Modeling and Analysis of Networked Discrete Event Systems
with Multiple Control Channels

We consider a DES modeled as a deterministic finite-state automaton (DFA)

$$\mathbf{G}=(Q,\Sigma,\delta,q_{0}),$$

where $Q$ is the finite set of states, $\Sigma$ is the finite set of events, $\delta:Q\times\Sigma\to Q$ is the (partial) transition function and $q_{0}\in Q$ is the initial state. The transition function can be extended to $\delta:Q\times\Sigma^{*}\to Q$ recursively in the usual manner; see, e.g., [4]. A string is a sequence of events in the form of $s=\sigma_{1}\sigma_{2}\cdots\sigma_{n},\sigma_{i}\in\Sigma$; a language is a set of strings. We denote by $\Sigma^{*}$ the set of all strings over $\Sigma$ including the empty string $\epsilon$. Then the language generated by DFA $\mathbf{G}$ is $\mathcal{L}(\mathbf{G})=\{s\in\Sigma^{*}:\delta(q_{0},s)!\}$, where $!$ means “is defined". For any positive natural number $N\in\mathbb{N}$, we denote by $[1,N]$ the set of natural numbers from $1$ to $N$.

In the supervisory control framework, the event set is partitioned as two disjoint sets

$$\Sigma=\Sigma_{c}\dot{\cup}\Sigma_{uc},$$

where $\Sigma_{c}$ is the set of controllable events and $\Sigma_{uc}$ is the set of uncontrollable events. A supervisor is a function

$$\textbf{{Sup}}:\mathcal{L}(G)\to 2^{\Sigma_{c}},$$

(1)

that dynamically decides which controllable events to enable. Note that uncontrollable events are always enabled by default. Then the closed-loop language under control, denoted by $\mathcal{L}(\textbf{{Sup}}/\mathbf{G})$, is defined recursively by

• •

  $\epsilon\in\mathcal{L}(\textbf{{Sup}}/\mathbf{G})$;

• •

  for any $s\in\Sigma^{*},\sigma\in\Sigma$, we have $s\sigma\in\mathcal{L}(\textbf{{Sup}}/\mathbf{G})$ iff

  $$[s\in\mathcal{L}(\textbf{{Sup}}/\mathbf{G})]\wedge[s\sigma\in\mathcal{L}(\mathbf{G})]\wedge[\sigma\in\textbf{{Sup}}(s)\cup\Sigma_{uc}].$$

Intuitively, the above definition says that $s\sigma$ is in the closed-loop language if (i) $s$ is in the closed-loop language; and (ii) $s\sigma$ is feasible in $\mathbf{G}$; and (iii) $\sigma$ is either uncontrollable or enabled by Sup upon the occurrence of $s$.

Note that we can write supervisor Sup in Equation (1) equivalently as

$$\textbf{{Sup}}:\mathcal{L}(\mathbf{G})\times\Sigma_{c}\to\{\textbf{0},\textbf{1}\},$$

(2)

where “0" means “disable" and “1" means “enable".

Although supervisors defined in Equations (1) and (2) are mathematically equivalent, their physical interpretations in terms of how the supervisor is implemented are quite different. Specifically, Equation (1) suggests that, at each instant, the supervisor sends control decision $\textbf{{Sup}}(s)$ as a package to the plant. However, in practice, each controllable event represents a single actuator and actuators in the plant may be physically distributed. Therefore, Equation (2) captures a more general scenario where the supervisor sends control decision $\textbf{{Sup}}(s)(\sigma)$ (disable or enable) to each $\sigma$ individually. This difference is crucial in the networked setting as the supervisor may send control decisions to different actuators by different communication channels whose network properties (delays or losses) are different and independent.

In general, the system has $|\Sigma_{c}|$ actuators. Hereafter, for the sake of simplicity, we assume that event set $\Sigma$ is ordered so that the first $|\Sigma_{c}|$ events in $\Sigma$ are controllable with $\Sigma_{c}=\{\sigma_{1},\sigma_{2},\cdots,\sigma_{|\Sigma_{c}|}\}$. For each $\sigma_{i}\in\Sigma_{c}$, the corresponding actuator is denoted by $\textbf{{Act}}_{i}$. Also, for the sake of simplicity, we write

$$\textbf{{Sup}}_{i}(s):=\textbf{{Sup}}(s)(\sigma_{i}),$$

which is the control decision (0 or 1) the supervisor sends to $\textbf{{Act}}_{i}$ upon the occurrence of string $s$ in the plant. In the non-networked setting, this decision will be received by $\textbf{{Act}}_{i}$ immediately and take effect. However, in the networked setting, the decision for $\sigma_{i}$ is transmitted to $\textbf{{Act}}_{i}$ via the corresponding control channel denoted by $\textbf{{Channel}}_{i}$, which may be subject to the following issues:

• •

  Control Delays: decision $\textbf{{Sup}}_{i}(s)$ may be delayed in the sense that it will go to $\textbf{{Channel}}_{i}$ and is not received by $\textbf{{Act}}_{i}$ immediately but after some delays.

• •

  Control Losses: decision $\textbf{{Sup}}_{i}(s)$ may be lost in the sense that it will not go to $\textbf{{Channel}}_{i}$, and therefore, it will never be received by $\textbf{{Act}}_{i}$.

Note that, in this work, we only focus on the issue of delays and losses in control channels. The observation channels are assumed to be perfect with full observation. To present our framework, we make the following assumptions on the control channels:

1. A1

   For each $\textbf{{Channel}}_{i}$, the number of consecutive decision losses is upper bounded by integer $N_{i}^{L}\in\mathbb{N}$;

2. A2

   For each $\textbf{{Channel}}_{i}$, the number of control delays is upper bounded by integer $N_{i}^{D}\in\mathbb{N}$;

3. A3

   The number of delays or losses are counted by event occurrences in the plant;

4. A4

   For each $\textbf{{Channel}}_{i}$, decisions in the channel are first-in-first-out (FIFO);

5. A5

   The initial control decisions are embedded in each $\textbf{{Act}}_{i}$; hence are not subject to control delays or losses;

6. A6

   The dynamic of each $\textbf{{Channel}}_{i},i=1,\dots,|\Sigma_{c}|$ are independent, i.e., delays or losses in one channel will not affect delays or losses in another channel.

In the following sections, we will formally describe the dynamics of such a networked supervisory control system.

In the system theory, states are referred to as variables with minimum number that are sufficient enough to represent the current status of the dynamic system. In the standard supervisory control framework, string $s\in\mathcal{L}(\textbf{{Sup}}/\mathbf{G})$ executed is sufficient enough to serve as the “state" of the system as it determines both the current-state $\delta(q_{0},s)$ in the plant and the current control decision $\textbf{{Sup}}(s)$ applied (when Sup is realized by a finite-state automaton, it further suffices to know the current-state in the supervisor automaton). However, this information is not sufficient to determine the current status of a networked control system as the current control decision issued and the current control decision applied may be different due to communication delays and losses.

Now, we provide a general state-space model for networked supervisory control systems. The “state" of the networked closed-loop system contains four components:

1. (i)

   The states of the plant and the supervisor; both can be determined by string $s\in\mathcal{L}(\mathbf{G})$ executed.

2. (ii)

   The configuration of each actuator $\textbf{{Act}}_{i}$ representing the current effective control decision (0 or 1) for $\sigma_{i}$.

3. (iii)

   The configuration of each control channel $\textbf{{Channel}}_{i}$, which captures what decisions are waiting in the channel and in what order. Formally, for each $\textbf{{Channel}}_{i}$, the channel configuration for $\sigma_{i}$ is a set of pairs in the form of

   $$\theta_{i}=\{(\gamma_{1},n_{1}),(\gamma_{2},n_{2}),\dots,(\gamma_{k},n_{k})\}$$

   where $\gamma_{j}\in\{\textbf{0},\textbf{1}\}$ and $n_{j}\in[1,N_{i}^{D}]$. We denote by $\Theta_{i}$ the set of all channel configurations for $\textbf{{Channel}}_{i}$. Intuitively, for each $(\gamma,n)\in\theta_{i}$, $\gamma\in\{\textbf{0},\textbf{1}\}$ represents a disable or enable decision while $n$ represents the maximum remaining time of $\gamma$ in the channel.

4. (iv)

   Finally, for the case of decision losses, we also need a counter to capture the number of consecutive decision losses for each channel.

Hereafter, we will discuss how this state-space evolves, i.e., the dynamics of the networked control system. We first discuss the issues of communication delays and losses separately and then show how to combine them together.

For the case of control delays, in addition to the current string executed $s\in\mathcal{L}(\mathbf{G})$, we also need to track the current channel configuration $\theta_{i}\in\Theta_{i}$ and the current actuator configuration $a_{i}\in\{\textbf{0},\textbf{1}\}$. For any $\theta_{i}\in\Theta_{i}$, we define operator $\textsc{NX}:\Theta_{i}\to\Theta_{i}$ by

$$\textsc{NX}(\theta_{i})=\{(\gamma,n-1):(\gamma,n)\in\theta_{i}\wedge n>1\}$$

which decreases the remaining time of each control decision in the channel by one unit.

Now, for controllable event $\sigma_{i}\in\Sigma_{c}$, suppose that the current actuator configuration for $\textbf{{Act}}_{i}$ is $a_{i}\in\{\textbf{0},\textbf{1}\}$ and the current channel configuration $\textbf{{Channel}}_{i}$ is $\theta_{i}=\{(\gamma_{1},n_{1}),(\gamma_{2},n_{2}),\dots,(\gamma_{k},n_{k})\}\in\Theta_{i}$. Suppose that $\textbf{{Sup}}_{i}$ sends a new decision $\gamma\in\{\textbf{0},\textbf{1}\}$. We are interested in what are the next actuator configuration and channel configuration for $\sigma_{i}$. To this end, we define operator

$$\texttt{Move}_{i}^{D}:\{\textbf{0},\textbf{1}\}\times\Theta_{i}\times\{\textbf{0},\textbf{1}\}\to 2^{\{\textbf{0},\textbf{1}\}\times\Theta_{i}}$$

by: for any $(a_{i},\theta_{i})\in\{\textbf{0},\textbf{1}\}\times\Theta_{i}$ and $\gamma\in\{\textbf{0},\textbf{1}\}$, we have

• •

  If $\theta_{i}=\emptyset$, then

  $$\texttt{Move}_{i}^{D}(a_{i},\theta_{i},\gamma)=\{(\gamma,\emptyset),(a_{i},\{(\gamma,N_{i}^{D})\})\}$$

  (3)

• •

  If $\theta_{i}\neq\emptyset$ and $n_{min}=1$, then

  $$\texttt{Move}_{i}^{D}(a_{i},\theta_{i},\gamma)=\{(\gamma_{min},\textsc{NX}(\theta_{i})\cup\{(\gamma,N_{i}^{D})\})\}$$

  (4)

• •

  If $\theta_{i}\neq\emptyset$ and $n_{min}>1$, then

  		$\displaystyle\texttt{Move}_{i}^{D}(a_{i},\theta_{i},\gamma)$		(5)
  	$\displaystyle=$	$\displaystyle\left\{\!\!\begin{array}[]{c c}(a_{i},\textsc{NX}(\theta_{i})\cup\{(\gamma,N_{i}^{D})\}),\\ (\gamma_{min},\textsc{NX}(\theta_{i}\setminus\{(\gamma_{min},n_{min})\})\cup\{(\gamma,N_{i}^{D})\})\end{array}\!\!\right\}$		(8)

  where $(\gamma_{min},n_{min})\in\theta$ is the pair such that $n_{min}=\min\{n_{1},\dots,n_{k}\}$.

Intuitively, $\texttt{Move}_{i}^{D}$ captures how the configurations of $\textbf{{Channel}}_{i}$ and $\textbf{{Act}}_{i}$ evolve when a new decision is issued. Note that $\texttt{Move}_{i}^{D}$ is a non-deterministic transition function as control delay may happen or not. More precisely:

• •

  Equation (3) captures the case when $\textbf{{Channel}}_{i}$ is empty. Depending on whether or not control delay occurs, there are two scenarios

  • –

    Decision $\gamma$ is received by $\textbf{{Act}}_{i}$ without delay: then the configuration of $\textbf{{Act}}_{i}$ will be updated to $\gamma$ and the configuration of $\textbf{{Channel}}_{i}$ is still $\emptyset$;

  • –

    Decision $\gamma$ is delayed in $\textbf{{Channel}}_{i}$: then the configuration of $\textbf{{Act}}_{i}$ will be unchanged while the configuration of $\textbf{{Channel}}_{i}$ will be updated to $\{(\gamma,N_{i}^{D})\}$, where $N_{i}^{D}$ is the upper bound for delays.

• •

  Equation (4) captures the case when $\textbf{{Channel}}_{i}$ is full in the sense that there exists a control decision $\gamma_{min}$ with only one unit left, i.e., $n_{min}=1$. Since we consider FIFO channel, $\textbf{{Act}}_{i}$ will receive $\gamma_{min}$ definitely and the remaining times of all other decisions in $\textbf{{Channel}}_{i}$ will be decreased by one unit with new decision $(\gamma,N_{i}^{D})$ plugged in.

• •

  Equation (5) captures the case when $\textbf{{Channel}}_{i}$ is non-empty but is also not full. Then the following two scenarios are possible

  • –

    No decision is received by $\textbf{{Act}}_{i}$. In this case, the actuator configuration remains unchanged and the channel configuration is updated to $\textsc{NX}(\theta_{i})\cup\{(\gamma,N_{i}^{D})\}$.

  • –

    Decision $\gamma_{min}$ is received by $\textbf{{Act}}_{i}$. In this case, the actuator configuration is updated to $\gamma_{min}$ and the channel configuration is updated by eliminating $\gamma_{min}$ and adding $\gamma$. Still, no decision whose remaining time is greater than $n_{min}$ can be received due to the FIFO property of the channel.

We illustrate the above concepts for the case of communication delays by the following example.

Now we proceed to model communication losses in the control channel. As we mentioned earlier, we assume that communication losses can happen only when the supervisor sends a control decision to a communication channel. In other words, once a decision goes into the communication channel, it will not be lost in between and will arrive at the actuator eventually.

To focus on the effect of control losses, we first do not consider control delays in this subsection. Therefore, we do not need the channel configuration as a part of the "state". However, to capture the assumption that $\textbf{{Channel}}_{i}$ cannot have more than $N_{i}^{L}$ consecutive losses, we need to introduce a “counter" component. We denote by $\mathbb{N}_{i}^{L}:=\{0,1,\cdots,N_{i}^{L}\}$ the counter set.

Now, for controllable event $\sigma_{i}\in\Sigma_{c}$, suppose that the current actuator configuration is $a_{i}\in\{\textbf{0},\textbf{1}\}$ and the counter number is $c_{i}\in\mathbb{N}_{i}^{L}$. Suppose that $\textbf{{Sup}}_{i}$ sends a new decision $\gamma\in\{\textbf{0},\textbf{1}\}$. We are also interested in what are the next actuator configuration and the next counter number for $\sigma_{i}$. To this end, we define operator

$$\texttt{Move}_{i}^{L}:\{\textbf{0},\textbf{1}\}\times\mathbb{N}_{i}^{L}\times\{\textbf{0},\textbf{1}\}\to 2^{\{\textbf{0},\textbf{1}\}\times\mathbb{N}_{i}^{L}}$$

by: for any $a_{i}\in\{\textbf{0},\textbf{1}\},c_{i}\in\mathbb{N}_{i}^{L}$ and $\gamma\in\{\textbf{0},\textbf{1}\}$, we have

• •

  If $c_{i}=N_{i}^{L}$, then

  $$\texttt{Move}_{i}^{L}(a_{i},c_{i},\gamma)=\{(\gamma,0)\}.$$

  (9)

• •

  If $c_{i}<N_{i}^{L}$, then

  $$\texttt{Move}_{i}^{L}(a_{i},c_{i},\gamma)=\{(\gamma,0),(a_{i},c_{i}+1)\}.$$

  (10)

Intuitively, Equation (9) captures the case that there are already $N_{i}^{L}$ consecutive losses for control decisions transmissions. Therefore, the current decision $\gamma$ will be received by $\textbf{{Act}}_{i}$ definitely and the counter is reset to $0$. On the other hand, Equation (10) captures the case that the accumulated consecutive losses $c_{i}$ has not yet exceeded the upper bound. Then decision $\gamma$ may either be received, which leads to $(\gamma,0)$, or be lost, which leads to $(a_{i},c_{i}+1)$. Therefore, $\texttt{Move}_{i}^{L}$ is still a non-deterministic function in general.

We illustrate the above concepts for the case of communication losses by the following example.

Now, we consider the general case in which both delays and losses may happen in $\textbf{{Channel}}_{i}$. In this case, we need to track the channel configuration $\theta_{i}\in\Theta_{i}$, the actuator configuration $a_{i}\in\textbf{{Act}}_{i}$ and the counter number $c_{i}\in\mathbb{N}_{i}^{L}$; these together constitute the "state" of the system, which are also referred to as the network configuration. To capture how the state evolves upon the occurrence of new decision $\gamma\in\{\textbf{0},\textbf{1}\}$, we define operator

$$\texttt{Move}_{i}^{DL}:\{\textbf{0},\textbf{1}\}\times\Theta_{i}\times\mathbb{N}_{i}^{L}\times\{\textbf{0},\textbf{1}\}\to 2^{\{\textbf{0},\textbf{1}\}\times\Theta_{i}\times\mathbb{N}_{i}^{L}},$$

which combines $\texttt{Move}_{i}^{D}$ and $\texttt{Move}_{i}^{L}$. Specifically, for any $a_{i}\in\{\textbf{0},\textbf{1}\},\theta_{i}\in\Theta,c_{i}\in\mathbb{N}_{i}^{L}$ and $\gamma\in\{\textbf{0},\textbf{1}\}$, we have

• •

  If $\theta_{i}=\emptyset$ and $c_{i}=N_{i}^{L}$, then

  $$\!\!\!\texttt{Move}_{i}^{DL}(a_{i},\theta_{i},c_{i},\gamma)=\left\{\begin{array}[]{c c}(\gamma,\emptyset,0),\\ (a_{i},\{(\gamma,N_{i}^{D})\},0)\end{array}\right\}.$$

  (11)

• •

  If $\theta_{i}=\emptyset$ and $c_{i}<N_{i}^{L}$, then

  $$\!\!\!\texttt{Move}_{i}^{DL}(a_{i},\theta_{i},c_{i},\gamma)=\left\{\begin{array}[]{c c}(a_{i},\emptyset,c_{i}+1),\\ (\gamma,\emptyset,0),\\ (a_{i},\{(\gamma,N_{i}^{D})\},0)\end{array}\right\}.$$

  (12)

• •

  If $\theta_{i}\neq\emptyset,n_{min}=1$ and $c_{i}=N_{i}^{L}$, then

  		$\displaystyle\texttt{Move}_{i}^{DL}(a_{i},\theta_{i},c_{i},\gamma)=$		(13)
  		$\displaystyle\left\{(\gamma_{min},\textsc{NX}(\theta_{i})\cup\{(\gamma,N_{i}^{D})\},0)\right\}.$

• •

  If $\theta_{i}\neq\emptyset,n_{min}=1$ and $c_{i}<N_{i}^{L}$, then

  		$\displaystyle\texttt{Move}_{i}^{DL}(a_{i},\theta_{i},c_{i},\gamma)=$		(14)
  		$\displaystyle\left\{\begin{array}[]{c c}(\gamma_{min},\textsc{NX}(\theta_{i})\cup\{(\gamma,N_{i}^{D})\},0),\\ (\gamma_{min},\textsc{NX}(\theta_{i}),c_{i}+1\})\end{array}\right\}.$		(17)

• •

  If $\theta_{i}\neq\emptyset,n_{min}>1$ and $c_{i}=N_{i}^{L}$, then

  		$\displaystyle\texttt{Move}_{i}^{DL}(a_{i},\theta_{i},c_{i},\gamma)=$		(18)
  		$\displaystyle\left\{\begin{array}[]{c c}(\gamma_{min},\textsc{NX}(\theta_{i}\setminus\{(\gamma_{min},n_{min})\})\cup\{(\gamma,N_{i}^{D})\},0),\\ (a_{i},\textsc{NX}(\theta_{i})\cup\{(\gamma,N_{i}^{D})\},0)\end{array}\right\}.$		(21)

• •

  If $\theta_{i}\neq\emptyset,n_{min}>1$ and $c_{i}<N_{i}^{L}$, then

  		$\displaystyle\texttt{Move}_{i}^{DL}(a_{i},\theta_{i},c_{i},\gamma)=$		(22)
  		$\displaystyle\left\{\begin{array}[]{c c}(a_{i},\textsc{NX}(\theta_{i}),c_{i}+1),\\ (\gamma_{min},\textsc{NX}(\theta_{i}\setminus\{(\gamma_{min},n_{min})\}),c_{i}+1),\\ (a_{i},\textsc{NX}(\theta_{i})\cup\{(\gamma,N_{i}^{D})\},0),\\ (\gamma_{min},\textsc{NX}(\theta_{i}\setminus\{(\gamma_{min},n_{min})\})\cup\{(\gamma,N_{i}^{D})\},0)\end{array}\right\}.$		(27)

The above defined $\texttt{Move}_{i}^{DL}$ boils down to $\texttt{Move}_{i}^{D}$ or $\texttt{Move}_{i}^{L}$, respectively, when $N_{i}^{L}=0$ or $N_{i}^{D}=0$. Specifically, when $N_{i}^{L}=0$, Equations (11), (13) and (18) becomes Equations (3), (4) and (5), respectively. When $N_{i}^{D}=0$, Equations (11) and (12) becomes Equations (9) and (10), respectively. However, Equations (14) and (22) capture the scenarios when delays and losses are combined. Specifically, the first element in Equation (14) corresponds to the case that $\gamma_{min}$ is received by $\textbf{{Act}}_{i}$ but $\gamma$ is plugged in $\textbf{{Channel}}_{i}$, while the the second element corresponds to the case that $\gamma_{min}$ is received by $\textbf{{Act}}_{i}$ but $\gamma$ is lost. For Equation (22), it captures the four possible combinations of "$\gamma_{min}$ is received by $\textbf{{Act}}_{i}$ or not" and "$\gamma$ is lost or not".

Since control channels are independent, the overall "state" of the systems is a vector in which each component reflects the status of a single channel. Formally, the actuator configuration of the overall system is a $|\Sigma_{c}|$-dimensional vector

$$\textbf{a}=(a_{1},a_{2},\dots,a_{|N_{c}|})\in\{\textbf{0},\textbf{1}\}^{|\Sigma_{c}|}.$$

We denote by $A$ the set of all actuator configurations. For each actuator configuration $\textbf{a}\in A$, we denote by $\Gamma(\textbf{a})$ the set of controllable events that are enabled, i.e.,

$$\Gamma(\textbf{a})=\{\sigma_{i}\in\Sigma_{c}:a_{i}=\textbf{1}\}.$$

Similarly, the channel configuration of the overall system is a $|\Sigma_{c}|$-dimensional vector

$$\boldsymbol{\theta}=(\theta_{1},\theta_{2},\dots,\theta_{|N_{c}|})\in\Theta_{1}\times\Theta_{2}\times\cdots\times\Theta_{|\Sigma_{c}|}.$$

We denote by $\Theta=\Theta_{1}\times\cdots\times\Theta_{|\Sigma_{c}|}$ the set of all channel configurations. Also, we define the overall consecutive losses counter as

$$\textbf{c}=(c_{1},c_{2},\cdots,c_{|\Sigma_{c}|})\in\mathbb{N}_{1}^{L}\times\mathbb{N}_{2}^{L}\times\cdots\times\mathbb{N}_{|\Sigma_{c}|}^{L}=:\mathbb{N}^{L}$$

The dynamics of the overall network configuration is defined by operator

$$\texttt{Move}:A\times\Theta\times\mathbb{N}^{L}\times\{\textbf{0},\textbf{1}\}^{|\Sigma_{c}|}\to 2^{A\times\Theta\times\mathbb{N}^{L}}$$

such that: for any $(\textbf{a},\boldsymbol{\theta},\textbf{c}),(\textbf{a}^{\prime},\boldsymbol{\theta}^{\prime},\textbf{c}^{\prime})\in A\times\Theta\times\mathbb{N}^{L}$ and $\boldsymbol{\gamma}\in\{\textbf{0},\textbf{1}\}^{|\Sigma_{c}|}$ , we have $(\textbf{a}^{\prime},\boldsymbol{\theta}^{\prime},\textbf{c}^{\prime})\in\texttt{Move}(\textbf{a},\boldsymbol{\theta},\textbf{c},\boldsymbol{\gamma})$ iff

$\displaystyle\forall i=1,\dots,|\Sigma_{c}|:(a_{i}^{\prime},\theta_{i}^{\prime},c_{i}^{\prime})\!\in\!\texttt{Move}_{i}^{DL}(a_{i},\theta_{i},c_{i},\gamma_{i}).$

(28)

In order to specify the complete dynamics of the closed-loop system, the last piece is to specify the initial configuration of the network. As we mentioned early, the initial control decision can neither be delayed nor lost. In practice, this means that the initial decisions are already embedded in the actuators before the plant runs. Therefore, the initial actuator configuration is given by

$$\textbf{a}_{0}=(\textbf{{Sup}}_{1}(\epsilon),\textbf{{Sup}}_{2}(\epsilon),\dots,\textbf{{Sup}}_{|\Sigma_{c}|}(\epsilon))\in A.$$

Since the communication channels are all empty initially, the initial channel configuration is given by

$$\boldsymbol{\theta}_{0}=(\emptyset,\emptyset,\dots,\emptyset)\in\Theta.$$

Also, the initial counter is given by

$$\textbf{c}_{0}=(0,0,\dots,0)\in\mathbb{N}^{L}.$$

Note that, although our main purpose is to calculate the actual strings generated by the system, to this end, we also need to track the configurations of the overall network simultaneously. This is captured by the concept of extended strings. Formally, an extended string is a string augmented with the network configuration in the form of

$$t=(s,\textbf{a},\boldsymbol{\theta},\textbf{c})\in\Sigma^{*}\times A\times\Theta\times\mathbb{N}^{L}.$$

A set of extended string is called an extended language. Therefore, a networked supervisory control system essentially generates an extended language $\mathcal{L}_{e}(\textbf{{Sup}}/\mathbf{G})$ defined recursively as follows:

• •

  $(\epsilon,\textbf{a}_{0},\boldsymbol{\theta}_{0},\textbf{c}_{0})\in\mathcal{L}_{e}(\textbf{{Sup}}/\mathbf{G})$;

• •

  For any $t=(s,\textbf{a},\boldsymbol{\theta},\textbf{c})$ and $\sigma\in\Sigma$, we have $t^{\prime}=(s^{\prime},\textbf{a}^{\prime},\boldsymbol{\theta}^{\prime},\textbf{c}^{\prime})\in\mathcal{L}_{e}(\textbf{{Sup}}/\mathbf{G})$ iff

  • –

    $t\in\mathcal{L}_{e}(\textbf{{Sup}}/\mathbf{G})$;

  • –

    $s^{\prime}=s\sigma\in\mathcal{L}(\mathbf{G})$;

  • –

    $\sigma\in\Gamma(\textbf{a})\cup\Sigma_{uc}$;

  • –

    $(\textbf{a}^{\prime},\boldsymbol{\theta}^{\prime},\textbf{c}^{\prime})\in\texttt{Move}(\textbf{a},\boldsymbol{\theta},\textbf{c},\boldsymbol{\gamma}(s^{\prime}))$, where $\boldsymbol{\gamma}(s^{\prime}):=(\textbf{{Sup}}_{1}(s^{\prime}),\dots,\textbf{{Sup}}_{|\Sigma_{c}|}(s^{\prime}))$

The above definition is explained as follows. Essentially, it defines all extended strings that can be generated by the system in an inductive manner. In the inductive step, the first condition says that the prefix should be an extended string generated by the system. The second condition says that new event $\sigma$ should be feasible in the original plant $\mathbf{G}$. The third condition says that event $\sigma$ should be enabled by the current actuator configuration a. Finally, the last condition captures how the network configuration evolves upon the new control decision $\boldsymbol{\gamma}(s^{\prime})$. The effect of $\boldsymbol{\gamma}(s^{\prime})$ will be recorded in the network configuration and may determine the occurrence of some future events when it arrives at actuators as a.

Based on the extended language, we can define the actual language generated by the closed-loop system in the networked setting as the projection of the extended language onto the first component, i.e.,

$$\mathcal{L}(\textbf{{Sup}}/\mathbf{G})=\{s\!\in\!\Sigma^{*}:\exists\textbf{a},\boldsymbol{\theta},\textbf{c}\text{ s.t.}(s,\textbf{a},\boldsymbol{\theta},\textbf{c})\!\in\!\mathcal{L}_{e}(\textbf{{Sup}}/\mathbf{G})\}.$$

This completes our general model for networked supervisory control system over multiple channel networks.

Note that, in all the above developments, it is assumed that each actuator uses its own control channel. In some applications, a group of actuators may share the same control channel because they are physically located together and the control decisions are sent as packages to each group of actuators. In such a scenario, we can assume that $\Sigma_{c}$ is partitioned as $m$ groups

$$\Sigma_{c}=\Sigma_{c,1}\dot{\cup}\Sigma_{c,2}\dot{\cup}\dots\dot{\cup}\Sigma_{c,m}$$

such that actuators in each $\Sigma_{c,i}$ share the same control channel $\textbf{{Channel}}_{i}$. Our framework and all previous developments can be easily adapted to the case. The only differences is that the control decision in each $\textbf{{Channel}}_{i}$ is then in the domain of $\{\textbf{0},\textbf{1}\}^{|\Sigma_{c,i}|}$ rather than in $\{\textbf{0},\textbf{1}\}$. In the most extreme case where all actuators share the same control channel, this general setting boils down to the single channel setting in the framework of Lin [7], where all decisions are sent as a single package. Here, instead of formally presenting the generalization which is rather straightforward, we use the following example to illustrate our point.