{R}^{v\times n}$. We assume that $(A,B)$ is stabilizable and $(A,C)$ is observable. $x_{i}(k\Delta)\in\mathbb{R}^{n}$ denotes the state of agent $i$ and $y_{i}(k\Delta)\in\mathbb{R}^{v}$ denotes the output. We assume that an upper bound of the initial condition $x_{i}(0)\in\mathbb{R}^{n}$ is known, i.e. $\|x_{i}(0)\|_{\infty}\leq C_{x_{0}}\in\mathbb{R}_{>0}$ [20, 21, 22]. Here, $C_{x_{0}}$ can be an arbitrarily large real as long as it satisfies this bound. This is for preventing the overflow of state quantization at the beginning. Let $u_{i}((k-1)\Delta)\in\mathbb{R}^{w}$ denote the control input, whose computation will be given in (12).

We introduce the control objective of this paper: State consensus. The average of the states is computed as $\overline{x}(k\Delta)=\frac{1}{N}\sum_{i=1}^{N}x_{i}(k\Delta)$ and state consensus is defined by

$\displaystyle\lim_{k\to\infty}x_{i}(k\Delta)-\overline{x}(k\Delta)=0,\,\,\,i=1,2,\cdots,N.$

(2)

It is trivial that (2) also implies the consensus of output $y_{i}(k\Delta)$. We assume that the spectral radius $\rho(A)\geq 1$. Otherwise, consensus (2) is trivially achieved by setting $u_{i}(k\Delta)=0$ for all $k$.

In this paper, the communication channel for information exchange is bandwidth limited and subject to DoS. We assume that the transmission attempts of each agent take place periodically at time $k\Delta$ with $k\in\mathbb{Z}_{\geq 1}$ and free of delay. Agent $i=1,2,\cdots,N$ can only exchange quantized information with its neighbor agents $j\in\mathcal{N}_{i}$ over the network $\mathcal{G}$ due to bandwidth constraints. In the presence of DoS, transmission attempts fail. We let $\{s_{r}\}\subseteq\{k\Delta\}$ represent the instants of successful transmissions. Note that $s_{0}\in\mathbb{R}_{\geq\Delta}$ is the instant when the first successful transmission occurs. Also, we let $s_{-1}$ denote the time instant $0$. In the remainder of the paper, we let $k$ represent instant $k\Delta$, and $s_{r}+p$ represent instant $s_{r}+p\Delta$ ($p\in\mathbb{Z}_{\geq 0}$) for the ease of notation.

Uniform quantizer. Let $\chi\in\mathbb{R}$ be the original scalar value before quantization and $q_{R}(\cdot)$ be the quantization function for scalar input values as

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

(7)

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

$\displaystyle|\chi-q_{R}(\chi)|\leq\sigma,\,\,\text{if}\,\,|\chi|\leq(2R+1)\sigma.$

(8)

Moreover, we define the vector version of the quantization function as $Q_{R}(\beta)=[\,q_{R}(\beta_{1})\,\,q_{R}(\beta_{2})\,\,\cdots\,\,q_{R}(\beta_{f})\,]^{T}\in\mathbb{R}^{f}$, where $\beta=[\beta_{1}\,\,\beta_{2}\,\,\cdots\beta_{f}]^{T}\in\mathbb{R}^{f}$ with $f\in\mathbb{Z}_{\geq 1}$. It is clear that $\beta$ can be properly quantized if $\|\beta\|_{\infty}\leq(2R+1)\sigma$. In the remainder of the paper, by quantizer overflow or saturation, we mean that at least a $\beta_{i}$ exceeds the range of quantization, i.e., $\|\beta\|_{\infty}>(2R+1)\sigma$ or equivalently $|\beta_{i}|>(2R+1)\sigma$.

In this paper, we refer to DoS as the event for which all the encoded signals cannot be received by the decoders and it affects all the agents. We consider a general DoS model developed in [5] that describes the attacker’s action by the frequency of DoS attacks and their duration. Let $\{h_{q}\}_{q\in\mathbb{Z}_{0}}$ with $h_{0}\geq\Delta$ denote the sequence of DoS off/on transitions, that is, the time instants at which DoS exhibits a transition from zero (transmissions are successful) to one (transmissions are not successful). Hence, $H_{q}:=\{h_{q}\}\cup[h_{q},h_{q}+\tau_{q}[$ represents the $q$-th DoS time-interval, of a length $\tau_{q}\in\mathbb{R}_{\geq 0}$, over which the network is in DoS status. If $\tau_{q}=0$, then $H_{q}$ takes the form of a single pulse at $h_{q}$. Given $\tau,t\in\mathbb{R}_{\geq 0}$ with $t\geq\tau$, let $n(\tau,t)$ denote the number of DoS off/on transitions over $[\tau,t]$, and let $\Xi(\tau,t):=\bigcup_{q\in\mathbb{Z}_{0}}H_{q}\,\cap\,[\tau,t]$ be the subset of $[\tau,t]$ where the network is in DoS status.

The lemma below presents the relation between the number of successful transmissions and the number of transmission attempts $k$. Let $T_{S}(1,k)$ and $T_{U}(1,k)$ denote the numbers of successful and unsuccessful transmissions between steps 1 and $k$, respectively.

To deal with output feedback, an observer estimating $x_{i}(k)$ from $y_{i}(k)$ is locally implemented at agent $i\in\mathcal{N}$ given by

	$\displaystyle\hat{x}_{i}(k)$	$\displaystyle=A\hat{x}_{i}(k-1)+Bu_{i}(k-1)$
		$\displaystyle\,\,\,+F(y_{i}(k-1)-C\hat{x}_{i}(k-1))$		(9)

where $\hat{x}_{i}(k)\in\mathbb{R}^{n}$ denotes the observer state and $\hat{x}_{i}(0)=0$. Since $(A,C)$ is observable, there exists a matrix $F\in\mathbb{R}^{n\times v}$ such that the spectral radius $\rho(A-FC)<1$. Then, $\hat{x}_{i}(k)$ is quantized and transmitted to the neighbors. Let $\tilde{x}_{i}^{j}(k)\in\mathbb{R}^{n}$ denote the decoded value of $\hat{x}_{i}(t)$ by agent $j\in\mathcal{N}_{i}$ and vice versa for $\tilde{x}_{j}^{i}(k)$, whose computations will be given later.

For agent $i\in\mathcal{N}$, the control input $u_{i}(k)$ has two modes aligned with the status of DoS:

$\displaystyle u_{i}(k)=\left\{\begin{array}[]{ll}K\sum_{j=1}^{N}a_{ij}(\tilde{x}_{j}^{i}(k)-\tilde{x}_{i}^{i}(k))&\text{if}\,\,k\notin H_{q}\\ 0&\text{if}\,\,k\in H_{q}\end{array}\right.$

(12)

for $k=1,2,\cdots$. Let $u_{i}(0)=0$. Note that each agent is able to passively know the status of DoS at $k$ by whether receiving neighbors’ transmissions or not. We assume that there exists a feedback gain $K\in\mathbb{R}^{w\times n}$ such that the spectral radius of

$\displaystyle J=\text{diag}(J_{2},\cdots,J_{N}),J_{i}=A-\lambda_{i}BK,\,\,\,i=2,...,N$

(13)

satisfies $\rho(J)<1$, where $\lambda_{i}$ denotes the eigenvalues of $L_{\mathcal{G}}$. Note that $\rho(J)<1$ is needed to achieve consensus even if DoS is absent. The assumption on the existence of $K$ is motivated by the consensusability of linear discrete-time multi-agent systems [20]. If DoS is absent and precise information is available, a sufficient condition to ensure the existence of $K$ is that $(A,B)~{}\textrm{stabilizable and}~{}\prod_{p}|\lambda_{p}^{u}(A)|<\frac{1+\lambda_{2}/\lambda_{N}}{1-\lambda_{2}/\lambda_{N}},$ where $\lambda_{p}^{u}(A)$ represents unstable eigenvalues of $A$.

In our paper, the encoder and decoder for each state have the same structure. In the following, we first explain the decoding process. In (12), $\tilde{x}_{j}^{i}(k)$ is the decoded value of $\hat{x}_{j}(k)$ ($j\in\mathcal{N}_{i}$) at the decoder of agent $i$ with initial condition $\tilde{x}_{j}^{i}(0)$. Its computation is given by

$\displaystyle\tilde{x}_{j}^{i}(k)=\left\{\begin{array}[]{ll}A\tilde{x}_{j}^{i}(k-1)+\theta(k-1)\hat{Q}_{j}^{i}(k)&\text{if $k\notin H_{q}$ }\\ A\tilde{x}_{j}^{i}(k-1)&\text{if $k\in H_{q}$}\end{array}\right.$

(16)

in which $k=1,2,\cdots$, and $\hat{Q}_{j}^{i}(k)$ is the value generated and transmitted by the encoder of agent $j$ given by

$\displaystyle\hat{Q}_{j}^{i}(k)=Q_{R}\left(\frac{\hat{x}_{j}(k)-A\tilde{x}_{j}^{i}(k-1)}{\theta(k-1)}\right).$

(17)

The computation of $\tilde{x}_{i}^{i}(k)$ in the decoder of agent $i$ follows

	$\displaystyle\tilde{x}_{i}^{i}(k)\!=\!\left\{\begin{array}[]{ll}A\tilde{x}_{i}^{i}(k-1)+\theta(k-1)\hat{Q}_{i}^{i}(k)&\text{if $k\notin H_{q}$ }\\ A\tilde{x}_{i}^{i}(k-1)&\text{if $k\in H_{q}$}\end{array}\right.$		(20)
	$\displaystyle\hat{Q}_{i}^{i}(k)=Q_{R}\left(\frac{\hat{x}_{i}(k)-A\tilde{x}_{i}^{i}(k-1)}{\theta(k-1)}\right).$		(21)

Now we explain the synchronization of a decoded state among agents, that is, for $i,l\in\mathcal{N}_{j}$, the values of $\tilde{x}_{j}^{i}(k)$, $\tilde{x}_{j}^{l}(k)$ and $\tilde{x}_{j}^{j}(k)$ are identical for all $k$. This is because the decoders of agents $i,j$ and $l$ have the same initial condition ($\tilde{x}_{j}^{i}(0)=\tilde{x}_{j}^{l}(0)=\tilde{x}_{j}^{j}(0)=0$), have the same $\theta(k)$ and receive the same $Q_{R}((\hat{x}_{j}(k)-A\tilde{x}_{j}^{i,l,j}(k-1))/\theta(k-1))$ in $\hat{Q}_{j}^{i}(k)$, $\hat{Q}_{j}^{l}(k)$ and $\hat{Q}_{j}^{j}(k)$ for all $k$. The synchronization of $\theta(k)$ in the decoders and also in the encoders will be explained after (31). Thus, the superscripts of $\tilde{x}_{j}^{i}(k)$, $\tilde{x}_{j}^{l}(k)$ and $\tilde{x}_{j}^{j}(k)$ can be omitted, and it is enough to let $\tilde{x}_{j}(k)$ represent the decoded value of $\hat{x}_{j}(k)$ in the decoders. At last, we point out that the encoder is a copy of the decoder, and thus one has that $\tilde{x}_{j}(k)$ in the encoder also has the same value as in the decoders.

Therefore (12) can be rewritten as

$\displaystyle u_{i}(k)=\left\{\begin{array}[]{ll}K\sum_{j=1}^{N}a_{ij}(\tilde{x}_{j}(k)-\tilde{x}_{i}(k))&\text{if}\,k\notin H_{q}\\ 0&\text{if}\,k\in H_{q}\end{array}\right.$

(24)

and (16) and (20) can be rewritten as

$\displaystyle\tilde{x}_{j}(k)=\left\{\begin{array}[]{ll}A\tilde{x}_{j}(k-1)+\theta(k-1)\hat{Q}_{j}(k)&\text{if $k\notin H_{q}$ }\\ A\tilde{x}_{j}(k-1)&\text{if $k\in H_{q}$}\end{array}\right.$

(27)

in which $k=1,2,\cdots$, $j\in\{i\}\cup\mathcal{N}_{i}$ with $\tilde{x}_{j}(0)=0$. Similarly, (17) and (21) can be written as

$\displaystyle\hat{Q}_{j}(k)=Q_{R}\left(\frac{\hat{x}_{j}(k)-A\tilde{x}_{j}(k-1)}{\theta(k-1)}\right),\,\,k=1,2,\cdots.$

(28)

The switched-type estimator in (27) has the following motivations. The first equation in (27) is for acquiring the quantized information of $\hat{x}_{j}(k)$ under successful transmissions, and then calculates $\tilde{x}_{j}(k)$. This step is also necessary for the DoS-free case [20]. The second equation in (27) is an open loop estimation, and it together with the second equation in (24) can decouple the state of the agents and the quantization errors (see Case d) later).

A key parameter in (28) is the scaling parameter $\theta(k-1)$. By adjusting its value dynamically, the variable to be quantized will be kept within the bounded quantization range without saturation. The scaling parameter $\theta(k)\in\mathbb{R}_{>0}$ is updated 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.\,\,\,k=1,2,\cdots$

(31)

with $\theta(0)=\theta_{0}\in\mathbb{R}_{>0}$. The parameters $\gamma_{1}$ and $\gamma_{2}$ are the so-called zooming-in and out factors in dynamic quantization, respectively. Since $\theta(k)$ in the encoders and decoders has the same initial condition $\theta(0)$, and $k\in H_{q}$ or $k\notin H_{q}$ is passively known as mentioned before, one can see that $\theta(k)$ is synchronized in all the encoders and decoders. The zooming-in and out mechanism in (31) is majorly inspired by [4, 7] studying centralized systems. The scaling parameter is a dynamic sequence whose increasing and decreasing depend on DoS attacks in order to mitigate the influence of DoS-induced packet losses. If one assumes that each agent is aware of its own $J_{i}$ and their neighbors’ $J_{j}$ ($j\in\mathcal{N}_{i}$), it is possible to design distributed zooming-in factors, namely $\gamma_{1}^{i}$. Then the new $\theta(k)$ and $\gamma_{1}$ are complicated and will be a vector and a matrix, respectively. This case will be left for future research. The design of zooming-out factor may not change since it is dependent on $A$ (see Lemma 2 later).

During DoS intervals, $\hat{x}_{j}(k)-A\tilde{x}_{j}(k-1)$ in (28) may diverge. Therefore, the scaling parameter $\theta(k-1)$ must increase using $\gamma_{2}$ so that $Q_{R}(\cdot)$ is not saturated. If the transmissions succeed, the quantizers zoom in and $\theta(k)$ decreases by using $\gamma_{1}$. The selections of $\gamma_{1}$ and $\gamma_{2}$ will be specified later, and one of the objectives of this paper is to find a $\gamma_{2}$ as tight as possible. If one assumes that the communication network is free from any DoS-induced or random packet losses, then the zooming-out mechanism is not necessary since $\hat{x}_{j}(k)-A\tilde{x}_{j}(k-1)$ in $Q_{R}(\cdot)$ does not diverge. In this case, one only needs the zooming-in mechanism to ensure the property of asymptotic convergence [20, 22].

By defining vectors $\tilde{x}(k)=[\tilde{x}_{1}^{T}(k)\cdots\tilde{x}_{N}^{T}(k)]^{T},\hat{x}(k)=[\hat{x}_{1}^{T}(k)\cdots\hat{x}_{N}^{T}(k)]^{T},Q(k)=[\hat{Q}_{1}^{T}(k)\cdots\hat{Q}_{N}^{T}(k)]^{T}\in\mathbb{R}^{nN}$, one can obtain the compact form of (27) as

$\displaystyle\tilde{x}(k)=\left\{\begin{array}[]{ll}A_{N}\tilde{x}(k-1)+\theta(k-1)Q(k)&\text{if $k\notin H_{q}$}\\ A_{N}\tilde{x}(k-1)&\text{if $k\in H_{q}$}\end{array}\right.$

(34)

for $k=1,2,\cdots$. The compact form of the observer in (III-A) is

	$\displaystyle\hat{x}(k)=$	$\displaystyle A_{N}\hat{x}(k-1)+B_{N}u(k-1)$
		$\displaystyle+F_{N}(y(k-1)-C_{N}\hat{x}(k-1))$		(35)

in which $A_{N}:=I_{N}\otimes A$, $B_{N}:=I_{N}\otimes B$, $C_{N}:=I_{N}\otimes C$, $F_{N}:=I_{N}\otimes F$, $u(k)=[u_{1}^{T}(k)\cdots u_{N}^{T}(k)]^{T}\in\mathbb{R}^{Nw}$ and $y(k)=[y_{1}^{T}(k)\cdots y_{N}^{T}(k)]^{T}\in\mathbb{R}^{Nv}$. Let $x(k)=[x_{1}^{T}(k)\cdots x_{N}^{T}(k)]^{T}\in\mathbb{R}^{nN}$. With the vectors $x(k),\tilde{x}(k)$ and $\hat{x}(k)$, we define the following errors in vector form

	$\displaystyle\!\!\!\!e_{c}(k)\!=\!\hat{x}(k)\!-\!\tilde{x}(k)\!=\![\hat{x}_{1}^{T}(k)\!-\!\tilde{x}_{1}^{T}(k)\!\cdots\!\hat{x}_{N}^{T}(k)\!-\!\tilde{x}_{N}^{T}(k)]^{T}$
	$\displaystyle\!\!\!\!e_{o}(k)\!=\!x(k)\!-\!\hat{x}(k)\!=\![x_{1}^{T}(k)\!-\!\hat{x}_{1}^{T}(k)\!\cdots\!x_{N}^{T}(k)\!-\!\hat{x}_{N}^{T}(k)]^{T}$

in which $e_{c}(k)$ denotes the error due to signal coding and $e_{o}(k)$ denotes the observer error. From (1), one can see that the update of $x(k)$ is independent on $k\in H_{q}$ or $k\notin H_{q}$. However, the update of $x(k)$ depends on $k-1\in H_{q}$ or $k-1\notin H_{q}$, due to the control law $u_{i}(k-1)$ in (24). Note that such dependence does not exist in [24]. One can obtain the compact form of the closed-loop dynamics of $x(k)$ as

		$\displaystyle x(k)=$
		$\displaystyle\left\{\!\!\!\!\begin{array}[]{ll}Gx(k-1)\!+\!L(e_{o}(k-1)\!+\!e_{c}(k-1))&\!\!\!\!\text{if}\,k\!-\!1\!\notin\!H_{q}\\ A_{N}x(k-1)&\!\!\!\!\text{if}\,k\!-\!1\!\in\!H_{q}\end{array}\right.$		(38)

where $G:=A_{N}-L_{\mathcal{G}}\otimes BK$ and $L:=L_{\mathcal{G}}\otimes BK.$ Let the discrepancy between the state of agent $i$ and $\overline{x}(k)$ be $\delta_{i}(k):=x_{i}(k)-\overline{x}(k)\in\mathbb{R}^{n}$ and let $\delta(k)=[\delta_{1}^{T}(k)\,\,\cdots\,\,\delta_{N}^{T}(k)]^{T}\in\mathbb{R}^{nN}$. Then one can obtain the compact dynamics of $\delta(k)$:

		$\displaystyle\delta(k)=$
		$\displaystyle\left\{\!\!\!\!\begin{array}[]{ll}G\delta(k-1)\!+\!L(e_{o}(k-1)\!+\!e_{c}(k-1))&\!\!\!\!\text{if}\,k\!-\!1\!\notin\!H_{q}\\ A_{N}\delta(k-1)&\!\!\!\!\text{if}\,k\!-\!1\!\in\!H_{q}.\end{array}\right.\!\!\!\!$		(41)

It is clear that if $\delta(k)\to 0$ as $k\to\infty$, consensus of the multi-agent system is achieved as in (2). If the multi-agent system is subject to DoS, $\delta(k)$ has a diverging mode by the second equation in (III-A), and the dynamics of $e_{c}(k)$ and $e_{o}(k)$ are not clear, which implies that consensus may not be achieved.

Note that the control system in this paper looks similar but different from the common “to zero” static controller, i.e., the control input $u_{i}(k)$ is set to zero under DoS attacks, see [10]. The controller design in our paper is also different from the “pure” prediction based controller in lossy networks (i.e., $u_{i}(k)$ is always computed based on the state prediction) in [24, 11]. In our paper, the decoder (27) generates $\hat{x}_{j}(k)$ ($j\in\{i\}\cup\mathcal{N}_{i}$) regardless of the presence of DoS. Meanwhile, the input $u_{i}(k)$ is set to zero and does not depend on the estimated state $\hat{x}_{j}(k)$ when $k\in H_{q}$, though $\hat{x}_{j}(k)$ is available. However, computing $\hat{x}_{j}(k)$ under DoS is necessary since its value is useful when DoS is over. Consequently, as will be shown later, the technical analysis involves four modes by expressing the overall system as a switched system. By contrast, the analysis in [24] needs to consider only two modes.

Though the update of $\delta(k)$ in (III-A) depends on the previous step, i.e., $k-1\notin H_{q}$ or $k-1\in H_{q}$, the update of $e_{c}(k)$ is affected by both the $k-1$ and $k$-th steps. This is because $x(k)$ in (III-A) depends on $k-1\notin H_{q}$ or $k-1\in H_{q}$, and $\hat{x}(k)$ in (34) depends on $k\notin H_{q}$ or $k\in H_{q}$. To make the technical analysis approachable, we conduct the analysis by four cases:

$\displaystyle\begin{array}[]{ll}\text{a})\,k\notin H_{q}$ and $k-1\notin H_{q}&\text{b})\,k\notin H_{q}$ and $k-1\in H_{q}\\ \text{c})\,k\in H_{q}$ and $k-1\notin H_{q}&\text{d})\,k\in H_{q}$ and $k-1\in H_{q}.\end{array}$

Case a) In view of $\tilde{x}(k)$ in the first equation of (34) and $\hat{x}(k)$ in (III-A), one can obtain $e_{c}(k)=\hat{x}(k)-A_{N}\tilde{x}(k-1)-\theta(k-1)Q_{R}\left(\frac{\hat{x}(k)-A_{N}\tilde{x}(k-1)}{\theta(k-1)}\right)$, in which we have

		$\displaystyle\hat{x}(k)-A_{N}\tilde{x}(k-1)$
		$\displaystyle=A_{N}e_{c}(k-1)+B_{N}u(k-1)+F_{N}C_{N}e_{o}(k-1)$
		$\displaystyle=He_{c}(k-1)-L\delta(k-1)+Pe_{o}(k-1)$		(42)

with $H:=A_{N}+L_{\mathcal{G}}\otimes BK$ and $P:=L-F_{N}C_{N}$. One can also compute the estimation error in the observer

$\displaystyle e_{o}(k)=(A_{N}-F_{N}C_{N})e_{o}(k-1)$

(43)

in which the spectral radius $\rho(A_{N}-F_{N}C_{N})<1$ due to $\rho(A-FC)<1$. Recall the update of $\delta(k)$ for $k-1\notin H_{q}$ in (III-A). Then by (III-A)–(43), one obtains the dynamics of $\delta(k)$, $e_{c}(k)$ and $e_{o}(k)$ as follows:

	$\displaystyle\delta(k)=G\delta(k-1)+L(e_{o}(k-1)+e_{c}(k-1))$		(44a)
	$\displaystyle e_{c}(k)=\,He_{c}(k-1)-L\delta(k-1)+Pe_{o}(k-1)$
	$\displaystyle\!\!-\!\theta(k\!-\!1)Q_{R}\!\left(\!\!\frac{He_{c}(k-1)\!-\!L\delta(k-1)\!+\!Pe_{o}(k-1)}{\theta(k-1)}\!\!\right)$		(44b)
	$\displaystyle e_{o}(k)=(A_{N}-F_{N}C_{N})e_{o}(k-1).$		(44c)

Case b) In view of $\tilde{x}(k)$ in the first equation in (34) and $\hat{x}(k)$ in (III-A), where $u(k-1)=0$ due to $k-1\in H_{q}$, one can obtain that $e_{c}(k)=A_{N}e_{c}(k-1)+F_{N}C_{N}e_{o}(k-1)-\theta(k-1)Q_{R}\left(\frac{A_{N}e_{c}(k-1)+F_{N}C_{N}e_{o}(k-1)}{\theta(k-1)}\right).$ The dynamics of $\delta(k)$ for Case b) follows from the second equation in (III-A). The control input applied to the observer is also $u(k-1)=0$ due to $k-1\in H_{q}$, which implies that (44c) still holds. With $e_{c}(k)$ above, one obtains the dynamics of $\delta(k)$, $e_{c}(k)$ and $e_{o}(k)$ as

	$\displaystyle\delta(k)=A_{N}\delta(k-1)$		(45a)
	$\displaystyle e_{c}(k)=A_{N}e_{c}(k-1)+F_{N}C_{N}e_{o}(k-1)$
	$\displaystyle\,\,-\theta(k-1)Q_{R}\left(\frac{A_{N}e_{c}(k-1)+F_{N}C_{N}e_{o}(k-1)}{\theta(k-1)}\right)$		(45b)
	$\displaystyle e_{o}(k)=(A_{N}-F_{N}C_{N})e_{o}(k-1).$		(45c)

Case c) By substituting the dynamics of $\tilde{x}(k)$ in the second equation of (34), one has the evolution of $e_{c}(k)$ as $e_{c}(k)=A_{N}e_{c}(k-1)+B_{N}u(k-1)+F_{N}C_{N}e_{o}(k-1)=He_{c}(k-1)-L\delta(k-1)+Pe_{o}(k-1).$ The dynamics of $e_{o}(k)$ in (43) also holds in this case. In view of the dynamics of $\delta(k)$ in the first equation of (III-A), overall one can obtain that

	$\displaystyle\delta(k)$	$\displaystyle=G\delta(k-1)+L(e_{o}(k-1)+e_{c}(k-1))$		(46a)
	$\displaystyle e_{c}(k)$	$\displaystyle=He_{c}(k-1)-L\delta(k-1)+Pe_{o}(k-1)$		(46b)
	$\displaystyle e_{o}(k)$	$\displaystyle=(A_{N}-F_{N}C_{N})e_{o}(k-1).$		(46c)

Case d) For computing $e_{c}(k)$, by $\tilde{x}(k)$ in the second equation of (34) and $\hat{x}(k)$ in (III-A) with $u(k-1)=0$ due to $k-1\in H_{q}$, one can have $e_{c}(k)=A_{N}\hat{x}(k-1)+F_{N}C_{N}e_{o}(k-1)-A_{N}\tilde{x}(k-1)=A_{N}e_{c}(k-1)+F_{N}C_{N}e_{o}(k-1).$ The dynamics of $e_{o}(k)$ in (43) still holds in this case. Then, by combining the dynamics of $\delta(k)$ in the second equation of (III-A), we can obtain that

	$\displaystyle\delta(k)$	$\displaystyle=A_{N}\delta(k-1)$		(47a)
	$\displaystyle e_{c}(k)$	$\displaystyle=A_{N}e_{c}(k-1)+F_{N}C_{N}e_{o}(k-1)$		(47b)
	$\displaystyle e_{o}(k)$	$\displaystyle=(A_{N}-F_{N}C_{N})e_{o}(k-1).$		(47c)