IEEE/CAA Journal of Automatica Sinica  2018, Vol. 5 Issue(5): 902-922   PDF    
Fundamental Issues in Networked Control Systems
Magdi S. Mahmoud, Mutaz M. Hamdan     
Systems Engineering Department, King Fahd University of Petroleum and Minerals, P. O. Box 5067, Dhahran 31261, Saudi Arabia
Abstract: This paper provides a survey on modeling and theories of networked control systems (NCS). In the first part, modeling of the different types of imperfections that affect NCS is discussed. These imperfections are quantization errors, packet dropouts, variable sampling/transmission intervals, variable transmission delays, and communication constraints. Then follows in the second part a presentation of several theories that have been applied for controlling networked systems. These theories include: input delay system approach, Markovian system approach, switched system approach, stochastic system approach, impulsive system approach, and predictive control approach. In the last part, some advanced issues in NCS including decentralized and distributed NCS, cloud control system, and co-design of NCS are reviewed.
Key words: Decentralized networked control systems (NCS)     distributed networked control systems     network constraints     networked control system     quantization     time delays    
Ⅰ. INTRODUCTION

Anetworked control system (NCS) is a system in which the traditional control loops are closed through a communication network such that signals of the system (control signals and feedback signals) can be exchanged among all components (sensors, controllers, and actuators) through a common network. Fig. 1 shows a typical structure of NCS. In comparison with traditional control system, NCS has several advantages including: less wiring, lower cost, and more flexibility and maintainability of the system. As a result, NCS have been used widely in the last decades in many fields such as: industrial control, process control, engineering systems, aerospace systems, intelligent systems, microgrids, and teleoperation, to name a few. However, it turns out that the inclusion of networks in dynamical systems introduce new challenges to the overall system due to the appearance of imperfections. These include quantization errors, varying delays, dropouts, etc. The imperfections essentially affect the behavior of the NCS by degrading the performance or causing instability. It is therefore essential to build up an appropriate dynamic representation of the NCS and design effective controllers that achieve stability under these circumstances.

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Fig. 1 A typical networked control system

The research of NCS can be classified into two main categories: 1) control of network, and 2) control over or through network. Control of network considers the problems of communication network such as communication protocols, routing control, congestion control etc. On the other hand control over or through network focuses on the design and control of systems that are using a network as a transmission media to obtain the desired performance. The second topic "control over network", which will be the subject of this paper, contains two main aspects: the quality of service (QoS) and the quality of control (QoC). Maintaining both of QoS and QoC is a major objective of research in NCS. Measures of the network like transmission rates and error rates are subjected to QoS, while QoC is concerned with the stability of the system subjected to different conditions.

Several survey papers exist in the literature for summarizing the updated result on NCS. Following a chronological order, [1] reviewed the stability of NCS in 2001. While in 2006 [2] provided a general survey on NCS, in which the effect of NCS over the control methodologies of conventional large scale system is reviewed. In 2007, the challenges of control and communication in networked real-time system were presented in [3], and [4] provided an overview on estimation, analysis, and controller synthesis for NCS. Some of the research topics and trends of NCS were presented in 2010 [5]. A survey on network-induced constraints in NCS was presented in 2013 [6]. In 2015, [7] has discussed several aspects of NCS such as: quantization, estimation, fault detection and networked predictive control. Also, it presented cloud control issues. Recently, an overview on the theoretical development of NCS was provided in [8] and [9]. An overview of the research investigations into the evolving area of NCS was provided in [10]. The interaction between control and computing theories was discussed [11]. Reference [12] provided a review on event-based control and filtering of NCS. Besides, some results were presented in [13]-[17], and a coverage of analysis, stability and design of NCS can be found in [18].

The objectives of this paper are: First, to provide a review on modeling of the imperfections in NCS. Second, review the theories applied for analyzing and achieving the stability of NCS. Finally, to present some advanced issues in NCS including decentralized and distributed NCS, Cloud NCS and Co-design of NCS.

Ⅱ. MODELING OF NCS

The components of NCSs are connected via communication systems as shown in Fig. 1, and this connection addresses new imperfections and constraints that have to be considered in the modeling of the complete system. As listed in [19], the imperfections and constraints in NCS are classified into five types:

1) quantization errors in the transmitted signals;

2) packet dropouts, because of the unreliable transmissions;

3) variable sampling/transmission intervals;

4) variable transmission delays; and

5) communication constraints, since not all of the signals of sensors and actuators can be transmitted at the same time.

These imperfections are summarized in Fig. 2.

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Fig. 2 Types of imperfections and constraints in NCS
A. Quantization Errors

Due to the existence of the communication network and its limited transmission capacity, signals have to be quantized before they are transmitted. The control signal and plant output signal both are quantized before they are sent to the network as shown in Fig. 3. A quantizer is a device that receives a real-valued signal and converts it to a piecewise constant one with a finite set of values.

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Fig. 3 System configuration of an NCS with quantizers

In the literature, there are two common types of quantization which are: logarithmic quantization and uniform quantization.

1) Logarithmic Quantization:

The logarithmic quantization is considered as static quantization, its performance near the origin is better in comparison with uniform quantization, and it could be either with infinite quantization level or with finite quantization level. The logarithmic quantization with infinite quantization level is modeled as

$ q(y)= \left\{\begin{matrix} v_i,&{\rm if} & \frac{v_i}{1+\delta} < y < \frac{v_i}{1-\delta}, \\ && y>0\\ 0,&{\rm if}&y=0\\ -q(-y),&{\rm if}&y < 0 \end{matrix}\right. $ (1)

where $U=\{ \pm v_i:v_i=\rho^i v_0, i=\pm1, \pm 2, \ldots\}$ $\cup \{\pm v_0\}\cup \{0\}, 0 < \rho < 1, v_0>0$ is the set of the quantization values. The logarithmic quantization with finite quantization level is modeled as

$ q(y)=\left\{\begin{matrix} v_i&{\rm if}&\frac{v_i}{1+\delta} < y < \frac{v_i}{1-\delta}, \\ && 0 < i < N-1\\ 0,&{\rm if}&0\leq y \leq \frac{v_{N-1}}{1+\delta}\\ v_0,&{\rm if} &y>\frac{v_0}{1+\delta}\\ -q(-y),&{\rm if}&y < 0 \end{matrix}\right. $ (2)

where $U=\{ \pm v_i:v_i=\rho^i v_0, i=\pm1, \pm 2, \ldots, \pm(N-1)\}$ $\cup \{\pm v_0\}\cup \{0\}, 0 < \rho < 1, v_0>0$ is the set of the quantization values.

2) Uniform Quantization:

Uniform quantizer is easier to be operated and it has the following conditions when it is applied with an arbitrarily-shaped quantitative area which satisfies:

1) if $\| y \|_2 \le M \mu$ , then $\| q(y)-y \|_2\le \Delta \mu$

2) if $\| y \|_2 > M \mu$ , then $\| q(y) \|_2 > M\mu - \Delta \mu$

where $M$ and $\Delta$ are the saturation value and the sensitivity, respectively. The upper bound of the quantization error when the quantization is not saturated is represented by the first condition. While, the second condition gives the way of testing the saturation of the quantization. The rectangular shaped quantitative area is modeled as:

$ q(y)=\left\{\begin{matrix} 0&{\rm if} & -\frac{1}{2} < y < \frac{1}{2}\\[2mm] i,&{\rm if} &\frac{2i-1}{2} < y < \frac{2i+1}{2}, \\ && i=1, 2, \ldots, K-1\\ K,&{\rm if} &y\geq \frac{2K-1}{2}\\ -q(-y),&{\rm if}&y\leq -\frac{1}{2}. \end{matrix}\right. $ (3)

More details about the two types of quantization could be found in [7] and [20].

Remark 1: The logarithmic quantizers are mainly used with linear systems with infinite quantization levels. While, the zoom strategy is a beneficial control policy when uniform quantization is applied, and it has two steps: "zoom-in" and "zoom-out" [7].

Remark 2: The implementation of zooming-in and zooming-out was initially discussed in [21], [22]. It was used to obtain the sufficient condition for the asymptotic stability for linear and nonlinear systems.

As a result of quantization, information loss will be introduced in the system. Therefore, the model of NCS has to take it into account. The quantization error is inversely proportional to the number of bits used for quantization, i.e., the small number of bits leads to a higher quantization error. Due to this fact, a significant research is directed to determine the minimum number of bits required for achieving the stability of the system, some examples could be found in [21]-[25].

Some researchers focus on controlling the quantization and its effects on the system. Reference [26] proposed a sector bounded approach for dealing with the quantization errors, so its effects on NCS could be investigated using the procedures of robustness analysis. Quantization and stochastic packet dropouts were considered in the study of the quadratic stability of NCS and finite quantization was used for implementing the controller [27]. The quantizer step size influence on NCS considering packet dropouts and finite-level quantization were studied in [28]. In [29] and [30], an adaptable "center" and "zoom" parameters of the quantizers with finite values were considered, the input-to-state stability was obtained by applying a strategy of switching the controller continuously between "zooming-out" and "zooming-in". The same strategy of "zooming-out" and "zooming-in" was used to obtain parametrized input to state stability of NCS subjected to packet dropout and unknown disturbances but with random lengths of quantization regions based on the packet dropout process [31]. Reference [32] has used sector bound and convex combination property of quantizer for determining the sufficient conditions to achieve the desired control of NCS subjected to several categories of asynchronous sampling and quantization. Quantization with the implementation of event triggering control was discussed in many literatures, some examples are: [33]-[37].

Remark 3: There are two phenomena caused by quantization:

1) Saturation, which occurs when the signal is larger than the quantization range and that leads to a higher quantization error causing instability in the closed loop system.

2) Deterioration of the performance around the original point, which occurs near the origin when the signal is not exactly quantized due to the limitation of the accuracy of the quantizers, and this will prevent approaching the asymptotic stability of the closed-loop system.

B. Packet Dropouts

Due to the use of the network for communication, the signals of the systems need to be grouped before transmitting, each group of signals is called "packet" and its size depends on the network used. The transmission of packets could be either single or multiple. In single packet transmission, all data are grouped from sensors or controller and transmitted together. On the other hand, in multiple packet transmission, the data are transmitted in several network packets, causing non simultaneous arriving of data to the controller or actuator. The limited size of the network is not the only reason of using parallel transmission, but also the distribution of sensors and actuators practically over a large area makes it difficult to lump the data into one network packet leading to use multiple transmission.

Occurrence of failures or message collisions on nodes cause packet dropout. To avoid that, most protocols use transmission retry mechanisms; However, if the retransmission fails within a limited time, the packets are dropped. Since the communication network is the source of the losses, this type of dropouts is called "network-induced packet dropout". Moreover, if a new packet sent earlier is available at the node later, it is more practical to discard it and use the recent one, and this type of packet dropouts called "active packet dropout". For tackling this issue, some techniques like logical zero-order-hold (ZOH) mechanisms [38] and message rejection [39] were proposed.

One methodology of dealing with packet dropouts is to design the controller to withstand with the upper bound of the dropouts in the system [40], [41], and [42]. Another famous approach is to represent the dropout in the system by a switch [43]-[45]. As shown in Fig. 4, when the switch is open ($T_2$ ) the dropout happens, while there is no dropout when it is closed ($T_1$ ). Then, the relation between the packet dropout rate and the $H_\infty$ controller is derived to guarantee the exponential stability of the system. Consider the following discrete-time linear system model of an NCS:

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Fig. 4 An NCS with quantizers and packet dropout represented by a switch
$ x(k+1)=A_{\sigma (k)}x(k)+B_{\sigma (k)}u(k) $ (4)

where $x(k)$ is the switched system state, $\sigma (k): R^+=\{0, 1, 2, \ldots\}\rightarrow \Gamma=\{1, 2, \ldots, N \}$ is the switching signal which is a piecewise constant function depending on time $k$ and/or state $x(k)$ . The subsystem $i$ is activated when $\sigma(k)=i$ , $A_i$ and $B_i$ are constant matrices. A hybrid state feedback controller $u(k)$ is used such that

$ u(k)=-K_{\hat{\sigma}(k)}\hat x(k) $ (5)

where $K$ is the controller gains to be designed, $\hat x_k$ is the switched system state, and $\hat{\sigma}(k)$ is the switching signal received by the hybrid controllers over the network. The subsystem $i$ is activated at time $k$ when $\sigma_i(k)=1$ , and $\sigma_i(k)=0$ otherwise, and thus $\sum_{i=1}^{N}\sigma_i(k)=1$ . So, the discrete time switched system is represented by

$ x_{k+1}=\sum\limits_{i=1}^{N}\sigma_i(k)(A_ix_k+B_iu_k). $ (6)

Now, let $s$ be the transmission indicator function defined as

$ s=\begin{cases} 1,&{\text{sample is transmitted} }\\ 2,&{\text{sample is not transmitted} } \end{cases} $ (7)

So, the dynamics of the switch system could be described by

$ \hat x(k)=\beta_s x(k)+(1-\beta_s)\hat x(k-1) $ (8)
$ \hat {\sigma}(k)=\beta_s \sigma(k)+(1-\beta_s)\hat {\sigma}(k-1) $ (9)

where $\beta_s$ are switch variables such that $\beta_1=1$ and $\beta_2=0$ . By combining (4)-(6), the closed loop networked switched control system with the hybrid state feedback controller is rewritten as

$ x(k+1)=\sum\limits_{i=1}^{N}\sigma_i(k)(A_ix(k)-B_i(\sum\limits_{i=1}^{N} \hat{\sigma}_i)K_i\hat x(k)). $ (10)

Now, let $\zeta (k)=[x(k)^T \hat x(k)^T]$ be the augmented state vector, then, the closed-loop networked switched control system with the network packet dropout effect is written as

$ \zeta (k+1)=\Phi_s\zeta (k) $ (11)

and here we have two cases, when the switch is positioned at $T_1, s=1, \beta_1=1$ , so

$ \begin{gather*} \hat x(k)=x(k) \\ \hat{\sigma}(k)=\sigma(k) \\ \Phi_1=\begin{bmatrix} \sum\limits_{i=1}^{N}\sigma_i(k)A_i&- \sum\limits_{i=1}^{N}\sigma_i(k)B_iK_i\\ \sum\limits_{i=1}^{N}\sigma_i(k)A_i&- \sum\limits_{i=1}^{N}\sigma_i(k)B_iK_i \end{bmatrix}. \end{gather*} $

When the switch is positioned at $T_2, s=2, \beta_2=0$ , so

$ \begin{gather*} \hat x(k)=x(k-1) \\ \hat{\sigma}(k)=\sigma(k-1) \\ \Phi_2=\begin{bmatrix} \sum\limits_{i=1}^{N}\sigma_i(k)A_i&- \sum\limits_{i=1}^{N}\sigma_i(k)\sum\limits_{l=1}^{N}\sigma_l(k-1)B_iK_l\\ 0&I \end{bmatrix}. \end{gather*} $

Theorem 1 [43]: For system (11), assume that the plant state and the switching signal in a single packet are transmitted at a rate of $r$ . If there exist symmetrical positive definite matrices $P_i, Q_i, i \in \Upsilon$ and scalars $\alpha_ 1, \alpha_2 > 0$ such that

$ \alpha_ 1^r \alpha_2^{r-1} > 1 $ (12)
$ \Phi_1^T P_j \Phi_1 + Q_i - \alpha_1^{-2}P_i \le 0 $ (13)
$ \Phi_2^T P_j \Phi_2 + Q_i - \alpha_2^{-2}P_i \le 0 $ (14)

hold, then system (11) is exponentially stable.

Remark 4: The proof of Theorem Ⅱ-B could be derived using the following candidate switched Lyapunov-Krasovskii functional:

$ \begin{align} V(\zeta_k)&\!\!=\!\!\zeta_k^T\left(\sum\limits_{i=1}^N \sigma_i(k) P_i\right)\zeta_k \\ &+ \zeta_{k-1}^T \left(\sum\limits_{i=1}^N \sigma_i(k-1)Q_i\right) \zeta_{k-1}. \end{align} $ (15)

The other methodology is to consider the packet dropout as a random process, then model it as a Markovian process as in [46] and [47], or as a Bernoulli distribution such as [48] and [49]. In [50], the stability analysis and controller synthesis problems were presented for NCS with time-varying delays and affected by nonstationary packet dropouts. The plant is described by the following discrete-time linear time-invariant system

$ x_p(k+1) = A x_p + B u_p, \;\;y_p = C x_p $ (16)

where $x_p(k) \in\mathbb{R}^n$ is the state vector of the plant and $u_p(k)$ $\in\mathbb{R}^m$ and $y_p(k) \in\mathbb{R}^p$ are the control input and output vectors of plant, respectively. $A$ , $B$ , and $C$ are real matrices with appropriate dimensions. The measurement received by the controller is affected by a randomly varying communication delay and represented by:

$ y_c(k) = \begin{cases} y_p(k - \tau_k^m), \;& \delta(k) = 1 \\ y_p(k), \;& \delta(k) = 0 \end{cases} $ (17)

where $\tau_k^m$ is the "measurement delay" which satisfies the Bernoulli distribution, and $\delta(k)$ is Bernoulli distributed white sequence representing the occurrence of packet dropouts in the NCS. Also, let

$ Prob \{ \delta(k) = 1 \} = p_k $

where $p_k$ assumes discrete values. So, there are two classes to be considered which are [50]:

Class 1: $p_k$ has the probability mass function where $q_r $ $- q_{r-1}=$ constant for $r = 2, \ldots, n$ . This covers a wide range of cases [51].

Class 2: $p_k= X/n, \; n>0\;$ and $ 0 \leq X \leq n$ is a random variable that follows the Binomial distribution ${ B}(q, n), \; q>0$ , that is

$ Prob\Bigg\{p_k = \frac{(ax+b)}{n}\Bigg\} = \left ( \begin{array}{l} n \\ x \end{array} \right ) q^x (1-q)^{n-x}, \;\;b>0\\ x=0, 1, 2, \ldots, n, \;\;an+b < n. $

The following observer-based controller is required to be designed in case that the full state information is not available and the time delay occurs on the actuation side [51]:

$ \begin{align} & {\rm Observer:} \\ & \hat{x}(k+1) = A \hat{x} + B u_p(k) + L(y_c(k) - \hat{y}_c(k)) \\ & \hat{y}_c(k) = \begin{cases}C \hat{x}(k), \; &\delta(k) = 0 \\ C \hat{x}(k - \tau^m_k), \;& \delta(k) = 1 \end{cases} \end{align} $ (18)
$ \begin{align} & {\rm Controller:}\\ &u_c(k) = K \hat{x}(k) \\ & u_p = \begin{cases} u_c(k), \; \qquad&\alpha(k) = 0 \qquad\qquad\\ u_c(k - \tau^a_k), \;&\alpha(k) =1 \end{cases} \end{align} $ (19)

where $\hat{x}(k) \in\mathbb{R}^n$ is the estimate of the system (16), $\hat{y}_c(k)$ $ \in\mathbb{R}^p$ is the observer output, and $L \in\mathbb{R}^{n \times p}$ and $K \in\mathbb{R}^{m \times n}$ are the observer and controller gains, respectively, and $\tau^a_k$ is the actuation delay.

Assume that the "actuation delay" $\tau_k^a$ and the "measurement delay" $\tau_k^m$ are time-varying with bounded conditions as follows:

$ {\tau}^{-}_m \leq \tau^m_k \leq {\tau}^{+}_m, \;\;\; {\tau}^{-}_a \leq \tau^a_k \leq {\tau}^{+}_a $ (20)

Also, let the estimation error $e(k)$ equal to $x_p (k) - \hat{x}(k)$ . Then

$ x_p (k+1) = \left \{ \begin{array}{l} A x_p(k) + B K x_p(k-\tau^\alpha_k)- B K e(k-\tau^\alpha_k), \\ \qquad \qquad \hfill \alpha(k) = 1 \\ (A+BK)x_p(k) - B K e(k), \hfill\alpha(k) = 0\end{array} \right. $ (21)
$ \begin{align} e(k+1) &\!\!=\!\! x_p(k+1) - \hat{x}(k+1)\\ &\!\!=\!\! \left \{ \begin{array}{l} A e(k) - L C e(k-\tau^m_k), \delta(k) = 1 \\ (A- LC)e(k), \qquad \delta(k) = 0. \end{array} \right. \end{align} $ (22)

In terms of $\xi(k) = [x_p^T(k) \quad e^T(k)]^T, \;$ system (21) and (22) can be written in the following form:

$ \xi(k+1) = { A}_j \xi(k) + {B}_j \xi(k - \tau^m_k) + {C}_j \xi(k - \tau^a_k) $ (23)

where the matrices $\{{ A}_j, \;{B}_j, \;{\pmb C}_j, \;j=1, \ldots, 4\}$ with $j$ is an index identifying one of the following pairs $\{(\delta(k) =1, $ $ \alpha(k) = 1), \;(\delta(k) = 1, \;\alpha(k) = 0), \;(\delta(k) = 0, \;\alpha(k)= 0), $ $(\delta(k) = 0, \;\alpha(k) = 1)\}$

$ \begin{align} { A}_1 &\!\!=\!\! \left[\begin{array}{cc} A&0 \\ 0&A \end{array} \right], \; { A}_2 = \left[\begin{array}{cc} A+BK &-BK \\ 0&A \end{array} \right]\\ { A}_3 &\!\!=\!\! \left[\begin{array}{cc} A+BK &-BK \\ 0&A-LC \end{array} \right], \; { A}_4 = \left[\begin{array}{cc} A&0 \\ 0&A-LC \end{array} \right]\\ {B}_1 &\!\!=\!\! \left[\begin{array}{cc} BK &-BK \\ 0&0 \end{array} \right], \; {B}_2 = \left[\begin{array}{cc} 0&0 \\ 0&0 \end{array} \right]\\ {B}_3 &\!\!=\!\! \left[\begin{array}{cc} 0&0 \\ 0&0 \end{array} \right], \; {B}_4 = \left[\begin{array}{cc} BK &-BK \\ 0&0 \end{array} \right]\\ {C}_1 &\!\!=\!\! \left[\begin{array}{cc} 0&0 \\ 0 &-LC \end{array} \right], \; {C}_2 = \left[\begin{array}{cc} 0&0 \\ 0 &-LC \end{array} \right]\\ {C}_3 &\!\!=\!\! \left[\begin{array}{cc} 0&0 \\ 0&0 \end{array} \right], \; {C}_4 = \left[\begin{array}{cc} 0&0 \\ 0&0 \end{array} \right]. \end{align} $ (24)

Now, it is desired to design an observer based feedback stabilizing controller in the form of (18) and (19) such that the closed loop system (23) is exponentially stable in the mean-square sense. The switched time-delay systems based approach is used to solve this problem [50].

Theorem 2 [50]: Let the controller and observer gain matrices $K$ and $L$ be given. The closed-loop system (23) is exponentially stable if there exist matrices $0 < P, \;0 < Q^T_j$ $=Q_j, j=1, \ldots, 4$ and matrices $R_i$ , $S_i$ , and $M_i$ , $i = 1, 2, $ such that the following matrix inequality holds

$ \begin{align} &\Lambda_j = \left[\begin{array}{cc} \Lambda_{1j}&\Lambda_{2j} \\ \ast& \Lambda_{3j} \end{array} \right] \; < \; 0\label {LMI01}\\ \end{align} $ (25)
$ \begin{align} &\Lambda_{1j} = \\ &\left[\!\!\begin{array}{ccc} \Psi_j + \Phi_{j1} \!&\!-R_1 + S^T_1 \!&\!-R_2 + S^T_2 \\ \ast \!&\!-S_1 -S_1^T -\hat{\sigma}_jQ_j \!&\! 0\\ \ast \!&\! \ast \!&\! -S_2 -S^T_2 -\hat{\sigma}_jQ_j \end{array} \right] \\ &\Lambda_{2j} = \left[\begin{array}{cc}-R_1 + M^T_1- \Phi_{j2} &-R_2 + M^T_2 -\Phi_{j3} \\ -S_1 -M^T_1 \!&\! 0 \\ 0 \!&\! -S_2 -M^T_2 \end{array} \right] \\ &\Lambda_{3j} = \left[\begin{array}{cc}-M_1- M^T_1 + \Phi_{j4} \!&\! \Phi_{j5} \\ \ast \!&\!-M_2 -M^T_2 + \Phi_{j6} \end{array} \right] \end{align} $ (26)

where

$ \begin{align} \Psi_j &\!\!=\!\! -P + \hat{\sigma}_j ( {\tau}^{+}_m - {\tau}^{-}_m + {\tau}^{+}_a - {\tau}^{-}_a + 2)Q_j \\ &+ R_1 + R_1^T + R_2 + R_2^T \\ \Phi_{j1} &\!\!=\!\! ({ A}_j + {B}_j + {C}_j)^T \hat{\sigma}_j P ({ A}_j + {B}_j + {C}_j)\\ \Phi_{j2} &\!\!=\!\! ({ A}_j + {B}_j + {C}_j)^T \hat{\sigma}_j P {B}_j \\ \Phi_{j3} &\!\!=\!\! ({ A}_j + {B}_j + {C}_j)^T \hat{\sigma}_j P {C}_j, \;\Phi_{j5}={B}^T_j P {C}_j \\ \Phi_{j4} &\!\!=\!\! {B}^T_j \hat{\sigma}_j P {B}_j, \;\Phi_{j6} = {C}^T_j \hat{\sigma}_j P {C}_j. \end{align} $
C. Variable Sampling/Transmission Intervals

The signals in NCS need to be sampled before the transmission through the network. The sampling periods are usually fixed in conventional systems due to its simplicity in design and analysis, and called: "time-triggered sampling", "periodic sampling", and "uniform sampling". On the other hand, it is varying in the recent NCS since it is waiting in a queue before the transmission process which will be based on the availability of the network and the protocol used. It is proved recently that sampling at varying time may have better performance than sampling at fixed intervals [52].

The other method of sampling is event-triggered sampling. It is also called: "Lebesgue sampling", "level-crossing sampling", and "magnitude-driven sampling", etc., in this case, the sampling and transmission occurs based on triggering of an event such as changing of one of the output signal to a specific value.

There are several approaches for modeling sampled/ transmission intervals [52], but the most famous one is the input delay approach due to the using of the linear matrix inequality. By applying this approach, it is easy to determine the maximum upper bound of two consecutive samplings and design the proper controller for the NCS. Let the system with sampled signal is given by

$ \begin{align} \dot{x}&\!\!=\!\!Ax(t)+Bu(t), \\ u(t)&\!\!=\!\!Kx(t_k), \quad t_k\le t < t_{k+1} \end{align} $ (27)

where $x(t)\in \mathbb{R}^n$ and $u(t)\in \mathbb{R}^m$ are the state vector of the system and control input vector, respectively. $\{t_1, t_2, \ldots, t_k, \ldots\}$ is a sequence of sampling such that $t_k$ $< t_{k+1}, \text{lim}_{k \rightarrow \infty} t_k=\infty$ and $\text{sup}_k \{t_{k+1}-t_k\}\le h_u$ for some known $h_u>0$ .

By applying the input delay approach, the above system is rewritten as [53]

$ \begin{align} \dot{x}&\!\!=\!\!Ax(t)+BKx(t-\tau (t)), t_k\le t < t_{k+1} \end{align} $ (28)

with piecewise time varying delay $\tau(t):=t-t_k, t_k\le t$ $ < t_{k+1}$ satisfying $0\le \tau (t) \le h_u \forall t\geq t_0$ . Using this model, the Lyapunov-Krasovskii functional approach could be used to obtain the stability conditions and formulate the linear matrix inequalities for calculating the admissible upper bound of $h_u$ and the corresponding controller gain $K$ [53]. Other examples of the input delay approach are found in [54]-[59].

D. Variable Transmission Delays

As shown in Fig. 5, the NCS has two main kinds of delays: 1) Sensor to controller delay, which represents the time between sampling the signal from sensors and receiving it by the controller; 2) Controller to actuator delay, which represents the time between generating the control signal and receiving it by the actuator. Some of the sources of these delays are the limited data bandwidth, network traffic, and the used protocols in the network [60]. In the early published work, only one of these delays was considered in the design of the controller and it is called one-mode controller. On the other hand, two-mode controller is used to show that both of the aforementioned delays were considered in the model. One of the earlier results on two-mode controller was discussed in [61]-[63], where both of the sensor-to-controller and controller-to-actuator random delays were considered and modeled as Markov chain.

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Fig. 5 Network induced delay

The networked induced delay is represented by

$ \tau(t_k) = \tau_{sc}(t_k) + \tau_{ca}(t_k) $ (29)

where $\tau(t_k)$ is the total networked induced delay at sampling time $t_k$ , $\tau_{sc}$ and $\tau_{ca}$ are the sensor to controller delay and controller to actuator delay, respectively. Fig. 6 shows the timing diagram of the signals in NCS.

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Fig. 6 Signals in an NCS with delays

The overall delay in the NCS is calculated by considering any possible delay in the system such as computational delays in the controller, actuator, and sensor nodes [64]. So, the complete delay in the system is represented by

$ \tau(t_k) = \tau_{sc}(t_k) + \tau_{ca}(t_k) + \tau_{c}(t_k)+\tau_{a}(t_k) +\tau_{s}(t_k) $ (30)

where $\tau_{c}, \tau_{a}$ , and $\tau_{s}$ are the computational delay in the controller, actuator, and sensor, respectively. Note that $u$ as shown in Fig. 5 could be defined as

$ u(t) = Kx(t-\tau(t_k)) $ (31)

where $K$ represents the feedback control gain matrix.

Remark 5: The induced delay in the NCS ($\tau$ ) could be extended to consider the dropouts in the system by representing it as a special case of time delay, such that

$ \tau_d = \tau(t_k)+dh $ (32)

where $\tau_d(t_k)$ is overall delay including the dropouts delay, $d$ is the number of dropouts, and $h$ is the sampling period.

There are four main models for random delays in NCS which are [65]:

1) Constant Delay Model

The NCS in this model is considered as a deterministic system with a constant time delay normally equal to the maximum delay in the system similar to (29) or (30). It is used when it is difficult to characterize the random delay in the system. Here, a receiver buffer is introduced at the controller (or actuator) node, and its size is equal to the maximum delay (sensor to controller delay or controller to actuator delay) [66] and [67]. Thus, the NCS can be treated as a deterministic system, after that, many deterministic control methods can be applied to achieve the stability of the NCS.

2) Mutually Independent Stochastic Delay Model

When the probabilistic dependence is unknown, the constant delay model and the deterministic control strategies could hardly achieve the required performance of the system. The reason is due to the presence of many stochastic factors in networks such as: load in network, competition between nodes, and network congestion, and these factors make the network delay to be stochastic. The delay could be modeled either as mutually independent or probabilistically dependent.

3) Markov Chain Model

This type considers the special dependency relationships among the delays which is the Markov chain. This model has two types:

a) One Markov chain including the sum of delays in the NCS, i.e., sensor to controller and controller to sensor.

b) Two Markov chains for modeling both sensor to controller delay and controller to actuator delay.

4) Hidden Markov Model

In this model, all of the stochastic factors such as: load in the network, competition of nodes, and network congestion are grouped into a hidden variable and defined as a network state, and this network state governs the distribution of delays. The network state cannot be observed directly but rather it can be estimated through observing network delays, and so, a hidden Markov model is applied to describe the relation between the network state and the network delay.

E. Communication Constraints

In NCS, the communication network is normally shared with sensors and actuators of multiple nodes, and because of the limitation of data transmission, only one or some of these nodes are active at a time and have access to the network. This is the reason behind the communication constraints or some times it is called as "medium access constraint". As a result, the network requires a protocol for allocating the access of each node to it. This protocol could be either deterministic or random [68]. And so, the model of constraints in NCS could be either deterministic or stochastic.

1) Deterministic Model of Communication Constraints

Previously, the problem was to choose a periodic communication sequence and after that, to design a suitable controller for that [69]. But, this method is NP-hard problem as shown in [70]. So, the later work was to design the controller first and then to find the suitable communication sequence either off-line [71], [72] or online [73]-[75]. Other examples of this type could be found in [76]-[80].

2) Stochastic Model of Communication Constraints

In this model a random media access control (MAC) protocol is used. One example is that a node makes sure that there is no other traffic before transmitting its data [68]. Examples of this model could be found in [81]-[86].

F. Discussions

As mentioned in the beginning of this section, there are five imperfections that could affect NCS. The stability analysis of NCS with two types of imperfections availed the highest level of efforts in the recent years, examples of that are [87]-[92]. Table Ⅰ shows the references that discussed three or four imperfections. To the best of the authors knowledge, no research has considered all of the five imperfections together.

Table Ⅰ
REFERENCES THAT STUDY MANY IMPERFECTIONS
Ⅲ. CONTROL OVER NETWORKS

Several methods were developed for stabilizing NCS while considering one or more of the imperfections which have been discussed in the previous section. These methods as listed in Fig. 7 include: Input delay system approach, switched system approach, Markovian system approach, impulsive system approach, stochastic system approach, and predictive control approach. A discussion on each of them follows.

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Fig. 7 Theories of control over networks
A. Input Delay System Approach

The NCS in this approach is modeled as a system with time varying delay including: the delay from sensor to controller, the delay from controller to actuator, and a representation of the dropout as a delay [1], [108]. Additionally, the computational delay could be considered in the model as mentioned earlier in (32) [109]. Moreover, this approach was developed for considering the signal sampling [55], and were applied for solving the problem of synchronization of complex network [110], [111].

As mentioned in Section Ⅱ-B, some researchers have considered the disorder of data. A logical ZOH is designed to identify most recently arrived control input signal by making comparison between time stamps of the received signals and then the newest one is used for controlling the process [38]. The NCS with the sampler and the logic ZOH is described by the following discrete time system with a delay on the input

$ x(k+1)=Ax(k)+Bu(k-\tau (k)) $ (33)

where $\tau (k)$ is the input delay which is bounded by $\tau _{\max}$ such that $0\le \tau (k)\le \tau _{\max}$ . By applying a state feedback controller $u=Kx$ the overall closed loop system is described by

$ x(k+1)=Ax(k)+BKx(k-\tau (k)), \;\;\; K\in \mathbb{Z}_+ $ (34)

where $K$ is to be designed and $\mathbb{Z}_+$ is the set of nonnegative integers. The sufficient condition of stability of the NCS is derived based on a proper Lyapunov function and presented using LMI method as follows [38].

Theorem 3 [38]: The NCS described in (34) is asymptotically stable if there exist matrices $P\in \mathbb{S}^+, R\in \mathbb{S}^+, W_1 $ $\in \mathbb{R}^{n\times n}$ , and $ W_2 \in \mathbb{R}^{n\times n}$ satisfying

$ \begin{bmatrix} \psi_{11} &\psi_{12}&W_{1} \\ \psi_{12}^{T}&\psi_{22}&W_{2}\\ W_{1}^{T}&W_{2}^{T}&\frac{-1}{\tau_{\max}}R \end{bmatrix} < 0 $ (35)

where

$ \begin{align} \psi_{11}&\!\!=\!\! A^{T}PA-P+\tau_{\max}(A^{T}-I)R(A-I)+W_{1}+W_{1}^{T}\\ \psi_{12}&\!\!=\!\! A^{T}PBK+\tau_{\max}(A^{T}-I)RBK-W_{1}+W_{2}^{T}\\ \psi_{22}&\!\!=\!\! K^{T}B^{T}PBK+\tau_{\max}K^{T}B^{T}RBK-W_{2}-W_{2}^{T} \end{align} $

Remark 6: The basis of proving Theorem 3 is the application of Lyapunov-Krasoviskii stability theory to the following function:

$ V(x_{k}, k)=V_{1}(x_{k}, k)+V_{2}(x_{k}, k)+V_{3}(x_{k}, k) $ (36)

where

$ \begin{align} V_{1}(x_{k}, k)&\!\!=\!\!x^{T}(k)Px(k)\\ V_{2}(x_{k}, k)&\!\!=\!\!\sum\limits_{l=k-\tau(k)}^{k-1}x^{T}(l)Qx(l)\\ V_{3}(x_{k}, k)&\!\!=\!\!\sum\limits_{l=-\tau_{\max}+1}^{0} \sum\limits_{h=k-1+l}^{k-1}\zeta^{T}(h)R\zeta(h) \end{align} $

and

$ \begin{align*} x_{k} &= \begin{bmatrix} x^{T}(k)&x^{T}(k-1)&\ldots&x^{T}(k- \tau_{\max}) \end{bmatrix}^{T}\\ \zeta(k) &= x(k+1)-x(k) \\ \zeta(k) &= (A-I)x(k)+BKx\left(k-\tau(k)\right)\\ \sum\limits_{h=k-\tau(k)}^{k-1}\zeta(h) &= x(k)-x\left(k-\tau(k)\right). \end{align*} $

Another way of analyzing NCS is by using a structure of a Master-Slave system [112]-[114]. The control of the system is performed in one PC considered as a Master which communicates through a network with a Slave that includes another PC and the process is as shown in Fig. 8. Four delay sources are considered here: communication delay, data sampling, transmitting delay, and the possible packet losses. The Slave is considered to have the following linear form:

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Fig. 8 An NCS with a master-slave structure
$ \begin{align} \dot{x}(t)&=Ax(t)+Bu(t-\tau_{1}(t))\\ y(t)&=Cx(t) \end{align} $ (37)

where $(A, B, C)$ is controllable and observable and $\tau_{1}(t)$ is the total Master-to-Slave delay.

For a given $k$ and for any $t \in [t_{1, k}+h_{1m}, t_{1, k+1}+h_{1m}]$ , there exists a $k'$ such that the proposed observer is represented by

$ \begin{eqnarray} \hat{x}(t)&\!\!=\!\!A \hat{x}(t)+Bu(t_{1, k})-L(y(t_{2, k'})-\hat{y} (t_{2, k'})) \\ \hat{y}(t)&\!\!=\!\!C\hat{x}(t) \end{eqnarray} $ (38)

where $k'$ corresponds to the newest output information received by the Master. Now, given a signal $g(t)$ and the global delay $\tau (t)$ and packet loss delay $h(t_k)$ occurred during the transmission of the packet containing the $k^{th}$ sample at time $t_k$ , $g(t)$ can be described by

$ \begin{eqnarray} g(t_{k}-h(t_{k})) &\!\!=\!\! g(t-h(t_{k})-(t-t_{k})) \\ &\!\!=\!\!g(t-\tau(t)) \\ t_{k}\leq t < t_{k+1}, \tau(r)&\!\!=\!\!h(t_{k})+t-t_{k}+d \end{eqnarray} $ (39)

Now, by using (39), (38) is rewritten as

$ \begin{align} \hat{x}(t)&\!\!=\!\!A\hat{x}(t)+Bu(t-\tau_{1}(t)) \\ & - L(y(t-\delta_{2}(t))-\hat{y}(t-\tau_{2}(t))) \\ \hat{y}(t)&\!\!=\!\!C\hat{x}(t) \end{align} $ (40)

where $\tau_{1}(t) = h_{1, k}+t-t_{1, k}+d_1$ and $\tau_{2}(t) = h_{2, k'}+t$ $-t_{2, k'}+d_2$ , The characteristics of the system lead to $\tau_{1}(t)$ $\leq h_{1m}+T+d_1$ and $\tau_{2}(t)\leq h_{2m}+T+d_2$ . Also, the error vector between the estimated state $\hat{x}(t)$ and the real state $x(t)$ is defined as

$ e(t)=x(t)-\hat{x}(t). $ (41)

Thus, the error is represented as

$ \dot{e}(t)=Ae(t)-LCe(t-\tau_{2}(t)). $ (42)

Finally, a delay dependent state feedback control is designed by the Master based on Lyapunov-Krasovskii functional and the LMI approach, and a remote observer is used for estimating the states of the Salve. So, a uniform stability is achieved for the system and it is characterized by the following two theorems [113].

Theorem 4 [112]: Suppose that, for some positive scalars $\alpha$ and $\varepsilon$ , there exists $n\times n$ matrices $0 < P_1, P, $ $ S, Y_1, Y_2, Z_1, Z_2, Z_3, R, R_a$ and a matrix $W$ with appropriate dimensions such that the following LMI conditions are satisfied for $j = 1, 2$

$ \begin{gather*} \begin{bmatrix} \Xi_{2}&\begin{bmatrix} \beta_{2j}WC-{Y}_{1}\\ \varepsilon_{2j}WC-{Y}_{2} \end{bmatrix}&\mu_{2}\beta_{2}j \begin{bmatrix}WC\\ \varepsilon WC \end{bmatrix}\\ \ast&-{S}&0\\ \ast&\ast&-\mu_{2}{R}_{a} \end{bmatrix} < 0 \\ \begin{bmatrix} R&Y\\ \ast&Z \end{bmatrix} \ge 0 \end{gather*} $

where $\beta_{ij}, \; i=1, 2, \; j=1, 2$ are given by

$ \begin{eqnarray} \beta_{11}&\!\!=\!\!{\rm e}^{\alpha(\tau_{1}-\mu_{1})}, \quad \beta_{12}={\rm e}^{\alpha(\tau_{1}+\mu_{1})} \\ \beta_{21}&\!\!=\!\!{\rm e}^{\alpha(\tau_{2}-\mu_{2})}, \quad \beta_{22}={\rm e}^{\alpha(\tau+\mu2)} \end{eqnarray} $ (43)

and the matrices $Y, Z$ and $\Xi_2$ are defined as

$ \begin{eqnarray} Y=[Y_{1} \quad Y_{2}], \quad Z=\begin{bmatrix}{Z_{1}}&Z_{2}\\ {\ast}&Z_{3}\end{bmatrix} \end{eqnarray} $ (44)
$ \begin{align} \Xi_{2}^{11}&\!\!=\!\!P^{{T}}(A_{0}+\alpha I)+(A_{0}+ \alpha I)^{{T}}P+ S\\ &+\tau_{2}Z_{1}+Y_{1}+Y_{1}^{{T}}\\ \Xi_{2}^{12}&\!\!=\!\!P_{1}-P+\varepsilon P^{{T}} (A_{0}+\alpha I)^{{T}}+\tau_{2}Z_{2}+Y_{2}\\ \Xi_{2}^{22}&\!\!=\!\!-\varepsilon(P+P^{{ T}})+\tau_{2}Z_{3}+2\ \mu_{2}R_{a}+\tau_{2}R. \end{align} $

Then, the gain

$ L = (P^T)^{-1}W $ (45)

makes the error (42) of observer (40) exponentially converge to the solution $e(t) = 0$ , with a decay rate $\alpha$ .

The solution of the LMI problem corresponding to this theorem is written

$ L = LMI_{obs}(\mu_2, \tau_2, \alpha) $ (46)

For the control design, consider the controller $u = Kx, i=$ $1, 2$ , i.e., the ideal situation $e(t) = 0, x(t) = \hat{x}(t)$ and

$ \dot{x}(t) = Ax(t) + BKx (t - \tau_1 (t) ). $ (47)

Theorem 5 [112]: Suppose that, for some positive scalars $\alpha$ and $\varepsilon$ , there exists a positive definite matrix $\bar{P}_1$ , matrices of size $n\times n: 0 < \bar{P}, \bar{U}, \bar{Y}_1, \bar{Y}_2, \bar{Z}_1, \bar{Z}_2, \bar{Z}_3$ similar to (44) and an $n \times m$ matrix $W$ , such that the following LMI conditions hold

$ \begin{gather*} \Gamma_{3i}= \begin{bmatrix} \Xi_{3}&\begin{bmatrix} \beta_{1i}BW-\bar{Y}_{1}^T\\ \varepsilon_{1i}BW-\bar{Y}_{2}^T \end{bmatrix}&\mu_{1} \begin{bmatrix} \beta_{1i}BW\\ \varepsilon \beta_{1i}BW \end{bmatrix}\\ \ast&-\bar{S}&0\\ \ast&\ast&-\mu_{1}\bar{R}_{a} \end{bmatrix} < 0 \\ \forall i=1, 2 \\ \begin{bmatrix} \bar{R}&\bar{Y_1}&\bar{Y_2}\\ \ast&\bar{Z_1}&\bar{Z_2} \\ \ast&\ast&\bar{Z_3} \end{bmatrix} \ge 0 \end{gather*} $

where $\beta_{1i}$ for $i=1, 2$ are defined by (43) and

$ \begin{align} \Xi_{3}^{11}&\!\!=\!\!(A_{0}+\alpha I)\bar{P}+\bar{P}^T(A_{0}+\alpha I)^T + \bar{S} \\ &+ \tau_{1}\bar{Z}_{1}+\bar{Y}_{1}+\bar{Y}_{1}^T\\ \Xi_{3}^{12}&\!\!=\!\!\bar{P}_{1}-\bar{P}+\varepsilon \bar{P}^T(A_{0}+\alpha I)^T+\tau_{1}\bar{Z}_{2}+\bar{Y}_{2}\\ \Xi_{3}^{22}&\!\!=\!\!-\varepsilon(\bar{P}+\bar{P}^T)+\tau_{1}\bar{Z}_{3}+2\ \mu_{1}\bar{R}_{a}+\tau_{1}\bar{R} \end{align} $

Then, the gain

$ K = W \bar{P}^{-1} $ (48)

exponentially stabilizes the system (47) with the decay rate $\alpha$ for all delay $\tau_1(t)$ .

The solution of the LMI problem corresponding to Theorem 5 is written as

$ \begin{eqnarray} K = LMI_{con}(\mu_1, \tau_1, \alpha) \end{eqnarray} $ (49)

Remark 7: The input delay system approach was applied in other researches with other protocols like round-robin (RR) protocol [115] and quadratic protocol (QP) [116] and the results of both of them were extended and applied on a discrete NCS with actuator constraints [117] and the results were also generated for systems with more than two nodes [118].

B. Markovian System Approach

In this approach, the Markovian model is applied to represent the closed loop NCS. In [46], the Markovian system approach was applied on a vehicle control problem in order to study the effect of packet dropouts on the system which uses a wireless local network, and it has the following description:

$ \begin{align} x(k+1)= &A_{\theta (k)}x(k)+B_{\theta (k)}u(k) \\ y(k)=&C_{\theta (k)}x(k) \\ x(0)= &x_0, \theta(0)=\theta _0 \end{align} $ (50)

where $\theta (k)$ is the time-varying dependence of the state matrices via the network packet losses parameters, and $\theta$ is either equal to (0) when the packet from sensor is dropped or equal to (1) when the packet is received. Thus, $\theta (k)$ is represented as $\theta (k)=\sigma \in \{0, 1\}$ . The state matrices are functions of a discrete-time Markov chain with values in a finite set $\mathcal{N}$ $=\{1, \ldots, N\}$ . The Markov chain has a transition probability matrix $\mathcal{P} = [p_{ij}]$ where $p_{ij}= Pr(\theta (k + 1)= j|\theta(k) = i)$ subject to the restrictions $p_{ij}\geq0$ and $\sum^N_{j=1} p_{ij}=1$ for any $i \in \mathcal{N}$ , which means that jumping probability is positive and that the Markov chain has to jump from mode $i$ into some state that has a probability of one. Using the above model, the stochastic stability is derived for system (50) and represented in an LMI format as follows:

Theorem 6 [119]: System (50) is mean square stable (MSS) iff there exists $G>0$ such that

$ G-\sum\limits_{j=1}^N p_jA_j^TGA_j. $ (51)

Now, the objective is to design a dynamic output feedback controller that has the following form:

$ \begin{align} x_c(k+1)=&A_{c, \theta (k)}x_c(k)+B_{c, \theta (k)}y_c(k) \\ u(k)= &C_{c, \theta (k)}x_c(k) \end{align} $ (52)

where $x_c(k) \in \mathbb{R}^n$ is the controller state and the subscript $c$ denotes the controller matrices/states. Again, for $\theta (k)=\sigma$ $ \in \{0, 1\}$ , $A_{c\sigma}$ , $B_{c\sigma}$ , and $C_{c\sigma}$ are used to denote the state space matrices of this two mode controller. The closed loop system is described by:

$ \begin{eqnarray} A_{cl, \sigma}=\begin{bmatrix} \bar{A}_{\sigma}&\bar{B}_{\sigma}C_{c\sigma} \\ B_{c\sigma}\bar{C}_{\sigma}&A_{c\sigma} \end{bmatrix} \end{eqnarray} $ (53)

where the subscript "$cl$ " denotes the closed loop matrices. By applying Theorem 6 and by using Schur complements, system (53) is MSS for the following condition [46]:

$ \begin{eqnarray} \begin{bmatrix} Z&\ast&\ast \\ \sqrt{p}A_{cl, 0}Z&Z&0\\ \sqrt{1-p}A_{cl, 1}Z&0&Z \end{bmatrix} > 0 \end{eqnarray} $ (54)

where $Z=G^-1$ .

In a similar way the model for a NCS with known packet loss was derived as a Markovian jumping system, and then an $H_{\infty}$ controller was designed for that system [120]. [121] studied the stochastic stability of discrete-time NCS with random delays. Two Markov chains were used to represent the sensor-to-controller and controller-to-actuator delays, and the obtained closed-loop systems are two modes jump linear systems. In [122] the stability of the discrete time NCS contains polytopic uncertainty was considered, where a smart controller is updated with the buffered sensor information at stochastic intervals and the amount of the buffered data received by the controller under the buffer capacity constraint is also random. The exponential stability of generic switched NCS and the exponential mean-square stability of Markov-chain driven NCS were assured by establishing the sufficient conditions.

Other examples of literature that had used Markovian system approach are [123]-[128].

Remark 8: A special class of hybrid and stochastic system is called Markovian jump system. This system is applicable in many real systems such as manufacturing systems, power, chemical, economic, communication and control systems. A Markovian jump time-delay system model considering external disturbances of an event triggered NCS was presented in [129], the $H_\infty$ control problem was solved and sufficient conditions to ensure stability of the closed-loop system were derived. More details about Markovian jump systems could be found in [130].

C. Switched System Approach

The NCS in switched system approach is represented by a discrete-time switched system with a finite number of subsystems. In [131], a discrete-time switched systems with arbitrary switching was formulated for an NCS while considering the effects of bounded uncertain access delay and packet losses. Then, both of asymptotic stability and $L_{\infty}$ persistent disturbance attenuation issues were investigated. The plant is represented by the following continuous linear time-invariant system:

$ \begin{eqnarray} \dot{x}(t)&\!\!\!=\!\!\!A_c x(t)+B_c u(t)+E_c d(t) \\ z(t)&\!\!\!=\!\!\!C_c x(t) \end{eqnarray} $ (55)

where $t\in \mathbb{R}^+$ , $\mathbb{R}^+$ is the positive real numbers, $x(t)\in \mathbb{R}^n$ is the state variable, $u(t)\in \mathbb{R}^m$ is the control input, and $z(t)\in \mathbb{R}^p$ is the controlled output. The disturbance input $d(t)$ is contained in $\mathcal{D}\subset{\mathbb{R}^r}$ , and $A_c, B_c, E_c, C_c$ are constant matrices. The control signal is assumed to be time-varying within a sampling period, and the sampling period is divided into a number of sub-intervals in which the controller reads its buffer at a higher frequency than the sampling frequency. The discrete time model of the system is described by

$ \begin{eqnarray} x(k+1)&\!\!=\!\!Ax(k)+B(u_1(k)+u_2(k)+\cdots+u_N(k)) \\ &+Ed(k) \end{eqnarray} $ (56)

where

$ \begin{align*}&A=e^{A_c T_s}\\& B=\int_0^{\frac{T_s}{N}} e^{A_c \eta} B_c d\eta\\&E=\int_0^{T_s} e^{A_c \eta} E_c d\eta\end{align*} $

and $C=C_c$ . Then, the time delay and dropouts effects were added to the model. There are three main scenarios which could occur during each sampling period.

1) If the delay $\tau = h \times T$ , where $T = {T_s}/{N}$ , and $h =1, 2, \ldots, d_{1_{\max}}$ , then $u_1[k] = u_2[k] = \cdots = u_h[k] = u[k-1], u_{h+1}[k] = u_{h+2}[k] = \cdots = u_N[k] = u[k]$ , and (56) can be written as

$ \begin{eqnarray} &x(k+1)=\!\!Ax[k]+h\cdot Bu[k-1]+(N-h)\cdot Bu[k] \\ &+ Ed[k]. \end{eqnarray} $ (57)

2) If a packet-dropout happens with delay less than $\tau_{\max}$ , then the actuator will implement the previous control signal, i.e., $u_1[k] = u_2[k] = \cdots = u_N[k] = u[k-1]$ . Therefor, the state transition equation (56) for this case can be written as

$ x[k+1]=Ax[k]+N\cdot Bu[k-1]+Ed[k]. $ (58)

3) The packets are dropped periodically, with period $T_m$ which is an integral multiple of the sampling period $T_s$ , i.e., $T_m = mT_s$ . In case of $m = {T_m}/{T_s}\geq 2$ , the first $(m_1)$ packets are dropped. Then, for these first $(m_1)$ steps, the previous control signal is used. Therefore

$ \begin{align} x(kT_m + T_s) &\!\!=\!\! Ax(kT_m) + NBu(kT_m - T_s) \\ &+ Ed(kT_m) \\ x(kT_m + 2T_s) &\!\!=\!\! A_2x(kT_m) \\ &+ N (AB + B)u(kT_m - T_s) \\ &+ AEd(kT_m) + Ed(kT_m + T_s) \\ &\vdots \\ x(kT_{m}+(m-1)T_{s})&\!\!=\!\!A_{m-1}x(kT_{m}) \\ &+ N\sum\limits_{i=0}^{m-2}A_{i}Bu(kT_{m}-T_{s}) \\ &+ [A_{m-2}E, \ldots, E] \\ &+ \begin{bmatrix} d(kT_{m})\\ \vdots \\ d(kT_{m} +(m-2)T_{s}) \end{bmatrix} \end{align} $

where the integer $N = {Ts}/{T}$ , and $T$ is the period of the controller reading its receiving buffers.

During the period $t \in [kT_m + (m- 1)T_s, (k +1)T_m)$ , the new packet is transmitted successfully with some delay, such that $\tau = h\frac{T_s}{N}$ , where $h = 0, 1, 2, \ldots, d_{\max}$ . And Assuming that $d(kT_m) = d(kT_m+1) =\cdots = d(kT_m+m-1)$ , and that the controller uses just the time-invariant linear feedback control law, $u(t) = Kx(t)$ . Then, we may obtain

$ \begin{eqnarray} x((k+1)T_{m})&\!\!=\!\![A_{m}+(N-h)BKA_{m-1}]x(kT_{m}) \\ &+ \Pi_1 + \Pi_2 \end{eqnarray} $ (59)

where

$ \begin{align} \Pi_1&\!\!=\!\![N\sum\limits_{i=1}^{m-1}A_{i}+(N-h)NBK\sum\limits_{i=0}^{m-2}A_{i}+h] \\ & \times BKx(kT_{m}-T_{s})\\ \Pi_2&\!\!=\!\![(N-h)BK\sum\limits_{i=0}^{m-2}A_{i}+\sum\limits_{i=0}^{m-1}A_{i}]Ed(kT_{m}). \end{align} $

Now, by defining $\hat{x}[k]=\begin{bmatrix} x(kT_m - Ts)\\x(kT_m) \end{bmatrix}$ , then the above equations can be written as

$ \begin{align} \hat{x}[k+1] &\!\!=\!\! \begin{bmatrix} x((k +1)T_{m} -T_{s})\\ x((k +1)T_{m}) \end{bmatrix} \\ &\!\!=\!\! \begin{bmatrix} \phi_1&\phi_2 \\ \phi_3&\phi_4\end{bmatrix} \begin{bmatrix} x(kT_{m}-T_{s})\\ x(kT_{m})\end{bmatrix} +E_{m}d(kT_{m}) \end{align} $

where

$ \begin{align} \phi_1 &\!\!=\!\! N\sum\limits_{i=0}^{m-2}A_{i}B{K} \\ \phi_2 &\!\!=\!\! A_{m-1}\\ \phi_3 &\!\!=\!\! N\sum\limits_{i=0}^{m-2}A_{i}B{K} + (N-h)NBK\sum\limits_{i=0}^{m-2}A_{i}+h)BK \\ \phi_4 &\!\!=\!\! A_{m}+(N-h)BKA_{m-1}A_{m-1} \\ E_{m}&\!\!=\!\!\begin{bmatrix} \sum\limits_{i=0}^{m-2}A_{i}E\\ (N-h)BK\sum\limits_{i=0}^{m-2}A_{i}E+\sum\limits_{i=0}^{m-1}A_{i}E \end{bmatrix}. \end{align} $

Here $m = {T_m}/{T_s}\geq 2$ in this case, and $h = 0, 1, \ldots, d_{\max}$ .

The aforementioned system could be represented in the following format:

$ \hat{x}[k+1] = \Phi_{q}\hat{x}[k]+E_{q}d[k]. $ (60)

Theorem 7 [131]: If the set ${\cal P}^{(k)} \subset int\{X_0(\mu)\}$ for some $k$ , then the switched system (60) does not admit a positive disturbance invariant set under arbitrary switching in $X_0(\mu)$ , In other words $\mu < \mu_{\infty}$ , where

${\cal P}$ is a positive disturbance invariant for the switched system (60) with arbitrary switching such that

$ \begin{align} {\cal P}^{(k)}&\!\!=\!\!{\cal P}^{(k-1)} \bigcap\underline{pre}({\cal P}^{(k-1)}), \quad k=0, 1, \ldots, m-2 \\ {\cal P}^{(0)}&\!\!=\!\!X_{0}(\mu). \end{align} $

$\mu_{\infty}$ is the $l_{\infty}$ induced norm from $d[k]$ to $z[k]$ and defined as

$ \begin{align} \mu_{inf}=\inf\{\mu\ \colon\ \Vert z[k]\Vert_{l^{\infty}}\leq\mu\quad \forall d[k], \Vert d[k]\Vert_{l^{\infty}}\leq 1\}. \end{align} $

Remark 9: In [131], a systematic proof and detailed stability analysis and $l^{\infty}$ persistent disturbance attenuation for the above NCS were developed. Similar works are presented in [132]-[136].

In [137], the continuous time NCS is modeled as an event based discrete-time model while allowing nonuniform sampling and varying delay larger than a sampling period, then the stability is achieved by solving a control problem for a switched polytopic system with an additive norm bounded uncertainty. A discrete-time switched linear uncertain system was used to model an NCS that includes time-varying transmission intervals, time-varying transmission delays, and communication constraints [138]. At each single transmission, only one known node is allowed to communicate with the network for transmitting its data. Then, a convex over-approximation method in the form of a polytopic system with norm-bounded additive uncertainty was used to determine the stability criteria for this NCS and represent it in an LMI format. The same procedure was followed but with consideration of the quantization of the NCS [107]. The asymptotic stability was assured based on quantizer with finite quantization level and the quantizer parameters were suitably adjusted [106].

Remark 10: A switched system approach is implemented for NCS with network-induced delays by defining a switching function. The closed-loop NCS is represented by a time-delay switched system with two switching modes and each mode has a controller with a different gain [139]. Stability analysis is carried out based on both the time-delay switched system model and the average dwell time technique. Similarly, the exponential stability is achieved [140].

Other examples of the switched system approach could be found in [141]-[147].

D. Stochastic System Approach

The stochastic system approach is applied when the network-induced delays and/or packet dropouts are random. [148] has discussed the stability of NCS with stochastic input delays, considering the following delayed NCS:

$ \dot{x}(t)=Ax(t)+Bu(t-\tau (t)) $ (61)

where $\tau (t)$ refers to a piecewise continuous and bounded time delay including the delay from sensor to actuator, the delay from actuator to sensor, and the effect of dropouts. Assuming that the probability distribution of $\tau (t)$ is known a priori which takes values in $[0, \hat{\tau}_1]$ and $(\hat{\tau}_1, \hat{\tau}_2]$ , and $0\le \hat{\tau}_1 \le \hat{\tau}_2$ , also, two random events $\mathcal{F}_1, \mathcal{F}_2$ are defined for the stochastic input delay $\tau (t)$ such that: $\mathcal{F}_1: \tau (t) \in [0, \hat{\tau}_1]$ and $\mathcal{F}_2: \tau (t) \in (\hat{\tau}_1, \hat{\tau}_2]$ , and a random variable $\delta (t)$ is defined such that $\delta (t)=1$ when $\mathcal{F}_1$ occurs and $\delta (t)=0$ when $\mathcal{F}_2$ occurs. Moreover, two functions are defined $\tau_1:\mathbb{R}^+\rightarrow [0, \hat{\tau}_1]$ and $\tau_2:\mathbb{R}^+\rightarrow [\hat{\tau}_1, \hat{\tau}_2]$ such that

$ \begin{eqnarray} \tau_1 (t)&\!\!=\!\! \left\{\begin{matrix} \tau(t),&{\rm if} \delta (t)=1 \\ \hat{\tau}_1,&{\rm if} \delta (t)=0 \end{matrix}\right. \\ \tau_2 (t)&\!\!=\!\! \left\{\begin{matrix} \hat{\tau}_1,&{\rm if} \delta (t)=1 \\ \tau(t),&{\rm if} \delta (t)=0. \end{matrix}\right. \end{eqnarray} $ (62)

Now, by using (62), (61) can be rewritten as

$ \begin{eqnarray} \dot{x}(t)&\!\!=\!\!Ax(t)+\delta(t)Bu(t-\tau_1 (t)) \\ &+(1-\delta(t))Bu(t-\tau_2 (t)) \end{eqnarray} $ (63)

where $u(t)=Kx(t)$ , $K$ is a constant matrix to be designed, and $\delta(t)$ is assumed to have a Bernoulli distribution sequence with $Prob\{\delta(t)=1\}=E\{\delta(t)\}=\delta_0$ and $Prob\{\delta(t)=0\}$ $=1-E\{\delta(t)\}=1-\delta_0$ and $0\le \delta_0 \le 1$ is a constant. The closed-loop system of (63) is described by the following formula:

$ \begin{eqnarray} \dot{x}(t)&\!\!=\!\!Ax(t)+\delta(t)BKx(t-\tau_1 (t)) \\ &+(1-\delta(t))BKx(t-\tau_2 (t)). \end{eqnarray} $ (64)

Theorem 8 [148]: System (64) is exponentially stable in the mean-square sense if, for given constants $\hat{\tau}_1$ , $\hat{\tau}_2$ , $\delta_0$ and matrix $K$ , there exist matrices $P>0, Q_i>0, R_i>0, Z_i>0 (i=1, 2), N_j, M_j, T_j, W_j (j=1, 2, 3, 4, 5, 6)$ , and $S_l (l$ $=1, 2, 3, 4)$ of appropriate dimensions such that the following LMIs hold:

$ \begin{eqnarray} \begin{bmatrix} \Xi_{11}&\ast &\ast \\ \Xi_{21}^l&\Xi_{22}&\ast\\ \Xi_{31}&0&\Xi_{33} \end{bmatrix} < 0, \quad l=1, 2, 3, 4 \end{eqnarray} $ (65)

where $\Xi_{11}$ , $\Xi_{21}^l$ , $\Xi_{22}$ , $\Xi_{31}$ , and $\Xi_{33}$ are functions of $\hat{\tau}_1$ , $\hat{\tau}_2$ , $\delta_0$ , $K$ , $P, Q_i, R_i, Z_i, N_j, M_j, T_j, W_j$ and $S_l$ .

Remark 11: Consider the following candidate function

$ V(x_t) = \sum\limits_{i=1}^7 V_i(x_t) $ (66)

with

$ \begin{align} V_1(x_t) &\!\!=\!\! x^T(t)Px(t) \\ V_2(x_t) &\!\!=\!\! \int_{t-\hat{\tau}_1}^{t}x^T(s)Q_1x(s)ds \\ V_3(x_t) &\!\!=\!\! \int_{t-\hat{\tau}_2}^{t}x^T(s)Q_2x(s)ds \\ V_4(x_t) &\!\!=\!\! \int_{t-\hat{\tau}_1}^{t} \int_{s}^{t} y^T(v)R_1y(v)dvds \\ V_5(x_t) &\!\!=\!\! \int_{t-\hat{\tau}_2}^{t-\hat{\tau}_1} \int_{s}^{t} y^T(v)R_2y(v)dvds \\ V_6(x_t) &\!\!=\!\! \delta_0(1-\delta_0) \int_{t-\hat{\tau}_1}^{t} \int_{s}^{t} \zeta^T(v)\beta^TZ_1\beta\zeta(v)dvds \\ V_7(x_t) &\!\!=\!\! \delta_0(1-\delta_0) \int_{t-\hat{\tau}_2}^{t-\hat{\tau}_1} \int_{s}^{t} \zeta^T(v)\beta^TZ_2\beta\zeta(v)dvds \end{align} $

Application of Lyapunov-Krasovskii stability theory to (66) forms the basis of the proof.

A similar result could be found in [149]. In [150], random communication delays from sensor to the controller and from controller to actuator through a limited bandwidth communication channel were represented by a linear function of the stochastic variable satisfying Bernoulli random binary distribution, then the exponential stability is achieved by applying an $H_\infty$ controller. The stability was discussed using a stochastic computational technique in NCS by representing transmission intervals and transmission delays as a sequence of continuous random variables [151]. Properties for some input-output stability were derived for a general class of nonlinear NCS with exogenous disturbances using stochastic protocols subject to random network-induced delays and packet losses [152]. Other examples of using stochastic approach in NCS are found in [153]-[158].

E. Impulsive System Approach

In this approach, the NCS is represented by a hybrid discrete/continuous model or in other words "impulsive system". An LTI system with uncertainties in the parameters of the process and intervals of sampling was modeled as a linear impulsive system described by the following equations [159]:

$ \begin{eqnarray} \dot{x}(t)&\!\!=\!\!f_k(x(t), t), t\neq s_k \forall k \in \mathbb{N} \\ x(s_k)&\!\!=\!\!g_k(x(s_k^-), s_k), t=s_k \forall k \in \mathbb{N} \end{eqnarray} $ (67)

where $f_k$ and $g_k$ are locally Lipschitz functions from $\mathbb{R}^n X \mathbb{R}$ to $\mathbb{R}^n$ such that $f_k(0, t)=0, g_k(0, t)=0, \forall t\geq 0$ , and the impulse time sequence $s_k$ forms a strictly increasing sequence in $[s_0, \infty)$ for some initial time $s_0 \geq 0$ . Then, a discontinuous Lyapunov function at the impulse times was used for achieving the exponential stability of system (67). For an NCS with variable sampling time, the objective is to find the maximum allowable time interval $\tau_{MATI}$ for the following general LTI system [159]

$ \dot{x}(t)= Ax(t)+Bu(t) $ (68)

where $x, u$ are the state and input of the process respectively. At the sampling time $s_k, k\in \mathbb{N}$ the state of the system $x(s_k)$ , is sent to the controller and the control input $u=Kx(s_k)$ is sent back to the actuator. The resulting closed loop system is modeled as an impulsive system with state $\xi(t):=[x'(t) z'(t)]$ where $z(t):=x(s_k), t \in [s_k, s_{k+1})$ . So, the impulsive system is describe by:

$ \begin{eqnarray} \dot{\xi}(t)&\!\!=\!\!F\xi(t), \qquad t\ne s_k \forall k \in \mathbb{N} \\ \xi(s_k)&\!\!=\!\!\left[\begin{array}{c} x(s_k^-)\\x(s_k^-) \end{array} \right], t=s_k \forall k \in \mathbb{N} \end{eqnarray} $ (69)

where $F=\left[\begin{array}{cc} A&BK \\ 0&0 \end{array} \right]$ .

Theorem 9 [160]: System (69) is stable if there exist symmetric positive definite matrices $P, R, X_1$ and a matrix $N$ such that

$ M_1+\tau_{MATI}M_2 < 0 $ (70)
$ \begin{align} \left[\begin{array}{cc} M_1&\tau_{MATI}N \\ \ast & -\tau_{MATI}R \end{array} \right] < 0 \end{align} $ (71)

where

$ \begin{align} M_1&\!\!=\!\!\left[\begin{array}{c} P \\ 0 \end{array} \right] [A BK]+\left[\begin{array}{c} A^T \\ (BK)^T \end{array} \right][P 0]\\ &-\left[\begin{array}{c} I \\-I \end{array} \right]X_1[I-I] -N[I-I]-\left[\begin{array}{c} I \\-I \end{array} \right]N^T\\ &+ \tau_{MATI} \overline{F}^TR\overline{F}, \\ M_2&\!\!=\!\!\left[\begin{array}{c} I \\-I \end{array}\right]X_1\overline{F} + \overline{F}X_1^T [I-I], \overline{F}=[A BK]. \end{align} $

Remark 12: Application of Lyapunov-Krasovskii stability theory to the following candidate function:

$ V(\xi, \rho) := V_1(x) + V_2(\xi, \rho) + V_3(\xi, \rho) $ (72)

where

$ \begin{align} V_1(x)&\!\!=\!\!x^TPx \\ V_2(\xi, \rho)&\!\!=\!\! \xi^T \left ( \int_{-\rho}^{0}(s+\tau_{MATI})(Fe^{Fs})^T \tilde{R}Fe^{Fs} ds \right ) \xi \\ V_3(\xi, \rho)&\!\!=\!\! (\tau_{MATI}-\rho)(x-z)^TX_1(x-z) \\ \tilde{R}&\!\!=\!\!\begin{bmatrix} R&0 \\ 0&0 \end{bmatrix} \\ \rho(t)&\!\!=\!\!t-s_k, \quad t\in [s_k, s_{k+1}) \forall k \in \mathbb{N} \end{align} $

with $R$ , $P$ , and $X_1$ as symmetric positive definite matrices.

The Razumikhin technique and Lyapunov functions are applied to achieve exponential stability of an NCS subject to variable bounded delay by applying impulsive control [161]. A model of a threshold-error-dependent augmented impulsive system with an interval time-varying delay is used to design a dissipative control for model-based NCS subject to event triggered communication [162]. A delay scheduled impulsive (DSI) controller is presented to achieve a robust stability of NCS subject to integral quadratic constraint and delays [162]. Other examples of this approach could be found in [163]-[167].

F. Predictive Control Approach

In this approach, a network predictive controller (NPC) is designed in order to compensate for the effect of time delays and packet dropouts in the network. The NPC scheme consists of two parts as shown in Fig. 9: a control prediction generator and a compensator. A set of future control predictions is generated, packed, and transmitted to the plant side by the control prediction generator based on the signals received from the sensor. Using the most recent control value from the latest control prediction sequence, the compensator is designed to compensate for the delays and dropouts that occur from sensor to controller or from controller to actuator channels to achieve the desired performance.

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Fig. 9 A network predictive controller scheme

In [168], an NPC was presented for compensating for the network-induced delay, for the following discrete-time system:

$ \begin{eqnarray} x_{k+1}&\!\!\!=\!\!\!Ax_k +Bu_k \\ y_k&\!\!\!=\!\!\!Cx_k \end{eqnarray} $ (73)

where $x_k\in \mathbb{R}^n$ is the state vector, $u_k$ and $y_k\in \mathbb{R}$ are the input and output vectors, respectively. $A$ , $B$ , and $C$ are system matrices. An output feedback controller is used such that $u_k$ $=K_ky_k$ . The NPC generates state predictions on the controller side up to time $t + \tau$ as follows:

$ \begin{eqnarray} \begin{matrix} \hat{x}_{k-d_k+1|k-d_k}=Ax_{k-d_k}+B\hat{u}_{k-d_k} \\ \vdots \\ \hat{x}_{k-d_k+i|k-d_k}=Ax_{k-d_k+i-1}+B\hat{u}_{k-d_k+i-1} \end{matrix} \end{eqnarray} $ (74)

where $i=2, 3, \ldots, d_k+\tau_k, \ldots, d_k+\tau$ and $\hat{x}_{k-d_k+1|k-d_k}$ is the state prediction for time $k-d_k+1$ on the basis of the information at time $k-d_k$ . And on the controller side, the following control signals are generated

$ \begin{eqnarray} U_{k|k-d_k}=\begin{bmatrix} \hat{u}_{k|k-d_k}, \hat{u}_{k+1|k-d_k}, \ldots, \hat{u}_{k+\tau|k-d_k} \end{bmatrix} \end{eqnarray} $ (75)

where

$ \begin{eqnarray} \hat{u}_{k+i|k-d_k}=K_{i, d_k}\hat{y}_{k+i|k-d_k} \end{eqnarray} $ (76)

and $\hat{u}_{k+i|k-d_k}$ is the control prediction and $\hat{y}_{k+i|k-d_k}$ is the output prediction. The control input on the actuator side is considered as

$ \begin{eqnarray} u_k=\hat{u}_{k|k-\tau_k-d_k}. \end{eqnarray} $ (77)

The output feedback control input of the system (73) after applying the NCS strategy is calculated as

$ \begin{eqnarray} u_{k} &\!\!=\!\! K_{d_{k}, \tau_{k}} y_{k - d_k - \tau_k} \\ &\!\!=\!\! K_{d_k, \tau_k}C\bar{U} \end{eqnarray} $ (78)

where

$ \begin{align} \bar{U}=\left(A^{d_k + \tau_k}x_{k - d_k - \tau_k} + \sum\limits_{i = 1}^{d_k + \tau_k}A^{d_{k} + \tau_k - i}B\hat{u}_{k - d_{k} - \tau_{k} + i - 1}\right). \end{align} $

Then, the Markovian jump systems with discrete and distributed delays could be written as

$ \begin{eqnarray} x_{k + 1} &\!\!=\!\! Ax_k + BK_{d_k, \tau_k}CA^{d_k + \tau_k}x_{k - d_k - \tau_k} \\ &+ BK_{d_k, \tau_k}\sum\limits_{\iota = 1}^{d_k + \tau_k}\mu_{\iota} \hat{u}_{k - \iota} \end{eqnarray} $ (79)

where $\iota=d_k+\tau_k-i+1$

Theorem 10 [168]: For the NCS in (73) with random delays $d_k$ and $\tau_k$ in the feedback and forward channels, where $d_k$ and $\tau_k$ are Markov processes, respectively, the closed-loop system in (79) with the predictive controller (78) is stochastically stable if there exist symmetric positive definite matrices $S_{i, j}$ , $R$ , and $Q$ such that the following matrix inequality holds

$ \begin{eqnarray} \Omega_{i, j} + \Lambda^{T}_{i, j}\bar{S}_{i, j}\Lambda_{i, j} + \bar{\beta}_{i, j}\Gamma^{T}_{i, j} Q \Gamma_{i, j} < 0 \end{eqnarray} $ (80)

where for all the $i$ and $j$

$ \begin{align} \Omega_{i, j} &\!\!=\!\! U_{1}^{T}\left(\bar{\alpha}R - S_{i, j} \right)U_{1} - U^{T}_{2}RU_{2} - {1\over \mu_{i + j}}U^{T}_{3}QU_{3} \\ \bar{\beta}_{i, j} &\!\!=\!\! \bar{\mu}_{i + j} + \bar{\mu}_{d + \tau}\left(1 - p_{ii}\pi_{jj}\right). \end{align} $

Remark 13: Consider the following candidate function

$ \begin{eqnarray} V(t) = V_{1}(k) + V_{2}(k) + V_{3}(k) + V_{4}(k) + V_{5}(k) \end{eqnarray} $ (81)

with

$ \begin{align*} &V_{1}(k) = x_k^{T}S(d_k, \tau_k)x_k\\ &V_{2}(k) = \sum\limits_{\nu = k - d_k - \tau_k}^{k - 1}x^{T}_{\nu}R x_{\nu}\\ &V_{3}(k) = \sum\limits_{\iota = 0}^{d + \tau - 1}\sum\limits_{\nu = k - \iota}^{k - 1}x_{\nu}^{T}\bar{R}x_{\nu}\\ &V_{4}(k) = \sum\limits_{\iota = 1}^{d_k + \tau_k}\mu_{\iota}\sum\limits_{\nu = k - \iota}^{k - 1}\hat{u}^{T}_{\nu}Q\hat{u}_{\nu} \\ &V_{5}(k) = \sum\limits_{\iota = 1}^{d + \tau}\mu_{\iota}\sum\limits_{\omega = 1}^{\iota - 1}\sum\limits_{\nu = k - \omega}^{k - 1}\hat{u}^{T}_{\nu}\bar Q\hat{u}_{\nu}. \end{align*} $

Application of Lyapunov-Krasovskii stability theory to (81) proves Theorem 10.

Other examples of implementing this approach could be found in [169]-[174].

Ⅳ. ADVANCED ISSUES IN NCS A. Decentralized and Distributed NCS

In addition to the centralized configuration of NCS in which a single controller communicates with the system, there are two other configurations which are: decentralized and distributed configurations. Large scale systems are usually modeled as a system of systems or system with subsystems with interconnection among them and several controllers are used for controlling the overall system, these controllers either have a communication among them as shown in Fig. 10 and it is called decentralized configuration or work separately as shown in Fig. 11 and it is called distributed configuration.

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Fig. 10 Model of a decentralized NCS
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Fig. 11 Model of a distributed NCS

The nodes of the controllers in the decentralized configuration do not share information with nearby nodes. Although the system may have one objective, this causes each controller to work locally. Because of lack of information, a sub-optimal control performance may be achieved. Moreover, a deterioration in the system performance and a limitation in the application scope of the decentralized configuration may be occurred in wireless NCS due to the absence of communication and cooperation between decentralized controllers [18]. Examples of decentralized NCS are discussed in [175]-[179].

The main features in distributed configuration is that, the exchange of information of each subsystem among components of the system and the plants in order to achieve the objective of the system and which usually contains large number of interacting physical units, can be physically distributed and interconnected to others to coordinate their tasks, and this leads to the so-called cooperative control [18]. Since sharing of local information is allowed among the distributed controllers in this configuration, they have the capability of coordination which leads to modularity, scalability and robustness. Examples of distributed NCS are found in [180]-[186].

B. Cloud Control System

Cloud computing is one of the recent essential tools in industry since it provides customers with a high powerful computation power and it reduces the requirements of the storage. And this opens new windows to the control techniques. Cloud control systems have become one of the most promising directions [187]. The structure of cloud control systems is shown in Fig. 12. The cloud computing system provides a medium of configurable resources including computation, software, data access, and storage services for practical systems while customers do not need to know the real location and configuration of the service provider during their using of it.

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Fig. 12 Cloud control systems

The requirements for computation and communication are increasing in cloud systems due to the increase of the system scale. Generally speaking, in the absence of powerful tools and appropriate system information, most of the complex systems cannot be controlled properly. But, a necessary platform for computability is provided by the development of new technologies, including recent innovations in software and hardware. Additionally, big data faces a lot of challenges such as: storage, search, capturing, transfer, sharing, visualization, analysis, etc. In the cloud control systems, big data will be transmitted to the centers of cloud computing to be treated firstly. Then, control signals, such as scheduling schemes, predictive control sequences and any other useful information will be generated instantly for cloud control systems. So, a powerful tool for controlling the complex system, which were not available before, is provided by cloud control systems.

Let us consider the following discrete dynamic system $S$ with unknown process disturbances and measurement noises

$ \begin{align} x(k+1) &=f(x(k), u(k), w(k)) \\ y(k) &=g(x(k), u(k), v(k)) \end{align} $ (82)

where $x(k)$ is the system state, $u(k)$ and $y(k)$ are the system input and output respectively; $f(x(k), u(k), w(k))$ and $g(x(k), u(k), v(k))$ are general models which can be linear or nonlinear. $w(k)$ and $v(k)$ are the unknown process disturbances and the unknown measurement noises respectively. For NCS where there are network induced time delays and data dropouts, it has been proven that the networked predictive method is a very effective method [15] and [187]. To estimate the state of the system (82) and then produce the predictive states with finite horizon $N_1$ , Kalman filter is adopted as follows:

$ \hat{x}(k\vert k)= KF(S, \hat{u}(k-1\vert k-1), y(k)) $ (83)
$ \hat{x}(k+i\vert k)= KF(S, \hat{u}(k\vert k), y(k)), i=1, 2, \ldots, N_{1} $ (84)
$ \hat{u}(k+i\vert k)= K(k+i)\hat{x}(k+i\vert k), i=1, 2, \ldots, N_{1} $ (85)

where $KF$ represents the compact form of Kalman filter expression and $K(k+i)$ is time-varying Kalman filter gain. A networked predictive control scheme consisting of a control prediction generator and a network delay compensator is proposed to overcome unknown random network transmission delays [187]. The network delay compensator considers the recent control value from the control prediction sequences available on the plant. When there is no time delay in the sensor-to-controller channel and the time delay from controller to actuator is $k_i$ then, the following predictive control sequences are received on the plant side:

$ \begin{eqnarray} & \left[u_{{t-k_{1}}\vert t-k_{1}}^{T}, u_{{t-k_{1}+1}\vert t-k_{1}}^{T}, \ldots, u_{t \vert t-k_{1}}^{T}, \ldots, u_{\iota_1}^T \right]^{T} \\ & \left[u_{{t-k_{2}}\vert t-k_{2}}^{T}, u_{{t-k_{2}+1}\vert t-k_{2}}^{T}, \ldots, u_{t \vert t-k_{2}}^{T}, \ldots, u_{ \iota_2}^T \right]^{T} \\ & \qquad \qquad \vdots \\ & \left[u_{{t-k_{t}}\vert t-k_{t}}^{T}, u_{{t-k_{t}+1}\vert t-k_{t}}^{T}, \ldots, u_{t \vert t-k_{t}}^{T}, \ldots, u_{ \iota_N}^T \right]^{T} \end{eqnarray} $ (86)

where $\iota_i=t+N-k_i \vert t-k-i$ and the control values $u_{t \vert t-k_i}$ for $i = 1, 2, \ldots, t$ , are available to be chosen as the control input of the plant at time $t$ , the output of the network delay compensator will be

$ u_{t}=u_{t\vert t-\min\{k_{1}, k_{2}, \ldots, k_{t}\}}. $ (87)

The controller sends packets to the plant node:

$ u(k + i\vert k)\vert i = 0, 1, \ldots, N. $ (88)

At each time instant $k$ , the actuator selects a preferable control signal to be the actual input of the plant:

$ u(k) = u(k\vert k - i) $ (89)

where $i ={\rm arg min}_i \{u(k|k - i) {\rm is available}\}$ .

In [188], a cloud predictive control scheme for networked multiagent systems (NMASs) via cloud computing was presented in order to achieve both consensus and stability simultaneously and for compensating actively the communication delays. More information about cloud control could be found in [189], [190].

Remark 14: The principle of cooperative cloud control system is used to describe the system in which two or more cloud controllers are used to achieve the control objectives in form of cooperation. Reasonable allocation for control objectives is however, a difficult dynamic problem [189].

C. Co-Design in NCS

The Co-design of NCS deals with the interaction between control and computing theories. Normally, digital networks are used for connecting sensors, controllers, and actuators in NCS. The concept of centralized control was applied in the previous years in which the sensors, controllers, and actuators are located in an area where the communication is peer to peer. As a result of the short distance, there is no time delay between the NCS parts and also, no packet losses. After that, the concept of decentralized systems was applied since manufactures tend to separate their large plants to subsystems with their own control system with indirect communication. Then, manufactures applied the hybrid systems which contain centralized and decentralized systems. As a result, Internet of things (IOT) became a rising research area in both of academia and industry and it established the future interaction between the computing and communications [11], [187].

Since IOT systems are complex, it is difficult to model them. Practically, the control system faces issues affecting the stability and the overall performance of the system which may lead to instability such as: critical time delay of data, irregularity, time-variance and packet dropout. Traffic congestion could cause losing of data and unreliable nature of the link or protocol malfunctioning [11]. On the other hand, the continuous growing of computing systems raises the capabilities to compute, store and process IOT data with high quality and reliable measurements [187].

The implementation of model estimation, optimization, and control approaches into into the progressive data control center was presented in [191] and the implementation of control theory in computing systems was discussed in [192]. Two methodologies for remote control systems were proposed in which control system design is provided as a cloud service and hence time and cost were reduced and the design of the plant-wide system became simpler [193]. In [194], both communication and control problems were solved simultaneously using co-design approach. [195] proposed a solution for a co-design problem of a mixed event-triggering mechanism (ETM) and state feedback controller for discrete-time linear parameter-varying (LPV) systems in a network environment. A parameter-dependent co-design condition for event-triggered $H_\infty$ control was given for the problem as a finite set of LMIs. The issue of performance limitation of networked systems was discussed by co-designing the controller and communication filter in [196]. And it was shown that using the controller and communication filter co-design could deteriorate the performance and revoke the effect of the channel noise.

Ⅴ. CONCLUSIONS

A survey on modeling and theories of networked control system (NCS) was provided in this paper. In the first part, the modeling of the different kind of imperfections that affect NCS was discussed. These imperfections are: quantization errors, packet dropouts, variable sampling/transmission intervals, variable transmission delays, and communication constraints. Then, several theories which were applied for controlling networked systems were presented. These theories include: input delay system approach, Markovian system approach, switched system approach, stochastic system approach, impulsive system approach, and predictive control approach. Finally, some advanced issues in NCS including decentralized and distributed NCS, cloud control system, and co-design of NCS were reviewed.

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