Estimation of network traffic status and switching control of networked control systems with data dropout

Abstract

The recent development of the communication technology accelerates studies of real‐time networked control systems using networks. The data dropout is essentially unavoidable, especially in wireless networks and it results from transmission errors and network traffic congestion. Multiple time‐varying network traffic status given by discrete‐time homogeneous Markov chains is assumed. The authors estimate the network traffic status characterised by the probability matrix of the Markov chain online from the data dropout history. According to the estimation of network traffic status, an appropriate controller is selected to improve the control performance. The effectiveness of the proposed method is verified through simulations and experiments.

INTRODUCTION

The recent development of the communication technology accelerates studies of real-time networked control systems (NCSs) using networks [1–6]. Unlike the traditional point-to-point wiring control, NCSs can reduce the wiring cost. In addition, installation and monitoring can be simplified, and system flexibility can also be improved. Therefore, NCSs have been found in a wide range of applications, such as teleoperated medical fields, robotics, unmanned aerial/guided vehicles, and many industries [7–16].

Real-time networks suffer from network-induced problems, such as data dropout [17, 18], delays [19, 20], quantization [21, 22], and cyber-attacks [23, 24], all of which deteriorate the control performance of the NCSs. In particular, data dropout is unavoidable, especially in wireless networks, and it results from transmission errors and network traffic congestion [25, 26]. The effect of data dropout is more critical than that of delayed data, since it requires the increase of updating intervals with a multiple of the sampling period [26]. In addition, occasional delayed data, which is not successfully transmitted within the sampling period, can be regarded as the data dropout [26]. Thus, we limit ourselves to the data dropout of NCSs in this study.

The stabilisation methods of NCSs with data dropout are categorised into online and offline approaches. In the offline approach, a controller is designed without consideration of practical data dropout. For example, in Ref. [27], $H_2$/$H_{\infty }$ controllers synthesis is considered under assumptions that the maximum data-dropout upper bound is known and the occurrence of the data dropout is modelled as an enlarged delay between consecutive data. The data dropout is assumed to follow the normal distribution, and state-feedback and output-feedback controllers are designed for the data dropout and quantization in Ref. [28]. In Ref. [29], a sufficient condition for stability and controller synthesis is addressed for a class of NCSs with delay, data disordering, and data dropout. In the work, a new data reordering method is considered to deal with data disordering and choose the latest control input. The second framework for stabilisation methods of NCSs with data dropout is the online framework, where the control input is calculated online based on the actual data dropout, even if the controller is designed offline. In Ref. [30], the designed $H_{\infty }$ filter is regarded as a Markov jump linear filter, and the filter gain is switched according to the data dropout. The model predictive control for the model of NCSs with data quantization and dropout is addressed [31]. In Ref. [32], a packetized predictive controller is applied to a non-linear plant with unbounded disturbance and a Markov data dropout process. However, NCSs with time-varying network traffic have not been sufficiently studied.

In this study, we address an online estimation of network traffic status for the NCSs with data dropout and its control without compensation of data dropout in the communication side. The network traffic status changes in real time according to its usage. Data dropout occurs due to the network traffic congestion in wireless networks [25, 33]. Switching to a suitable controller in accordance with the network traffic status improves the control performance of the NCSs. We also propose a switching control based on the network traffic status estimation. Our work is mainly motivated by Ref. [34], which addresses the data dropout process and the controller synthesis. The data dropout can be modelled as a stochastic process. In this study, the occurrence of the data dropout is modelled as a Markov process. The Markov process is modelled as a discrete-time homogeneous Markov chain composed of a finite number of states and a constant transition probability matrix. A data-dropout-dependent approach allows us to derive the linear matrix inequalities (LMIs) conditions for the controller design for the NCSs with the Markov data dropout process. In Ref. [34], the control performance of the NCSs is not addressed. For practical use, we also consider the $H_2$ control performance of the NCSs. More specifically, we assume multiple time-varying network traffic status given by discrete-time homogeneous Markov chains. From the data dropout history, a probability transition matrix of the Markov chain is estimated. Using the probability transition matrices of the Markov chains, we adopt a template-matching method in the field of image processing to measure the degree of similarity or dissimilarity between the matrices so that we estimate the network traffic status characterised by the matrix online. According to the estimation of network traffic status, an appropriate controller is selected so that the control performance is improved in the presence of the data dropout. The effectiveness of the proposed method is verified through simulations and experiments. The main contribution of this study is summarised as follows: (1) online estimation of network traffic status using data dropout history, (2) switching control according to the network traffic guaranteeing stability of the closed-loop system.

The rest of this paper is organised as follows. The NCSs, a data dropout process model, and the related LMIs are described in Section 2. In Section 3, we propose a network traffic status estimation focussing on the probability transition matrices of Markov chains and selection of appropriate pre-designed controllers. The effectiveness of the proposed method is verified through simulations and experiments in Section 4. Section 5 summarises this study.

PRELIMINARIES

In this study, $\mathbb{R}$ and $\mathbb{Z}$ stand for the set of real numbers and integers, respectively. ${\mathbb{R}}_{\ge 0}$ and ${\mathbb{Z}}_{\ge 0}$ stand for the set of non-negative real numbers and non-negative integers, respectively. ${\mathbb{R}}^n$ stands for the $n$-dimensional Euclidean space and ${\mathbb{R}}^{m\times n}$ for the set of $m\times n$ real matrices. The set of $n\times n$ real positive definite matrices is denoted by ${\mathbb{S}}^{+}$. For real symmetric matrices $X$ and $Y$, $X>Y$ denotes that $X-Y$ is positive definite. The $n\times n$ identity and $m\times n$ zero matrices are denoted by $I_n$ and $O_{m,n}$, respectively, which are also denoted by $I$ and $O$ if they are clear from the context. For any matrix $A$, ${[A]}_{ij}$ denotes the $(i,j)$-th entry of $A$. The symmetric entry of any Hermitian matrix is abbreviated by $*$. For any $x\in {\mathbb{R}}^n$, a discrete-time transfer function $G$, $\Vert x\Vert$ and $\Vert G{\Vert }_2$ denote the Euclidean norm of $x$ and $H_2$-norm of $G$, respectively.

Networked control system

Figure 1 shows the NCS considered in this study. In the NCS shown in Figure 1, the plant and the controller are connected via the network, and the data is transmitted via the network between them. Though the controller-to-plant network can be addressed in our setting, an additional assumption is required that the data dropout in the two networks can be measurable, which in turn complicates the discussion. A similar setting of a one-way network can be found in the literature [35, 36] to simplify the discussion. Thus, we consider the network only from the plant to the controller. This setting is reasonable when the network from the controller to the plant is sufficiently reliable. The controller shown in Figure 1 is composed of the estimation of network traffic status and the switching control, which is described in the next section. In Figure 1, the plant is represented as a linear discrete-time system as follows: 1 $x(t+1)=Ax(t)+Bu(t),$ 2 $z(t)=Cx(t)+Du(t),$ where $t\in {\mathbb{Z}}_{\ge 0}$, $x(t)\in {\mathbb{R}}^n$, $u(t)\in {\mathbb{R}}^m$, and $z(t)$ are the state, the input, and the evaluation output, respectively. In Equation (1), $A$ and $B$ are matrices with appropriate dimensions and ($A$, $B$)-controllable, and in Equation (2), $C={\left[\begin{array}{cc}\hfill Q_x^{\top }\hfill & \hfill O_{m,n}^{\top }\hfill \end{array}\right]}^{\top }$ and $D={\left[\begin{array}{cc}\hfill O_{n,m}^{\top }\hfill & \hfill Q_u^{\top }\hfill \end{array}\right]}^{\top }$ where $Q_x\in {\mathbb{R}}^{n\times n}$ and $Q_u\in {\mathbb{R}}^{m\times m}$.

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In Figure 1, when $x(t)$ is transmitted via the network, $x(t)$ drops out the subject to a certain probability distribution as will be described later. The estimator in Figure 1 estimates the networks traffic status $\hat{\sigma }(t)$ using the data dropout history so that an appropriate controller $K_{\hat{\sigma }(t)}$ is selected among pre-designed $K_{\sigma }$ ($\sigma \in \mathcal{Q}$), which will be described later.

We describe the data dropout of transmitted data via the networks in the NCS shown in Figure 1. The instance when data is successfully transmitted from the plant to the controller is recorded as the time series $\mathcal{T}≔⟨t_1,t_2,\text{…}⟩$. Defining $[\forall k>0]d(t_k)≔t_k-t_{k-1}$, where $t_0=-1$ for convenience, we model the data dropout process [34] as 3 $\mathcal{D}≔⟨d(t_1),d(t_2),\text{…}⟩.$

The length of the successive data dropout is assumed to be bounded by 4 $s≔\underset{k>0}{\max }d(t_k).$

If $s=1$, no data dropout occurs, while $s>1$ implies that data drops out and the number of successive data dropout is at most $s-1$. Note that $[\forall k>0]d(t_k)\in \mathcal{S}$, where $\mathcal{S}≔\{1,2,\text{…},s\}$. Figure 2 shows an example of the data dropout series. In Figure 2, $\mathcal{T}=⟨0,1,4,8,10,\text{…}⟩$ and $\mathcal{D}=⟨1,1,3,4,2,\text{…}⟩$

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The data dropout process in Equation (3) is regarded as a Markov process [34] if it is a discrete-time homogeneous Markov chain of which the state space is $\mathcal{S}$ on a complete probability space with a constant transition probability matrix $\mathrm{\Pi }\in {\mathbb{R}}^{s\times s}$ satisfying 5 $[\forall i\in \mathcal{S}]\sum _{j=1}^s{[\mathrm{\Pi }]}_{ij}=1,{[\mathrm{\Pi }]}_{ij}\ge 0.$

If $[\forall i\in \mathcal{S}]d(t_k)=i$, then $[\forall k>0][\forall i,j\in \mathcal{S}]\mathrm{Pr}(d(t_{k+1})=j|d(t_k)=i)={[\mathrm{\Pi }]}_{ij}$.

Assume that the data dropout follows the Markov chain of which transition probability is 6 $\mathrm{\Pi }=\left[\begin{array}{ccc}\hfill 0.6\hfill & \hfill 0.1\hfill & \hfill 0.3\hfill \\ \hfill 0.7\hfill & \hfill 0.0\hfill & \hfill 0.3\hfill \\ \hfill 0.6\hfill & \hfill 0.1\hfill & \hfill 0.3\hfill \end{array}\right].$

Then, for example, we have $[\forall k>0]\mathrm{Pr}(d(t_{k+1})=2|d(t_k)=1)=0.1$ and $[\forall k>0]\mathrm{Pr}(d(t_{k+1})=1|d(t_k)=2)=0.7$.

For a transition matrix $\mathrm{\Pi }\in {\mathrm{R}}^{s\times s}$, consider $v\in {\mathrm{R}}^s$, which satisfies ${\mathrm{\Pi }}^{\top }v=v$ and ${\sum }_{i=1}^s{v}_i=1$ and $[\forall i\in \mathcal{S}]v_i\ge 0$. From the property of $\mathrm{\Pi }$, $v$ is the steady-state probability distribution of the interval of data dropout. Note that $v_1$ is the probability of no data dropout while $v_i$ ($i=2,\text{…},s$) indicates the probability that the interval of the two contiguous successfully received data is $i$ in the steady state. Thus, for a given $\mathrm{\Pi }\in {\mathrm{R}}^{s\times s}$, the average interval of the received data, denoted by $\mu$, is calculated by $\mu ={\sum }_{i=1}^{s}i{v}_i$. For example, for Equation (6), $v={\left[\begin{array}{ccc}\hfill 0.61\hfill & \hfill 0.09\hfill & \hfill 0.30\hfill \end{array}\right]}^{\top }$ and $\mu =1.69$.

In this study, we consider $q$ data dropout processes to represent different network traffic status. Each data dropout process is modelled by its corresponding Markov chain. The network traffic status changes according to its usage. Such a change is represented by the transition from one Markov model to another. The transition rule, such as a transition probability, is not necessarily given in advance.

For convenience of later discussion, we define $\mathcal{Q}≔\{1,2,\text{…},q\}$, and 7 $\mathcal{A}(\sigma )≔\{\begin{array}{cc}\{1,2\}\hfill & (\sigma =1),\hfill \\ \{\sigma -1,\sigma ,\sigma +1\}\hfill & (\sigma \in \mathcal{Q}\backslash \{1,q\}),\hfill \\ \{q-1,q\}\hfill & (\sigma =q).\hfill \end{array}$

We assume the following conditions on the network traffic status:

$1$

Assumption

(Network traffic status transition)

• The data dropout is subject to one among $q$ ($q>1$) finite Markov chains. The network traffic status at $t$ is denoted by $\sigma (t)\in \mathcal{Q}$

• The network traffic status is given by the Markov model for which ${\mathrm{\Pi }}_{\sigma }$ ($\sigma \in \mathcal{Q}$) is given in advance

• For ${\mathrm{\Pi }}_{\sigma }$ ($\sigma \in \mathcal{Q}$), ${\mu }_1<{\mu }_2<\text{⋯}<{\mu }_q$, which implies that the average interval of the successively received data ${\mu }_{\sigma }$ ($\sigma \in \mathcal{Q}$) becomes large, that is, the data is likely to drop as $\sigma$ becomes large

• For any $\sigma \in \mathcal{Q}$, if $\sigma (t)=\sigma$, then $\sigma (t+1)\in \mathcal{A}(\sigma )$, which implies that the network traffic status remains or shifts to the neighbouring one, that is, the data dropout tendency changes gradually

• If $\sigma (t+1)\ne \sigma (t)$, then $\sigma (t+1)=\sigma (t+2)=\text{⋯}=\sigma (t+\tau )$, where $\tau$ is a sufficiently large positive integer, which implies a not-so-frequent change of the network traffic status, that is, the network status remains for a sufficiently long time after the change

$2$

Assumption

(Online monitoring of the data dropout) The past data dropout history, which is a subsequence of $\mathcal{D}$, is available at $t_k$ as

8 ${\mathcal{D}}_{t_k}≔⟨d(t_{k-m_c-1}),\text{…},d(t_{k-1}),d(t_k)⟩$

where $m_c>1$.

In practice, data dropout can be measured from time-stamped data when the receiving node is synchronized through the network [26]. The online monitoring of data dropout in Assumption 2 can be easily implemented by a small-size memory in the controller shown in Figure 1.

The estimator in Figure 1 estimates the network traffic status by updating the transition probability matrix $\mathrm{\Pi }(t)\in {\mathbb{R}}^{s\times s}$ for $t_k\le t<t_{k+1}$, using ${\mathcal{D}}_{t_k}$.

In Equation (1), we consider a static state-feedback controller $u(t)=Kx(t)$ when no data drops occur. If the data drops, we have $u(t)=u(t_k)=Kx(t_k)$ for $t_k\le t<t_{k+1}$. It is reasonable that $u(t)=0$ ($0\le t<t_1$). Therefore, the closed-loop NCS becomes 9 $\begin{array}{l}x(t+1)=Ax(t)+BKx(t_k),\hfill \\ ([\forall t_k,t_{k+1}\in \mathcal{T}]t_k\le t<t_{k+1}).\hfill \end{array}$

Stabilisation of NCS with the Markov data dropout process

We review a stabilisation approach for the NCS, in which the Markov chain models the data dropout process [34]. Given $\mathrm{\Pi }$, which is the transition probability matrix of the Markov chain, there exists a state-feedback controller $K$ in Equation (9) such that NCS is mean square stable if there exist matrices $[\forall i\in \mathcal{S}]X_i\in {\mathbb{S}}^{+}$, $G\in {\mathbb{R}}^{n\times n}$, and $Y\in {\mathbb{R}}^{m\times n}$, satisfying the following LMI: 10 $[\forall i\in \mathcal{S}]\left[\begin{array}{cc}\hfill -G-G^{\top }+X_i\hfill & \hfill M_i\hfill \\ \hfill *\hfill & \hfill -\mathrm{\Lambda }\hfill \end{array}\right]<0,$ where 11 $\begin{array}{ll}\hfill & M_i=\left[\begin{array}{ccc}\hfill \sqrt{{[\mathrm{\Pi }]}_{i1}}{(AG+B_{1}Y)}^{\top }\hfill & \hfill \text{…}\hfill & \hfill \sqrt{{[\mathrm{\Pi }]}_{is}}{(A^{s}G+B_{s}Y)}^{\top }\hfill \end{array}\right],\hfill \\ \hfill & \mathrm{\Lambda }=\mathrm{diag}(X_1,\text{…},X_s),\hfill \\ \hfill & [\forall j\in \mathcal{S}]B_j=\sum _{r=0}^{j-1}{A}^{r}B.\hfill \end{array}$

The solution of Equation (10) yields $K=Y{G}^{-1}$ in Equation (9) (See [34] for details).

For convenience, we denote Equation (10) simply by 12 $[\forall i\in \mathcal{S}]T_d(X_i,G,Y;\mathrm{\Pi })<0.$

$H_2$ state-feedback controller

We consider the data-dropout-free closed-loop system of Equation (1) with a static state-feedback controller $K$, which is independent of the network, as 13 $x(t+1)=(A+BK)x(t)+B_{w}w(t),$ 14 $z(t)=(C+DK)x(t),$ where $B_w$ is considered as an initial state and $w(0)=1$ and $w(t)=0$ ($t>0$), and $z$ is the evaluation output. In Equations (13) and (14), the $H_2$ norm of $G_c$, which is the transfer function from $w$ to $z$, is a suitable index to evaluate the transient response of Equation (13). There exists a $K$ in Equations (13) and (14) such that $G_c$ is stable and $\Vert G_c{\Vert }_2<\gamma$ ($\gamma >0$) is satisfied if and only if there exist $G,Z\in {\mathbb{S}}^{+},Y\in {\mathbb{R}}^{m\times n}$, satisfying the following LMIs: 15 $\left[\begin{array}{ccc}\hfill -G\hfill & \hfill *\hfill & \hfill *\hfill \\ \hfill AG+BY\hfill & \hfill -G\hfill & \hfill *\hfill \\ \hfill CG+DY\hfill & \hfill O\hfill & \hfill -\gamma I\hfill \end{array}\right]<0,$ 16 $-\left[\begin{array}{cc}\hfill Z\hfill & \hfill *\hfill \\ \hfill B_w\hfill & \hfill G\hfill \end{array}\right]<0,$ 17 $\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(Z)-\gamma <0.$

In this case, we have $K=Y{G}^{-1}$ in Equation (14) (see Ref. [37] for details). For convenience, the LMIs in Equation (15) to (17) are simply denoted by $T_2(G,Y,Z,\gamma )<0$. The controller, which minimises $H_2$-norm of $G_c$, can be obtained by solving the following optimization: 18 $\underset{G,Z\in {\mathbb{S}}^{+},Y\in {\mathbb{R}}^{m\times n}}{\inf }\gamma$ 19 $\begin{array}{ll}\hfill & \text{subject}\text{to}\hfill \\ \hfill & T_2(G,Y,Z,\gamma )<0.\hfill \end{array}$

CONTROLLER SYNTHESIS, ESTIMATION OF NETWORK TRAFFIC STATUS AND SWITCHING CONTROL

The network traffic varies in accordance with the use of the communication condition. The data dropout is likely to occur when the traffic is congested. In accordance with the network traffic status, it is necessary to select a suitable controller to improve the control performance. For this purpose, we propose an estimation of the network traffic status. The switching control based on the estimation method is also proposed.

Controller synthesis for each network traffic status

As described in 2.2, for a given $\mathrm{\Pi }$, a feedback controller $K$ is obtained such that Equation (9) is stable. Recall that we consider multiple ${\mathrm{\Pi }}_{\sigma }$ ($\sigma \in \mathcal{Q}$) in this study. Although $K_{\sigma }$ can be designed for a given ${\mathrm{\Pi }}_{\sigma }$ ($\sigma \in \mathcal{Q}$), it is not guaranteed that $K_{\sigma }$ stabilises Equation (9) for a different ${\mathrm{\Pi }}_{\overline{\sigma }}$ ($\overline{\sigma }\in \mathcal{Q}\backslash \{\sigma \}$). Since $\sigma (t)=\sigma$, then $\sigma (t+1)\in \mathcal{A}(\sigma )$ from Assumption 1 and Equation (9) is stabilised if it is stabilised for ${\mathrm{\Pi }}_{\overline{\sigma }}$ ($\overline{\sigma }\in \mathcal{Q}\backslash \{\sigma \}$) by $K_{\sigma }$ even if the network traffic changes from ${\mathrm{\Pi }}_{\sigma }$ to ${\mathrm{\Pi }}_{\overline{\sigma }}$ ($\overline{\sigma }\in \mathcal{Q}\backslash \{\sigma \}$). Such a controller can be obtained by considering the simultaneous LMIs in Equation (12).

Apart from the switching control according to the network traffic status, the control performance of the NCSs should be considered. This control performance was not addressed in Ref. [34], since the systems may be affected by modelling uncertainties in practice. Therefore, the $H_2$ index is incorporated as an additional control performance index.

Formally, $[\forall \sigma \in \mathcal{Q}]K_{\sigma }$ can be obtained by the following optimization problem with simultaneous LMIs: 20 $\underset{[\forall i\in \mathcal{S}]X_i,G,Z\in {\mathbb{S}}^{+},Y\in {\mathbb{R}}^{m\times n}}{\inf }\gamma$ 21 $\begin{array}{ll}\hfill & \text{subject}\text{to}\hfill \\ \hfill & [\forall \stackrel{\sim }{\sigma }\in \mathcal{A}(\sigma )][\forall i\in \mathcal{S}]T_d(X_i,G,Y;{\mathrm{\Pi }}_{\stackrel{\sim }{\sigma }})<0,\hfill \end{array}$ 22 $T_2(G,Y,Z,\gamma )<0.$

If the above optimization problem is feasible, the controller is given by $K_{\sigma }=Y{G}^{-1}$. Note that the controller $K_{\sigma }$ stabilises Equation (9) if the data dropout is subject to not only ${\mathrm{\Pi }}_{\sigma }$ but also ${\mathrm{\Pi }}_{\overline{\sigma }}$ ($\overline{\sigma }\in \mathcal{A}(\sigma )\backslash \{\sigma \}$). For example, $K_2$ stabilises Equation (9) even if the data dropout is subject to not only ${\mathrm{\Pi }}_2$ but also ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_3$.

Estimation of network traffic status

Using Equation (8), which is the past data dropout history and is available at $t$ from Assumption 2, we estimate the network traffic status at $t$ characterised by $\hat{\mathrm{\Pi }}(t)$ as a probability transition of a Markov process. To simplify discussion, $\sigma (0)=1$. For Equation (8), let $[\forall i,j\in \mathcal{S}]n_{ij}$ be the total number of the tuple of $(d(t_{\kappa -1}),d(t_{\kappa }))=(i,j)$ ($\kappa =k-m_c,\text{…},k$). We estimate the network traffic status at $t$ that is characterised by $\hat{\mathrm{\Pi }}(t)$ for $t_k\le t<t_{k+1}$ as 23 $[\forall i,j\in \mathcal{S}]{[\hat{\mathrm{\Pi }}(t)]}_{ij}=\frac{n_{ij}}{m_c-1}.$

We introduce the zero-means normalized cross-correlation (ZNCC) that is widely used in image processing [38], between $\hat{\mathrm{\Pi }}(k)$ and ${\mathrm{\Pi }}_{\sigma }$ ($\sigma \in \mathcal{Q}$) to evaluate the similarity between them. For two probability transition matrices ${\mathrm{\Pi }}^{(1)},{\mathrm{\Pi }}^{(2)}\in {\mathbb{R}}^{s\times s}$, the ZNCC of them is defined as follows: 24 $\begin{array}{l}{z}_{\text{ZNCC}}({\mathrm{\Pi }}^{(1)},{\mathrm{\Pi }}^{(2)})\hfill \\ ≔\frac{\sum _{j=1}^s\sum _{i=1}^s\left({[{\mathrm{\Pi }}^{(1)}]}_{ij}-{\overline{\mathrm{\Pi }}}^{(1)}\right)\left({[{\mathrm{\Pi }}^{(2)}]}_{ij}-{\overline{\mathrm{\Pi }}}^{(2)}\right)}{\sqrt{\sum _{j=1}^s\sum _{i=1}^s{\left({[{\mathrm{\Pi }}^{(1)}]}_{ij}-{\overline{\mathrm{\Pi }}}^{(1)}\right)}^2}\sqrt{\sum _{j=1}^s\sum _{i=1}^s{\left({[{\mathrm{\Pi }}^{(2)}]}_{ij}-{\overline{\mathrm{\Pi }}}^{(2)}\right)}^2}},\hfill \end{array}$ where $\overline{\mathrm{\Pi }}=\frac{1}{s^2}\sum _{j=1}^s\sum _{i=1}^s{[\mathrm{\Pi }]}_{ij}$. It is clear that $z_{\text{ZNCC}}({\mathrm{\Pi }}^{(1)},{\mathrm{\Pi }}^{(2)})=z_{\text{ZNCC}}({\mathrm{\Pi }}^{(2)},{\mathrm{\Pi }}^{(1)})$. In particular, $z_{\text{ZNCC}}(\mathrm{\Pi },\mathrm{\Pi })=1$. Notice that $-1\le z_{\text{ZNCC}}(\cdot ,\cdot )\le 1$. When the value of $z_{\text{ZNCC}}({\mathrm{\Pi }}^{(1)},{\mathrm{\Pi }}^{(2)})$ is close to 1, ${\mathrm{\Pi }}^{(1)}$ and ${\mathrm{\Pi }}^{(2)}$ can be regarded as similar to each other.

We also introduce another index for two probability transition matrices ${\mathrm{\Pi }}^{(1)},{\mathrm{\Pi }}^{(2)}\in {\mathbb{R}}^{s\times s}$, that is, the sum of absolute difference (SAD) between them is as follows: 25 $z_{\text{SAD}}({\mathrm{\Pi }}^{(1)},{\mathrm{\Pi }}^{(2)})≔\sum _{j=1}^s\sum _{i=1}^s\left|{[{\mathrm{\Pi }}^{(1)}]}_{ij}-{[{\mathrm{\Pi }}^{(2)}]}_{ij}\right|.$

It is clear that $z_{\text{SAD}}({\mathrm{\Pi }}^{(1)},{\mathrm{\Pi }}^{(2)})=z_{\text{SAD}}({\mathrm{\Pi }}^{(2)},{\mathrm{\Pi }}^{(1)})$. In particular, $z_{\text{SAD}}(\mathrm{\Pi },\mathrm{\Pi })=0$. Notice that $0\le z_{\text{SAD}}(\cdot ,\cdot )\le 2s$. Unlike $z_{\text{ZNCC}}(\cdot ,\cdot )$, if $z_{\text{SAD}}(\cdot ,\cdot )$ is close to 0, then ${\mathrm{\Pi }}^{(1)}$ and ${\mathrm{\Pi }}^{(2)}$ can be regarded as similar to each other, while the two matrices become dissimilar to each other as $z_{\text{SAD}}(\cdot ,\cdot )$ becomes large.

In image processing, the ZNCC is widely used as an index of collation of two images under the influence of shading, gain, and brightness offset fluctuation [39]. Since the network traffic status is characterised by the transition probability matrix of the Markov chain, the ZNCC can be a candidate of the similarity index between different transition probability matrices by addressing the transition probability matrices as images. The SAD, which is also being used in digital image processing, can be used as a dissimilarity index between different transition probability matrices.

Consider the following $q=3$ data dropout processes of which transition probability matrices are given by 26 $\begin{array}{ll}\hfill & {\mathrm{\Pi }}_1=\left[\begin{array}{cccc}\hfill 0.6\hfill & \hfill 0.1\hfill & \hfill 0.3\hfill & \hfill 0\hfill \\ \hfill 0.7\hfill & \hfill 0\hfill & \hfill 0.3\hfill & \hfill 0\hfill \\ \hfill 0.6\hfill & \hfill 0.1\hfill & \hfill 0.3\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],{\mathrm{\Pi }}_2=\left[\begin{array}{cccc}\hfill 0.4\hfill & \hfill 0.25\hfill & \hfill 0.1\hfill & \hfill 0.25\hfill \\ \hfill 0.4\hfill & \hfill 0.25\hfill & \hfill 0.1\hfill & \hfill 0.25\hfill \\ \hfill 0.4\hfill & \hfill 0.25\hfill & \hfill 0.1\hfill & \hfill 0.25\hfill \\ \hfill 0.4\hfill & \hfill 0.25\hfill & \hfill 0.1\hfill & \hfill 0.25\hfill \end{array}\right],\hfill \\ \hfill & {\mathrm{\Pi }}_3=\left[\begin{array}{cccc}\hfill 0.2\hfill & \hfill 0.25\hfill & \hfill 0.4\hfill & \hfill 0.15\hfill \\ \hfill 0.2\hfill & \hfill 0.25\hfill & \hfill 0.4\hfill & \hfill 0.15\hfill \\ \hfill 0.2\hfill & \hfill 0.25\hfill & \hfill 0.4\hfill & \hfill 0.15\hfill \\ \hfill 0.2\hfill & \hfill 0.25\hfill & \hfill 0.4\hfill & \hfill 0.15\hfill \end{array}\right],\hfill \end{array}$ where the averages of the data dropout are given as ${\mu }_1=1.7$, ${\mu }_2=2.2$, and ${\mu }_3=2.5$. ${[Z_{\text{ZNCC}}]}_{ij}≔z_{\text{ZNCC}}({\mathrm{\Pi }}_i,{\mathrm{\Pi }}_j)$ and ${[Z_{\text{SAD}}]}_{ij}≔z_{\text{SAD}}({\mathrm{\Pi }}_i,{\mathrm{\Pi }}_j)$ are calculated as follows: 27 $Z_{\text{ZNCC}}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0.58\hfill & \hfill -0.22\hfill \\ \hfill *\hfill & \hfill 1\hfill & \hfill -0.76\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill 1\hfill \end{array}\right],Z_{\text{SAD}}=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 3.8\hfill & \hfill 4.2\hfill \\ \hfill *\hfill & \hfill 0\hfill & \hfill 2.4\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill 0\hfill \end{array}\right].$

From $Z_{\text{ZNCC}}$ in Equation (27), for example, we judge that ${\mathrm{\Pi }}_1$ is more similar to ${\mathrm{\Pi }}_2$ than ${\mathrm{\Pi }}_3$ in Equation (26). Noticing that $Z_{\text{SAD}}$ in Equation (27) is used as a dissimilarity index, we judge that ${\mathrm{\Pi }}_1$ is less dissimilar to ${\mathrm{\Pi }}_2$ than ${\mathrm{\Pi }}_3$ in Equation (26).

We select one among ${\mathrm{\Pi }}_{\sigma }$ ($\sigma \in \mathcal{Q}$) at $t$ to which $\hat{\mathrm{\Pi }}(t)$ is most similar, using $z_{\text{ZNCC}}({\mathrm{\Pi }}_{\sigma },\hat{\mathrm{\Pi }}(t))$ or $z_{\text{SAD}}({\mathrm{\Pi }}_{\sigma },\hat{\mathrm{\Pi }}(t))$. When $z_{\text{ZNCC}}(\cdot ,\cdot )$ is adopted, the estimated network traffic status index at $t$, denoted by $\hat{\sigma }(t)\in \mathcal{Q}$, is given as follows: 28 $\begin{array}{l}\hat{\sigma }(t)\hfill \\ =\left\{\begin{array}{cc}\sigma \hfill & \text{if}[\sigma \in \mathcal{A}(\hat{\sigma }(t-1))]\hfill \\ \hfill & \wedge [[\forall \overline{\sigma }\in \mathcal{A}(\hat{\sigma }(t-1))\backslash \{\sigma \}]\hfill \\ \hfill & z_{\text{ZNCC}}({\mathrm{\Pi }}_{\sigma },\hat{\mathrm{\Pi }}(t))>{\alpha }_{\text{ZNCC}}{z}_{\text{ZNCC}}({\mathrm{\Pi }}_{\overline{\sigma }},\hat{\mathrm{\Pi }}(t))],\hfill \\ \hat{\sigma }(t-1)\hfill & \text{otherwise},\hfill \end{array}\right.\hfill \end{array}$ where ${\alpha }_{\text{ZNCC}}>1$ is a user-defined constant.

$z_{\text{SAD}}(\cdot ,\cdot )$ is adopted, noticing that it reflects the dissimilarity between the matrices, 29 $\begin{array}{l}\hat{\sigma }(t)\hfill \\ =\left\{\begin{array}{cc}\sigma \hfill & \text{if}[\sigma \in \mathcal{A}(\hat{\sigma }(t-1))]\hfill \\ \hfill & \wedge [[\forall \overline{\sigma }\in \mathcal{A}(\hat{\sigma }(t-1))\backslash \{\sigma \}]\hfill \\ \hfill & z_{\text{SAD}}({\mathrm{\Pi }}_{\sigma },\hat{\mathrm{\Pi }}(t))<{\alpha }_{\text{SAD}}{z}_{\text{SAD}}({\mathrm{\Pi }}_{\overline{\sigma }},\hat{\mathrm{\Pi }}(t))],\hfill \\ \hat{\sigma }(t-1)\hfill & \text{otherwise},\hfill \end{array}\right.\hfill \end{array}$ where ${\alpha }_{\text{SAD}}<1$ is a user-defined constant.

According to the estimation, the controller at $t$, which is obtained by Equation (20), is $K=K_{\widehat{\sigma }(t)}$.

The procedure of the design, estimation and control described in this section is summarised as follows:

• Give a plant model in Equations (1) and (2)

• Give $q$ network traffic models in Equation (5), that is, ${\mathrm{\Pi }}_{\sigma }$ ($\sigma \in \mathcal{Q}=\{1,2,\text{…},q\}$)

• Design $q$ feedback controllers, that is, $K_{\sigma }$ ($\sigma \in \mathcal{Q}$) for ${\mathrm{\Pi }}_{\sigma }$ using Equations (20)–(22)

• Set the closed-loop system of the plant and $K_1$ including networks where ${\mathrm{\Pi }}_1$ is used at $t=0$.

• Monitor the data dropout history of the closed-loop system, record it as Equation (8) and obtain $\hat{\mathrm{\Pi }}(t)$ in Equation (23).

• Calculate

  • (a)

    the similarities of $\hat{\mathrm{\Pi }}(t)$ and ${\mathrm{\Pi }}_{\sigma }$ ($\sigma \in \mathcal{Q}$) using ZNCC in Equation (24), or

  • (b)

    the dissimilarities of $\hat{\mathrm{\Pi }}(t)$ and ${\mathrm{\Pi }}_{\sigma }$ ($\sigma \in \mathcal{Q}$) using SAD in Equation (25)

• Estimate the network traffic status $\hat{\sigma }(t)$ using

  • (a)

    Equation (28) if Equation (24) is used, or

  • (b)

    Equation (29) if Equation (25) is used.

• Apply the feedback controller $K=K_{\widehat{\sigma }(t)}$. Go to Step 5.

SIMULATION AND EXPERIMENT

This section verifies the effectiveness of the proposed method through simulation and experiment.

Figure 3 shows the rotary inverted pendulum driven by a dc motor, which is used as a plant for verification. Let $x={\left[\begin{array}{cccc}\hfill {\theta }_m\hfill & \hfill {\theta }_p\hfill & \hfill {\omega }_m\hfill & \hfill {\omega }_p\hfill \end{array}\right]}^{\top }$ be the state vector and $u$ be the input voltage to the dc motor. Specifically, ${\theta }_m$, ${\theta }_p$, ${\omega }_m$, and ${\omega }_p$ are the angle [rad] of the dc-motor, the angle [rad] of the pendulum, the angular velocity [rad/s] of the dc-motor, and the angular velocity [rad/s] of the pendulum, respectively. The system matrices in Equation (1) of the plant with control period $T_s=0.03$ s are given by 30 $A=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0.020\hfill & \hfill 0.023\hfill & \hfill -0.000\hfill \\ \hfill 0\hfill & \hfill 1.040\hfill & \hfill -0.007\hfill & \hfill 0.030\hfill \\ \hfill 0\hfill & \hfill 1.205\hfill & \hfill 0.556\hfill & \hfill 0.005\hfill \\ \hfill 0\hfill & \hfill 2.557\hfill & \hfill -0.443\hfill & \hfill 1.009\hfill \end{array}\right],B=\left[\begin{array}{c}\hfill 0.013\hfill \\ \hfill 0.013\hfill \\ \hfill 0.799\hfill \\ \hfill 0.796\hfill \end{array}\right]$

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For $z$ in Equation (14), $C={\left[\begin{array}{cc}\hfill Q_x^{\top }\hfill & \hfill O_{1,4}^{\top }\hfill \end{array}\right]}^{\top }$ and $D={\left[\begin{array}{cc}\hfill O_{4,1}^{\top }\hfill & \hfill Q_u^{\top }\hfill \end{array}\right]}^{\top }$, where $Q_x=\mathrm{diag}\left[\begin{array}{cccc}\hfill 0.25\hfill & \hfill 0.1\hfill & \hfill 0.08\hfill & \hfill 0.1\hfill \end{array}\right]$ and $Q_u=1$.

In simulation, the modelling error is regarded as an input disturbance $w(k)$ in Equation (13) of which the amplitude is in low frequency and $|w(k)|\le 0.4$.

In all simulations and experiments, the proposed method is compared with the conventional method in which a single controller is used for an arbitrary data dropout process in Ref. [34]. For the other parameters, $s=6$ in Equation (4), $q=5$ in Assumption 1, $m_c=200$ in Assumption 2, $x(0)=B_w={\left[\begin{array}{cccc}\hfill 0\hfill & \hfill -0.035\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]}^{\top }$ in Equation (13), ${\alpha }_{\text{ZNCC}}=1.08$ in Equation (28) and ${\alpha }_{\text{SAD}}=0.65$ in Equation (29) are used.

The control performance is evaluated in the simulation and experiment by 31 $J≔T_s\sqrt{\sum _{t=0}^{T_f}\Vert z(t){\Vert }^2},$ where $T_f=\frac{900}{T_s}$, that is, the control performance is evaluated over 900 s.

Simulations and experiments are executed using MATLAB 7.5.0 on an Intel Core i5 650, 3.20 GHz processor.

Case 1

Each ${\mathrm{\Pi }}_{\sigma }$ ($\sigma \in \mathcal{Q}$) is assumed such that it reflects the bursty property of the data dropout, that is, the data dropout probability after a data dropout is higher than that after a successful data transmission [34]. In this case, $q=5$ (see Appendix for the transition probability matrices). For $\sigma \in \mathcal{Q}$, ${\mu }_{\sigma }$, which is the average of the data dropout, is given, respectively, as follows: ${\mu }_1=1.167$, ${\mu }_2=2.009$, ${\mu }_3=2.659$, ${\mu }_4=3.444$, and ${\mu }_5=4.509$. ${[Z_{\text{ZNCC}}]}_{ij}≔z_{\text{ZNCC}}({\mathrm{\Pi }}_i,{\mathrm{\Pi }}_j)$ and ${[Z_{\text{SAD}}]}_{ij}≔z_{\text{SAD}}({\mathrm{\Pi }}_i,{\mathrm{\Pi }}_j)$ are, respectively, as follows: 32 $Z_{\text{ZNCC}}=\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill 0.84\hfill & \hfill 0.59\hfill & \hfill 0.25\hfill & \hfill -0.25\hfill \\ \hfill *\hfill & \hfill 1\hfill & \hfill 0.74\hfill & \hfill 0.36\hfill & \hfill -0.19\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill 1\hfill & \hfill 0.66\hfill & \hfill -0.04\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill 1\hfill & \hfill 0.27\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill 1\hfill \end{array}\right],$ 33 $Z_{\text{SAD}}=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 3.0\hfill & \hfill 5.2\hfill & \hfill 7.8\hfill & \hfill 10.2\hfill \\ \hfill *\hfill & \hfill 0\hfill & \hfill 3.7\hfill & \hfill 6.4\hfill & \hfill 8.7\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill 0\hfill & \hfill 3.5\hfill & \hfill 6.4\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill 0\hfill & \hfill 4.7\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill 0\hfill \end{array}\right].$

Similarly, $K_{\sigma }$ ($\sigma \in \mathcal{Q}$) is obtained by solving the optimization Equations (20)–(22) as follows: 34 $K_1=\left[\begin{array}{cccc}\hfill 0.623\hfill & \hfill -20.813\hfill & \hfill 1.934\hfill & \hfill -2.595\hfill \end{array}\right],$ 35 $K_2=\left[\begin{array}{cccc}\hfill 0.414\hfill & \hfill -15.969\hfill & \hfill 1.476\hfill & \hfill -1.989\hfill \end{array}\right],$ 36 $K_3=\left[\begin{array}{cccc}\hfill 0.313\hfill & \hfill -13.645\hfill & \hfill 1.257\hfill & \hfill -1.699\hfill \end{array}\right],$ 37 $K_4=\left[\begin{array}{cccc}\hfill 0.222\hfill & \hfill -11.490\hfill & \hfill 1.053\hfill & \hfill -1.429\hfill \end{array}\right],$ 38 $K_5=\left[\begin{array}{cccc}\hfill 0.229\hfill & \hfill -11.499\hfill & \hfill 1.054\hfill & \hfill -1.429\hfill \end{array}\right],$

The single $H_2$ controller used in the conventional method for NCS with the arbitrary data dropout process in Ref. [34] is 39 $K_0=\left[\begin{array}{cccc}\hfill 0.177\hfill & \hfill -10.801\hfill & \hfill 0.986\hfill & \hfill -1.341\hfill \end{array}\right].$

Figures 4 and 5 show the simulation and experimental results using the ZNCC and the SAD estimations, respectively. Figure 4a and 5a show the ZNCC and SAD, respectively. The network traffic status can be successfully estimated from both Figure 4b and 5b. According to the estimated network traffic status, $\hat{\sigma }(t)$, $K=K_{\widehat{\sigma }(t)}$ is used in Equation (9). As a result, from Figure 4c,d and 5c,d, it can be observed that the control performance by the proposed estimation and switching control is better than that by the conventional method in both simulation and experiment.

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Table 1 lists the quantitative control performance of each evaluation of $J$ in Equation (31) and the estimation accuracy of the network traffic status in simulation and experiment. From Table 1, $J$ by the ZNCC and the SAD are less than that by the conventional method in the simulation and experiment, which implies that the proposed method can improve the control performance. In this case, the ZNCC-based estimation is superior to the SAD-based estimation in terms of the control performance and estimation accuracy.

TABLE 1 Control performance (Case 1)

	$J$ in sim.	$J$ in exp.	Estimation accuracy
Zero-means normalized cross-correlation	6.96	15.85	0.89
Sum of absolute difference	6.96	18.48	0.83
Conventional	9.35	31.01	-

Case 2

In this case, the transition probability matrices have a similar tendency addressed in Ref. [18, 40]. See Appendix for $[\forall \sigma \in \mathcal{Q}]{\mathrm{\Pi }}_{\sigma }$. The property that the data dropout probability after a data dropout is relatively high is characterised by $[\forall \sigma \in \mathcal{Q}]{\mathrm{\Pi }}_{\sigma }$. For $\sigma \in \mathcal{Q}$, ${\mu }_{\sigma }$ is calculated as ${\mu }_1=1.182$, ${\mu }_2=1.556$, ${\mu }_3=2.038$, ${\mu }_4=2.700$, and ${\mu }_5=3.112$. ${[Z_{\text{ZNCC}}]}_{ij}≔z_{\text{ZNCC}}({\mathrm{\Pi }}_i,{\mathrm{\Pi }}_j)$ and ${[Z_{\text{SAD}}]}_{ij}≔z_{\text{SAD}}({\mathrm{\Pi }}_i,{\mathrm{\Pi }}_j)$ are as follows, respectively: 40 $Z_{\text{ZNCC}}=\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill 0.95\hfill & \hfill 0.91\hfill & \hfill 0.74\hfill & \hfill 0.58\hfill \\ \hfill *\hfill & \hfill 1\hfill & \hfill 0.94\hfill & \hfill 0.77\hfill & \hfill 0.53\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill 1\hfill & \hfill 0.75\hfill & \hfill 0.51\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill 1\hfill & \hfill 0.33\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill 1\hfill \end{array}\right],$ 41 $Z_{\text{SAD}}=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 1.8\hfill & \hfill 3.2\hfill & \hfill 5.6\hfill & \hfill 7.8\hfill \\ \hfill *\hfill & \hfill 0\hfill & \hfill 2.0\hfill & \hfill 4.2\hfill & \hfill 7.0\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill 0\hfill & \hfill 4.0\hfill & \hfill 6.0\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill 0\hfill & \hfill 5.8\hfill \\ \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill 0\hfill \end{array}\right].$

$K_{\sigma }$ ($\sigma \in \mathcal{Q}$) is obtained by Equation (20) to (22) as 42 $K_1=\left[\begin{array}{cccc}\hfill 0.642\hfill & \hfill -21.278\hfill & \hfill 1.978\hfill & \hfill -2.652\hfill \end{array}\right],$ 43 $K_2=\left[\begin{array}{cccc}\hfill 0.441\hfill & \hfill -16.855\hfill & \hfill 1.558\hfill & \hfill -2.010\hfill \end{array}\right],$ 44 $K_3=\left[\begin{array}{cccc}\hfill 0.306\hfill & \hfill -13.674\hfill & \hfill 1.259\hfill & \hfill -1.701\hfill \end{array}\right],$ 45 $K_4=\left[\begin{array}{cccc}\hfill 0.246\hfill & \hfill -12.165\hfill & \hfill 1.117\hfill & \hfill -1.512\hfill \end{array}\right],$ 46 $K_5=\left[\begin{array}{cccc}\hfill 0.248\hfill & \hfill -12.179\hfill & \hfill 1.118\hfill & \hfill -1.514\hfill \end{array}\right].$

The $H_2$ controller is given in Equation (39) and is the same as with the Case 1.

Figure 6 shows the simulation and experimental results using the estimation by $z_{\text{ZNCC}}$ in Equation (24). The network traffic status is estimated according to $z_{\text{ZNCC}}$ and Equation (28). It cannot be observed that the network traffic status is estimated from Figure 6b. Specifically, the estimation fails when the actual network traffic status changes from $\sigma =4$ to $\sigma =5$ at around $t=455$. This incorrect estimation results from ${[Z_{\text{ZNCC}}]}_{35}>{[Z_{\text{ZNCC}}]}_{45}$ and ${[Z_{\text{ZNCC}}]}_{24}>{[Z_{\text{ZNCC}}]}_{34}$ in Equation (40) implies that ${\mathrm{\Pi }}_3$ is closer to ${\mathrm{\Pi }}_5$ than ${\mathrm{\Pi }}_4$ and ${\mathrm{\Pi }}_2$ is closer to ${\mathrm{\Pi }}_4$ than ${\mathrm{\Pi }}_3$. As a result, this misestimation leads to instability, as seen in Figure 6c,d. Figure 7 shows the simulation and experimental results using the estimation by $z_{\text{SAD}}$ in Equation (25). From Figure 7b, we can observe that the network traffic status is effectively estimated. In this case, ${[Z_{\text{SAD}}]}_{35}>{[Z_{\text{SAD}}]}_{45}$ and ${[Z_{\text{SAD}}]}_{24}>{[Z_{\text{SAD}}]}_{34}$ in Equation (40). $z_{\text{SAD}}$ is used as a dissimilarity index, we can comprehend that ${\mathrm{\Pi }}_3$ is farther from ${\mathrm{\Pi }}_5$ than ${\mathrm{\Pi }}_4$ and ${\mathrm{\Pi }}_2$ is farther from ${\mathrm{\Pi }}_4$ than ${\mathrm{\Pi }}_3$. As a result, Figure 7c,d demonstrate that the control performance by the proposed estimation and switching control is better than that by the conventional method in both simulation and experiment.

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Table 2 lists the evaluation of $J$ in Equation (31) and the estimation accuracy of the network traffic status (the values by the ZNCC are not listed in Table 2 due to the instability). As listed in Table 2, the evaluations of $J$ by the SAD-based estimation is less than those by the conventional method in both the simulation and experiment, which implies that the proposed method improves the control performance of Equation (9) in the presence of data dropout.

TABLE 2 Control performance (Case 2)

	$J$ in sim.	$J$ in exp.	Estimation accuracy
Sum of absolute difference	6.97	18.62	0.85
Conv.	9.20	35.20	-

These results conclude that NCS with data dropout can be stabilised using the proposed estimation of the network traffic status if the transition matrices of the network traffic status are arranged properly.

CONCLUSION

This study addressed an estimation of the network traffic status for NCSs with the data dropout and its control. The data dropout was modelled as a discrete-time homogeneous finite-state Markov chain. We proposed a traffic estimation of the network. In the proposed estimation, the ZNCC and the SAD were used as similarity and dissimilarity indices. Using these indices, the network traffic status was estimated, and an appropriate controller was selected such that the plant was effectively stabilised. The effectiveness of the proposed estimation and the switching control strategy was demonstrated through simulations and experiments.

In this study, the data dropout model was assumed to be available in advance. Future work includes more advanced estimation of network traffic status even if the probability transition matrix is not given.

ACKNOWLEDGEMENTS

This work was supported by JSPS KAKENHI Grant Number 20H02167.

CONFLICT OF INTEREST

None.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

APPENDIX

The transition probability matrices in Section 4 are as follows:

Case 1:

$\begin{array}{ll}\hfill & {\mathrm{\Pi }}_1=\left[\begin{array}{cccccc}\hfill 0.9\hfill & \hfill 0.1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.5\hfill & \hfill 0.5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],{\mathrm{\Pi }}_2=\left[\begin{array}{cccccc}\hfill 0.5\hfill & \hfill 0.45\hfill & \hfill 0.05\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.1\hfill & \hfill 0.6\hfill & \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.3\hfill & \hfill 0.3\hfill & \hfill 0.4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\hfill \\ \hfill & {\mathrm{\Pi }}_3=\left[\begin{array}{cccccc}\hfill 0.4\hfill & \hfill 0.2\hfill & \hfill 0.2\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.2\hfill & \hfill 0.5\hfill & \hfill 0.2\hfill & \hfill 0.1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.1\hfill & \hfill 0.8\hfill & \hfill 0.1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.3\hfill & \hfill 0.5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],{\mathrm{\Pi }}_4=\left[\begin{array}{cccccc}\hfill 0.4\hfill & \hfill 0.15\hfill & \hfill 0.15\hfill & \hfill 0.15\hfill & \hfill 0.15\hfill & \hfill 0\hfill \\ \hfill 0.2\hfill & \hfill 0.4\hfill & \hfill 0.2\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0\hfill \\ \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.5\hfill & \hfill 0.2\hfill & \hfill 0.1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.1\hfill & \hfill 0.8\hfill & \hfill 0.1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.25\hfill & \hfill 0.25\hfill & \hfill 0.5\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],\hfill \\ \hfill & {\mathrm{\Pi }}_5=\left[\begin{array}{cccccc}\hfill 0.3\hfill & \hfill 0.2\hfill & \hfill 0.2\hfill & \hfill 0.2\hfill & \hfill 0.1\hfill & \hfill 0\hfill \\ \hfill 0.2\hfill & \hfill 0.3\hfill & \hfill 0.2\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0.1\hfill & \hfill 0.2\hfill & \hfill 0.3\hfill & \hfill 0.2\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill & \hfill 0.3\hfill & \hfill 0.2\hfill & \hfill 0.1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.1\hfill & \hfill 0.8\hfill & \hfill 0.1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0.2\hfill & \hfill 0.6\hfill \end{array}\right].\hfill \end{array}$

Case 2:

$\begin{array}{ll}\hfill & {\mathrm{\Pi }}_1=\left[\begin{array}{cccccc}\hfill 0.8\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.9\hfill & \hfill 0.1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],{\mathrm{\Pi }}_2=\left[\begin{array}{cccccc}\hfill 0.7\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.6\hfill & \hfill 0.2\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.6\hfill & \hfill 0.1\hfill & \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],\hfill \\ \hfill & {\mathrm{\Pi }}_3=\left[\begin{array}{cccccc}\hfill 0.5\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.5\hfill & \hfill 0.2\hfill & \hfill 0.2\hfill & \hfill 0.1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.5\hfill & \hfill 0.2\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.6\hfill & \hfill 0.1\hfill & \hfill 0.3\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],{\mathrm{\Pi }}_4=\left[\begin{array}{cccccc}\hfill 0.4\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.3\hfill & \hfill 0\hfill \\ \hfill 0.4\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.3\hfill & \hfill 0\hfill \\ \hfill 0.4\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.3\hfill & \hfill 0\hfill \\ \hfill 0.4\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.3\hfill & \hfill 0\hfill \\ \hfill 0.4\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.3\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],\hfill \\ \hfill & {\mathrm{\Pi }}_5=\left[\begin{array}{cccccc}\hfill 0.5\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.3\hfill \\ \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0.1\hfill & \hfill 0.3\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill \\ \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.4\hfill & \hfill 0\hfill & \hfill 0.3\hfill \\ \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0.2\hfill & \hfill 0.3\hfill \\ \hfill 0.3\hfill & \hfill 0.2\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill \end{array}\right].\hfill \end{array}$

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