Abstract
Wireless mesh network control systems (WMNCSs) are typical Cyberphysical Systems (CPSs) widely used in industries that need to meet stringent performance requirements. These WMNCSs are characteristic of stochasticity at different levels as system behavior, network performance, and wireless signal propagation, which grievously increases the difficulties of system modeling and analysis. In this paper, we propose a three-layered modeling framework to capture the stochastic properties of WMNCSs at different levels with stochastic hybrid system (SHS), stochastic network calculus (SNC), and physical layer models. We also bridge the gaps between these methods with an upper bound approach. All the efforts give us a methodology of modeling and analysis WMNCSs with stochastic methods, so we can know how the factors as channel conditions, network topology, etc. affect the stability and performance of the system. To the best of our knowledge, it is the first work that provides such a unified and flexible framework to model and analyze WMNCSs with stochastic methods.
1. Introduction
Wireless communication has been practiced in the industrial market for many years. Today, more and more plants begin to use wireless mesh networks for industrial automation and control system applications. These systems can be considered as wireless mesh network control systems (WMNCSs) [1]. The migration of traditionally wired industrial infrastructure to wireless technologies can improve flexibility, scalability, and efficiency. Typical wireless network technologies suit for WMNCSs include WirelessHART [2], ISA100 [3], and WIA-PA [4].
WMNCSs are typical Cyberphysical Systems (CPSs) that are characterized by the tight interactions between computational components, communication networks, and physical dynamics. These components and their interactions are characteristic with stochasticity. The stochasticity arises from the factors as wireless channel condition, network topology, network protocol, routing, data flow configurations, control protocol, and so on. How to model and analyze WMNCSs is still a challenging task.
Methods have been proposed to model stochastic processes. Within these methods, stochastic hybrid system (SHS) [5] is a system modeling method that integrates the continuous dynamics with discrete variables and is intensively used to capture the uncertainty in the control system. Stochastic network calculus (SNC) [6] is a theory for network performance analysis that is based on min-plus convolution and stochastic process principles. Statistical models of fading channel [7] are used to capture the variation of wireless channel quality. These methods and models, however, are developed for different application areas, and it still remains gaps for them to be used together.
From the perspective of CPSs, we want these methods and tools developed at different area can be used together to solve complex problems in WMNCSs, which, we believe, is a pervasive principle for CPSs. This paper focuses on how to model and analyze WMNCSs with stochastic methods. We model WMNCSs at different abstract levels with SHS, SNC, and statistical models of fading channel. We also bridge the gaps between these methods, so they can be used together to model and analyze WMNCSs seamlessly.
The main contributions of this paper are as follows: we propose a three-layered modeling framework that can be used to capture the stochastic properties of WMNCSs at different levels including system behavior, network performance, and wireless signal propagation. To the best of our knowledge, it is the first work that provides such a unified and flexible framework to model and analyze WMNCSs with stochastic methods; with an upper bound approach based on MGFs (moment generating functions) and stochastic orders, we bridge the gaps between SHS, SNC, and statistical models of wireless fading channel. They can be used together to model and analyze WMNCSs seamlessly; we also give an example of how to model and analyze WMNCSs with this framework. The synthesis of above contribution is a methodology for modeling and analysis WMNCSs with stochastic methods.
The organization of this paper is as follows. Section 2 introduces the background knowledge and describes the problem in detail; Section 3 shows how to model WMNCSs with SHS; Section 4 tailors the SNC for WMNCSs; Section 5 describes the statistical models of wireless fading channel. Section 6 describes how to bridge the gaps between SHS, SNC, and statistical models of fading channel; Section 7 provides an example; Section 8 gives a summary of related works; Section 9 concludes this work.
2. Preliminaries
In this section, we first introduce the basic theories and math tools which are used to model WMNCSs, and then we give a description of the problem we are facing when modeling WMNCSs with these methods. Although previous works such as SHS, MGF, and SNC are not new methods, our framework is the first work that tries to combine these methods together to provide a unified and flexible modeling framework for WMNCSs. To bridge the gaps between these methods, we also adopt an upper bound approach based on MGFs and stochastic orders in our framework.
2.1. Stochastic Hybrid System
A stochastic hybrid system (SHS) is a system modeling method that integrates the continuous dynamics with discrete variables.
Stochastic Hybrid System (SHS). A stochastic hybrid system is defined by four ingredients.
A set of discrete modes and the system mode changes through instantaneous resets or impulses:
A set of (locally Lipschitz) vector fields that describes the evolution of the continuous state x(t) in each mode q:
A set of reset maps that determine how the system mode changes (jumps) and when resets happen:
A set of reset time distributions that define the system reset and impulse times:Here, the families of reset maps and reset time distributions are parameterized by the same set . When , the transition does not trigger in the mode q. We use a stochastic transition counter to count the number of times that the corresponding discrete transition/reset map is activated:
Figure 1 shows a graphic representation of an SHS, where each ellipse corresponds to a discrete mode and each edge to a transition between discrete modes. Ellipses are labeled with the corresponding discrete modes, and the vector field that governs the evolution of continuous states is written under it. Edges are labeled with reset maps.
When simulating, the SHS starts at an initial condition with . Let denote the maximum interval of the system, the process is constructed as follows:(1)Set and .(2)Determine the time staying at this mode, the jump taking to the next mode. Taking time triggered SHS (TTSHS) [5] as an example, at this step each random number is the draw for all with the reset-time distribution . Let , if any of is equal to 0 or more than two draws have the same minimum value, the procedure fails. Otherwise, let .(3)Solve the initial value problem with , , at the time interval . Then the state of the system evolves according to the differential equation corresponding to mode for the time interval .(4)If , the process terminated after the time point T.(5)At , the system jumps to a new mode and a reset transition happened according to the following equation: Then a new cycle begins and the process goes back to with new draws for all .
2.2. Moment Generating Function
For any random variable or stochastic process, we use or to denote its cumulative distribution function (CDF), and we use or to denote its complementary cumulative distribution function (CCDF).
Moment Generating Function. The MGF of any random variable X is defined for any :Similarly, for any stochastic processes X(t), it has for any .
MGF is a useful mathematical tool for probability and stochastic process. We list some most useful properties of MGF as below:Here, and are random variables, and are constants, and .
For the min-plus convolution operation and deconvolution operation defined below [8]where and are real-valued, bivariate functions which are used to describe independent random processes and , respectively, and the corresponding min-plus convolution and deconvolution operations for MGFs are defined asWe have the following important relationships between these operations [8]:Here and are bivariate functions defined at .
2.3. Stochastic Network Calculus
Stochastic network calculus (SNC) is a theory for network performance analysis based on min-plus convolution and stochastic process principles. SNC provides network performance guarantees that have the following form:Here x represents the targeted delay or loss with the permissible violation probability .
In SNC, the data arrival process and departure process of a network node are described by bivariate functions and , respectively. gives the amount of data arrives in an interval . gives the amount of data departs in an interval . It is easy to see that and are nonnegative functions, increasing in , decreasing in s, and and for all t.
Dynamic Server. In SNC, the dynamic service provided by the node is described by a random process , and the following inequality holds:
Leftover Service. Consider two flows at the same node with arrival processes and that are scheduled as a work-conserving server with the service process , where is nonnegative, increasing in t and . Then the service provided to the flow isAnd the MGF of is bounded for and by
Concatenation. Consider two dynamic servers and in series. There exists an equivalent, single dynamic server for whereAnd the MGF of , the equivalent single dynamic server is upper bounded for according to
Backlog and Delay Bounds. The backlog and delay are defined as:And they are bounds given by
2.4. Problem Description
Figure 2 shows an example of WMNCs. Here three sensors are used to measure the information from two plants, five routers are used to translate information, control tasks are computed on the controller and control commands are sent to the actuators to regulate the plants.
Many WMNCSs have stringent performance requirements for feedback control. However, these WMNCSs are characteristic with stochasticity which arises from the factors as wireless channel condition, network topology, network protocol, routing, data flow configuration, control protocol, etc. How to model and analyze WMNCSs is still a challenging task.
In this paper, we set up a three-layered modeling framework to capture the stochastic properties of WMNCSs at different abstract levels. We use SHS to set up a system layer model, SNC to set up a bit layer model, and statistical model for physical layer model. This framework can help us to figure out how the factors as network topology, channel condition, other network traffics, etc. affect the performance of multiple hopped wireless control systems.
SHS, SNC, and statistical models of fading channel are developed for different application areas. It has the following gaps for these methods to be used together:(1)For the periodic sampling events at the sensor node, how to set up the stochastic model and integrate with SNC. It gives the input of SNC model.(2)When considering the channel fading and coding, how to use statistical models at the physical layer to calculate the service process of SNC model. It is how to integrate the physical layer model with bit layer model.(3)To analyze the stability of SHS, it usually needs to know their set-time distributions, but the SNC model only provides the probability guarantees. It is how to integrate bit layer model with the system model.
In this work, we also bridge these gaps. We emphasize the unified and flexible properties of this framework. We take a unified upper bound approach based on MGFs and stochastic orders, so these models can be used together to analyze system performance. The flexibility embodied in the methods we selected at each layer, so we can solve the same problem with little effort when the conditions as network topology, channel condition, or data flow configuration are changed. The unified and flexible properties can help us to try different design combinations easily and support quick design decisions.
3. System Layer Model
In this section, we concentrate on how to model the feedback control behavior of WMNCSs at the system level with SHS. We use SHS to capture both the continuous dynamics and the discrete logic of WMNCSs with a simulation approach.
3.1. Modeling Controller with Stochastic Hybrid System
We consider an SISO (single input and single output) WMNCS that consists of a sensor, an actuator, and a controller that are connected through a shared wireless network. Figure 3 shows the structure of this system. Here the sensor and actuator are connected to the plant directly, and the controller communicates with the sensor and actuator through a shared wireless network.
When considering the linear time-invariant models for the plant and controller, the system can be described bywhere, in the plant model, denotes the state of the plant, denotes the most recently received control variable, and y denotes the output of the plant. In the controller model, denotes the state of the controller, denotes the controller output, and denotes the most recently received output of the plant. The controller is assumed to yield the closed-loop stable when the plant and the controller are directly connected, i.e., when and . We use to denote the times at which the k-th data of sensor data y or control command u ( or u) is received by the receiver, and the time intervals are assumed to be identically distributed with distribution . Between these times, and are held constant.
We use SHS to model above WMNCS. The key to the model network control system with SHS is to capture the sequence of events as a finite state machine, which is mapped to the discrete modes of SHS. We follow a simulation way and give two examples [5].
Case 1. When the controller and actuator are event-triggered, packet drops are not considered.
Here we consider a WMNCS with one sensor, one controller, and one actuator which are all events triggered. The control loop length is set to . Under the normal state, sensor node samples at the beginning of every control loop . For event triggered scheme, the controller and the actuator update their internal states when they receive new data, and the actuator takes action when receiving command. Further, we assume that is long enough and the following two assumptions are satisfied.
Assumption 1. As packet drops are not considered, the data packets will reach the destination finally. The value of sensor to controller delay and controller to actuator delay are lower and upper bounded. They are formulated as Here, and are the lower bounds of and , respectively, for the minimum time spent on computation and communication. and are the upper bounds of and , which can be guaranteed by network technologies as WirelessHART and ISA100 without the considering of packet losses.
Assumption 2. We assume that the following inequation always holds: This assumption guarantees the time sequence of mode jumps can be preserved. As is shown in the Figure 4, if , at the third control loop, the controller may receive sampling data before the actuator receives the command of the previous control loop. For a violation example, let us look at the state jump indicated by dotted arrow line, which means a jump may happen at the mode .
Under the above two assumptions, the discrete modes and transitions can be listed as below.
Discrete Modes. Two modes are required to model this case.
Mode 1, after the actuator updated its inner state, the controller is waiting for a new sample data.
Mode 2, after the controller received a new sampling data and updated its inner state, the actuator is waiting for a new command from the controller.
Jumps. Here we need two mode jumps.
Jump 1, for the sensor sampling data arrival controller, the controller updates the state estimator by received , it has
Jump 2, for the command packet arrival controller, the actuator updates its state and uses the latest value to actuate the plant. Under this situation, it has the jump:
In any interval between jumps, the system dynamic can be written as
The above equations express the fact that and are continuous signals and, at sampling times, and are updated to the value of or remained with the same value depending on jump condition.
Figure 5 shows the graphical representation of this SHS. The system has two discrete states (models), and two different reset maps defined by the matrices and . The reset maps are and and set-time distributions and are associated with them, respectively. This SHS can be regarded as a special case of TTSHS, for which there is only one jump at each mode.
For simulation, the sensor takes a sample at the beginning of the control loop. After , the controller receives the sampling data, updates its inner state, and the system states jump to . Then controller computes and sends control commands to the actuator. After , the actuator receivers the control command, updates its inner states, and the system jump to . Then a new cycle begins with the sensor taking a new sample at the beginning of the next control loop.
Case 2. When the controller and actuator are event-triggered, packet drops are considered.
In this situation, we use two more discrete modes to present packet drops at the controller and actuator. Figure 6 shows the SHS model. Its modes and jumps are listed as below.
Discrete Modes. Four modes are required to model this case.
Mode 1, after the actuator successfully received the command and updated its inner state, the controller is waiting for a new sample data from the sensor.
Mode 2, after the controller successfully received a new sampling data and updated its inner state, the actuator is waiting for new commands.
Mode 3, the actuator failed to receive the command due to the packet drop, and the controller is waiting for a new sample data from the sensor.
Mode 4, the controller failed to receive a new sampling data, and the actuator is waiting for a new command from the controller.
Jumps. There are eight jumps between discrete modes, which can be classified into three groups:
Group 1, jumps triggered by the controller receiving a new sampling data, and the controller updates its inner state with .
Group 2, jumps triggered by the actuator receiving a new sampling data, and the actuator updates its inner state with .
Group 3, jumps triggered by the controller or the actuator failed to receive a new sampling data, and the system is updated with
In Figure 6, every mode including two jump outs, one for successful transportation and one for packet drop. This process can be modeled by a Markov chain:We use to indicate the probability packet drops at the controller and to indicate the probability packet drops at the actuator, and the matrix of the Markov chain is given by
In any interval between jumps, the system dynamic follows the law of .
For simulation, the sensor takes a sample at the beginning of the control loop. Then after , it draws a coin with a probability for packet drop. If a successful transmission occurs, the controller receives the sampling data and updates its inner states, and the system jumps to , or else the system jumps to . Then controller computes and sends control commands to the actuator. After , it draws a coin with a probability for packet drop. If a successful transmission occurs, the actuator receives the control command and updates its inner states, and the system states jump to ; otherwise the system jumps to . When the system has entered into mode 3 (mode 4), it waits for ( ) time, draws a coin, and jumps to or ( or ).
3.2. Stability Analysis of SHS
We consider impulsive systems with several reset maps triggered by independent renewal processes; i.e., the intervals between jumps associated with a given reset map are identically distributed and independent of the other jump intervals. Considering the linear dynamic and reset maps, we establish that mean exponential stability (MES) is equivalent to the spectral radius of an integral operator being less than one.
Mean Exponentially Stable (MES). A SHS is mean exponentially stable if there exist constants and such that, for any initial condition of the system, we have
Assumption 3 (see [9]). The transition distributions are measurable functions.
and , that the following inequations are satisfied:where is a continuous function, and are constants, and is the projection function of to ( ).
Itô Equation and Extended Generator [9]. Under Assumption 3, we have Itô equation and extended generator for SHS. They are tools to compute the expectations of SHS. The Itô equation is The extended generator of SHS is defined as
The Expectation of SHS [9]. For every function that continuously differentiable with respect to x, t, and initial condition , the expectation on the state of SHS is given by
Theorem 4. For the TTSHS, the system is MSS if only if the inequation holds, where equals 1 if and 0 otherwise, is the spectral radius of operator .
Please refer to [10] for proof.
4. Bit Layer Model
In typical network-layer performance models, traffic and service are usually measured in bits; we refer to these models as bit layer models. In this section, we tailor SNC to model and analyze the performance of WMNCSs. We choose SNC because it satisfies the basic properties for network performance analysis as service guarantees, output characterization, concatenation property, leftover service property, and superposition property [6]. These properties will provide enough flexibility for our modeling and analysis.
We consider a data flow across a wireless multihop network and compute its end-to-end delay. As is shown in Figure 7, in a wireless N-node tandem network, there is a data flow traversing the entire network which may encounter cross traffic at each node. Here, the cross-traffic at a node is the aggregate of all traffic that traverses the node but does not belong to the considered flow. For the most common and useful strategy, all these data flows are assigned with a fixed priority and follow a FIFO scheduling at servers.
We use a random process to describe the service given to the considered flow at a node. This service process is governed by the instantaneous channel capacity and the cross traffic at the node. Here we use a discrete-time domain where is the index of time lot and is the time slot length. The system is assumed to start with empty queues at time t = 0.
For a flow indexed by l, we use random process to represent the cumulative arrivals at node n, to represent the service offered by node n, and to represent the departures from the node n. We firstly setup a single service node model, and then the multihop model. Throughout, we assume that the arrival and service processes satisfy stationary bounds.
4.1. Single Service Node Model
Here, considering a single service node, the cumulative arrivals and departures of data flow l within are formulated aswhere is the arrivals and the departures in the i-th time slot. Due to causality, we always have The dynamic service of the node given to the flow l in the time interval is represented by the random process .
For a stable queuing system, the average arrival rate is smaller than the average service rate. Stationary performance bounds can be obtained only when the system satisfies following stability condition:
The service that relates the departures of a system to its arrivals is
The backlog of the flow l at node n at time is given by
The delay of the flow l at node n is given by
As and are stochastic bivariate functions, the MGFs of and for any are given as
Theorem 5. Given a single service node where its arrival flows are described by , and its available service is given by a dynamic server . Fix and defineThen we have the following performance bounds.
Output burstiness: , whereBacklog: , whereDelay: , where is the smallest number satisfying
Proof. From the inequation , we have Comparing with inequation (14), it is easy to get the results for output burstiness.
From , we have Comparing with (14) and replacing s with t, we have the output backlog bounds.
From the definition of , it is easy to getAs , we can getand the result is obvious.
4.2. Multiple Hop Flow Model
For multiple hop situations, the stability condition is expressed as
For flow l, we want to compute its performance over multiple hops. At first, we introduce the following theorem.
Theorem 6. Considering a flow l passes a work-conserving server with the service process that is nonnegative, increasing in t, and . Then the service provided to flow l is given byHere Flow(n) gives all the data follows that pass the node n, and is the flows in the flow set with the priority higher than that of l.
The MGF of is bounded for , by
Proof. For a fixed priority flow l, we only consider the flows with a priority higher than that of l. The result (58) is proved by applying (17) several times. The MGF of can be deduced directly from (18).
Theorem 7. Consider a flow l charactering by passes nodes in series. There exists an equivalent, single dynamic server for whereAnd the MGF of , the equivalent, single dynamic server is upper bounded for according to
Proof. Equation (60) can be deduced directly by applying (19) several times and the MGF upper bound from (20).
Corollary 8. Consider a flow passes a cascade of servers ; it has
Proof. From the definition of operators and , it hasThe result is gotten directly from (61) by applying above equation several times.
The computation network performance analysis process for a WMNCS is given as Algorithm 1.
| ||||||||||||||||||||||||
Here, Node(l) denotes the node set that flow l passes, and denotes the leave over service of node i.
5. Physical Layer Model
For wireless networks, the condition of wireless channel and channel coding method will impact its network performance. At the physical layer, we are interested in how the channel fading and channel coding affect the network delays.
5.1. Channel Fading
As the radio-wave propagation through wireless channel is a complicated phenomenon characterized by various effects, such as multipath and shadowing. Wireless channel is characterized by the rapid variation of channel quality. Channel fading is used to refer to the deviation in the attenuation experienced by the transmitted signal when traversing a wireless channel.
To get a precise mathematical description of channel fading is impossible, and if can be obtained, it will also be too complex for tractable communications systems analyses. A range of relatively simple and accurate statistical models for fading channel which depend on the particular propagation environment and the underlying communication scenario has been proposed. A lot of models are proposed to describe the gain of fading channel depending on the type of fading (slow or fast) and the environment. For industrial application environment, we focus on multipath fading which is due to the combined effects of randomly delayed, reflected, scattered, and diffracted signal components. Multipath fading is relatively fast and is therefore responsible for the short-term signal variations. Table 1 lists some most commonly used multipath fading models [7].
Here, is the fading PDF function, is the instantaneous signal-to-noise power ratio (SNR) per symbol, and is the average signal-to-noise power ratio (SNR) per symbol.
5.2. Channel Coding
When error-correction coding is applied to the transmitted modulation and decisions are made based on an observation of the received signal much longer than a single symbol interval, it becomes necessary to consider the variation of the fading channel from symbol interval to symbol interval.
Papers [7, 11, 12] study the performance of coded communications over fading channel; the results are given as upper bounds on the average bit error probability. These models can also be integrated into our framework. As the length limit, we do not discuss these models in the work. For more detailed descriptions about coding effects for fading channel, please refer to [7, 11, 12].
6. Integrate Three Layer Models
In this section, we show how to overcome the gaps between above three-layer models with upper bound approach and stochastic orders. By changing and combining the models at different layers, our framework can adapt various modeling requirements.
6.1. Modeling the Input Streams
In WMNCSs, typically, sensors would take measurements in a periodic fashion. However, when data is sent through a shared network, the sensors are unable to send data if the network is busy and may need to wait until data can be transmitted, so this introduces jitters for input stream as is shown in Figure 8.
For these properties, we use the constrained arrival process to model the traffic caused by sensor sampling and controller commanding. The MGFs of the cumulative arrival processes of sensing and command data flows in the bit domain are bounded byA wide variety of traffic, including periodic sources, fluid sources, on-off sources, and regulated sources can be described using the constrained arrival process [13–16].
6.2. Modeling the Server Ability
In this subsection, we show how the fading channel affects the performance of wireless networks and how to use these physical layer models to refine the bit layer model.
In bit layer, we model the wireless network as tandem queues of servers. In the real world, these servers are fading channels with variable capacity. Here we ignore the impact of coding and assume the transmission rates over the fading channel are equal to their information-theoretic capacity limit. So the instantaneous, information-theoretic channel capacity of a fading channel is given by [17]Here is a constant of bandwidth and is used to describe the feature of fading channel.
For the fast-fading channel, where the latency requirement is greater than the coherence time and the codeword length spans many coherence periods, one can average over many independent channels fades by coding over a large number of coherence time intervals. Thus, it is possible to achieve a reliable rate of communication ofAnd it is meaningful to speak of this value as the capacity of the fast-fading channel. Here is random channel gain [17].
To calculate the service model for a fast-fading fading channel, we assume that the channel state is sampled by the receiver at a fixed time interval and use this sampled data to present the instantaneous SNR in the i-th time interval. It is obvious that is a random variable that has the probability distribution of the underlying fading model. When time interval is longer than the channel coherence, can be treated as independent random variables with identical distribution. Under the above assumptions and setting , the corresponding service process at bit layer is given byAnd it is easy to get
Theorem 9. Consider a flow l passes a cascade of independent and identically distributed fading channels, where each channel is described by g(γ). The MGF of this concatenation server is up bounded by
Proof. From Theorem 7, it hasThe theorem is proven.
Corollary 10. The end-to-end MGF bounds for a cascade of N fading channels with constrained arrivals process are given by
Proof. From Theorem 5, it hasIn the last step, we have used the combination equation
Corollary 11. To compute the delay bounds, it has the following inequation:
Proof. From Theorem 7 and the combination inequationit hasWe end the proof.
6.3. Analyzing System Stability
In this section, we first introduce the definition of stochastic order of SHSs and then give an upper bounded method to analyze the stability of SHS.
Stochastic Order. Given two random variables X and Y, we say if X is stochastically smaller than Y, , for all x, or in other words , for all , is written as
Similarly, for any two stochastic processes, it has
For stochastic order, it has the following properties.
If , then for any increasing function f.
Let and be independent. IF , then for any wide-sense increasing function on ( when ); then it has
Stochastic Order of SHSs. For an SHS , we define an SHS and all the other ingredients of remain the same with , except for reset time distributions. Then we define that is stochastically smaller than , denoted byif all the reset time distributions of (denoted by ) are stochastically smaller than the corresponding reset time distributions of (denoted by ) that
Theorem 12. For two SHSs and that , and is MES, then is also MES if it meets the following requirements: for all , and . Here and is projection of to ().
Proof. Let the function : be the continuously differentiable with respect to x and t. From its Itô equation (37), dividing the result by dt and take expectations on and , it haswhere represent the operations on and respectively. Here is a stochastic transition counter for jump at mode q. As , it obviously hasWith and the assumption , it hasfor all , and . As is MES, it is obvious thatFrom definition (35), it can be concluded that is MES.
7. Case Study
In this section, we show an example to model and analyze WMNCSs using our framework. As is shown in Figure 9, a WMNCS that controls a batch reactor is deployed in a chemical factory, where one sensor node and one actor node are connected to the remote controller through relay nodes. The channel fading effect is taken into account as the complexity of the environment.
7.1. System Models
A linear state-space model of this process is introduced by Taylor expansion near the working point. The plant of this reactor is described by
The open loop system is unstable, and we use a feedback controller to stabilize the system; i.e., Then the system is stable when the reactor and the controller are directly connected.
The system is implemented with wireless mesh network technology. Without considering packet drops, it can be modeled by an SHS with two discrete modes , as the TTSHS model given in Figure 5.
For the wireless mesh network architecture, the sensors and actuators are I/O device that provides or consumes data. The controller is deployed at the gateway, which communicates with the sensor node and actuator node through a backbone router and a cascade of route nodes respectively. For the simplicity of analysis, we assume that the routers for sensor information and actuator command are different, except for the backbone router, which is usually more capable and connected to the gateway directly through the wired network. In this paper, we only consider the delays because wireless communications, time for wired communications, and computation of control task are thought to be negligible. We use two cascades of stochastic network servers to model the wireless communications, for the sensor node to the backbone router, and for a backbone router to the actuator node. We use and to represent the number of servers for and , respectively.
Here, we consider the dynamic SNR server description for a Rayleigh fading channel with fast fading. The fading gain for Rayleigh fading is a random variable with distribution and the MGF for a single server is given by where is the upper incomplete Gamma function:
Another concern is node conflict, which is due to the half-duplex radio; two transmissions conflict with each other if they share a node (sender or receiver). In a statistical sense, the service is provided by a single server will be half off. We have the following setting:MGF bound for a cascade of N Rayleigh fading channels is given by
From the relationships at Figure 4, we can get the MGF up bounds of the set-time distributions and , which are given aswhere and can be evaluated through the method described in the next subsection.
7.2. Numerical Results
The values of all the important parameters for the above models are listed in Table 2. Here, the network parameters are set to , . For traffic, bounded input arrivals are set to ρ() = 20 kbps, σ() = 20 kbps for sampling data flows, and σ() = 50 kbps for command data flows. The control loop length is set to .
We next present how the delay violation probability for a given end-to-end delay bound is impacted by the node numbers and the SNR wireless channel. We compare multiple hop networks with 2, 4, and 6 server nodes under , SNR or 15dB, respectively. The results are obtained from Theorem 5 by setting the value of violation probability , solving on the left side of the equation, and minimizing over . We use numerical methods to compute the end to end delay bounds for the complexity of the bound expression [18]. Observing that the end-to-end bounds are scale linearly with the number of nodes. The complexity of computing the end-to-end bounds of multiple nodes is not more difficult than that for a single node.
Figures 10 and 11 show the results of the set-time violation bounds for and respectively. By getting the end-to-end delay bound for 0.99 violation probability at Figure 10 for a given SNR and dividing it by the number of nodes, we get the minimum delay for single node , and the parameters and at (94) under different SNR can be evaluated by multiply and with , respectively. The results show that the end-to-end delays are mainly affected by the number of nodes at high SNR, but low SNR will greatly increase the end-to-end delays, as well as affect the aggregation of delay distribution functions. By setting the end-to-end delay bound greater than and getting the corresponding violation probability, it gives us an observation of the violation probability for keeping the system event sequence order. Another observation is that all these bound traces are nearly piecewise linear for Rayleigh fading and can be further upper bounded by a piecewise linear function with two sections, corresponding to an exponential distributionHere, and can be evaluated through estimating the return model parameters. This can be used to further simplify the stability analysis computation at the next step.
Then the multiple hop distribution upper bounds are used to investigate the stability of the system under different network topology and channel condition. For control loop length set , and SNR . We test the mean exponential stability of the closed loop using the numerical method described in [19] by theorem 1. The results obtained are summarized in Figure 12.
In Figure 12, the horizontal coordinate is the number of nodes for the link between sensor and controller, and the vertical coordinate is the number of nodes for the link between the controller and actuator. The solid circles present the pass cases under this test, which means the system is guaranteed to be MES for the upper bounded methods we have used. The hollow circles present the failed cases, which mean the system is identified to be MES unstable with our methods, but maybe actually stable. The overall results presented above give us a quantitative inspection of how the channel condition and multiple hops affect the stability of the WMNCSs. Currently, the scalability of this framework is impeded by the used numerical method. In future research, we will explore numerical methods that can support faster, more exact, and various analysis.
8. Related Works
There are lots of previous researches closely related to this work. Paper [5] introduces two different models of SHS and illustrates how to use SHS to models NCSs. For a detailed introduction to SHS, please refer to [20–23]. The SHS models we considered in this paper are closely related to the Piecewise Deterministic Markov Process (PDMPs) introduced by Davis [24]. The stability of PDMPs has been studied intensively in many materials, including [10, 25–29]. Paper [30] gives a survey of stability analysis for SHS.
This idea of network calculus was initially introduced by Cruz in the seminal work [31]. Network calculus has developed along two tracks, deterministic and stochastic. Book [32] gives a detailed introduction to network calculus. An excellent book [6] is available for deterministic network calculus. Early representative works of SNCs include [33–37]. Recently, many crucial properties have been proved for SNC [38–42]. Paper [8] establishes a concise, probabilistic network calculus with moment generating functions. Paper [18] proposes an SNR domain approach to analyze the performance of multihop fading channels.
An introduction to physical models in digital communication is available [17]. Book [7] gives an MGF upper bounded approach to fading channels over digital communication. A detailed description of Rayleigh fading channels in mobile digital communication systems is given in [43, 44].
9. Conclusions
Recent progress at the area of network performance analysis and stochastic control theory enlighten this work. In the paper, we propose a three-layer modeling framework to capture the properties of WMNCSs at different levels with SHS, SNC, and physical layer models. We also bridge the gaps between these methods with an upper bound approach. These contributions result in a framework for modeling and analysis WMNCSs with stochastic methods and provide us a way to get detailed and unified results. To the best of our knowledge, it is the first work that provides such a unified and flexible framework to model and analyze WMNCSs with stochastic methods.
There are lots of interesting topics for further work. A topic is to extend the existing framework to support more controller models, network scheduling strategies. Another interesting topic is to set up a standard model library for rapid modeling and analysis. Finally, accompanied with this framework, a set of numerical methods can be set up to support faster, more exact, and various analyses.
Data Availability
The code and raw data used to support the findings of this study are restricted by the SIEMENS medium voltage switch Co., Ltd. at Wuxi, China, to protect patient. Some processed data are available from Jing Liu (liujing@ss.pku.edu.cn) and for researchers who meet the criteria for access to confidential data.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Jing Liu and Yixu Yao contributed equally to this paper.