# Transmission Scheduling Schemes of Industrial Wireless Sensors for Heterogeneous Multiple Control Systems

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. System Model

- While each sensor operates in a time-driven fashion, both the controller and actuator operate in an event-driven fashion. In general, wireless sensors transmit data in each assigned time slot dependent on the transmission scheduling scheme. However, both the controller and actuator only respond to newly received data over unreliable wireless links. We assume that the controllers are collocated with the actuators since the control signal is more critical than the sensing information in many practical NCSs [7].
- In [1], we defined three major metrics of WNCSs, namely sampling interval, packet dropout, and packet delay. Two main reasons of packet dropouts are packet discard due to the control algorithm and packet loss due to the wireless network itself. Most works of control systems model the dropouts as prolongations of the sampling interval [6,19]. The reason is that a new packet is transmitted at the next transmission time with new data if a packet is dropped. Hence, both the controller and actuator observe the time-varying sampling interval even if the sensing and actuating links operate in a fixed time interval. The time-varying sampling interval of successfully received information called transmission interval (TI) effectively captures the essential characteristics of packet dropout and sampling interval [8,19]. The delays are generally assumed to be smaller than the transmission intervals.

#### 3.1. Control Aspect

#### 3.2. Communication Aspect

## 4. Fundamental Observation

- uncertain time-varying TIs $h\in [\underline{h},\overline{h}]$; and
- uncertain time-varying network delays $d\in [\underline{d},\mathrm{min}(h,\overline{d})]$.

## 5. Optimization Problem Formulation

#### 5.1. Extended Transmission Interval

#### 5.2. Optimization Problem

## 6. Centralized Random Access Scheme

**Proposition**

**1.**

**Proof.**

#### Optimal Solution

- Stationarity with ${\alpha}_{i}$$$\begin{array}{c}{\nabla}_{{\alpha}_{i}}L(\eta ,{\alpha}_{i},{\lambda}_{i},\gamma )=0\end{array}$$
- Stationarity with $\eta $$$\begin{array}{c}{\nabla}_{\eta}L(\eta ,{\alpha}_{i},{\lambda}_{i},\gamma )=0\end{array}$$
- Complementary slackness of Equation (8b)$$\begin{array}{c}{\lambda}_{i}\left(\right)open="("\; close=")">\frac{1}{{p}_{i}{\alpha}_{i}}-{\overline{h}}_{i}-\eta =0\end{array}$$
- Complementary slackness of Equation (8c)$$\begin{array}{c}\gamma \left(\right)open="("\; close=")">\sum _{i=1}^{N}{\alpha}_{i}-1=0\end{array}$$
- Primal feasibility$$\begin{array}{c}\frac{1}{{p}_{i}{\alpha}_{i}}-{\overline{h}}_{i}-\eta \le 0\end{array}$$$$\begin{array}{c}\sum _{i=1}^{N}{\alpha}_{i}\le 1\phantom{\rule{3.3em}{0ex}}\end{array}$$
- Dual feasibility$$\begin{array}{c}{\lambda}_{i}\ge 0,\gamma \ge 0\end{array}$$

## 7. Distributed Random Access Scheme

**Proposition**

**2.**

**Proof.**

#### Optimal Solution

## 8. Centralized Lyapunov-Based Scheduling Scheme

#### Optimal Solution

**Proposition**

**3.**

**Proof.**

## 9. Performance Evaluation

## 10. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

**Proof.**

## Appendix B. Proof of Proposition 2

**Proof.**

## Appendix C. Proof of Proposition 3

**Proof.**

## References

- Park, P.; Ergen, S.C.; Fischione, C.; Lu, C.; Johansson, K.H. Wireless network design for control systems: A survey. IEEE Commun. Surv. Tutor.
**2018**, 20, 978–1013. [Google Scholar] [CrossRef] - Sztipanovits, J.; Koutsoukos, X.; Karsai, G.; Kottenstette, N.; Antsaklis, P.; Gupta, V.; Goodwine, B.; Baras, J.; Wang, S. Toward a science of cyber-physical system integration. Proc. IEEE
**2012**, 100, 29–44. [Google Scholar] [CrossRef] - O’donovan, T.; Brown, J.; Büsching, F.; Cardoso, A.; Cecílio, J.; Furtado, P.; Gil, P.; Jugel, A.; Pöttner, W.-B.; Roedig, U.; et al. The GINSENG System for Wireless Monitoring and Control: Design and Deployment Experiences. ACM Trans. Sens. Netw.
**2013**, 10, 1–40. [Google Scholar] [CrossRef] - Cecílio, J.; Furtado, P. Planning for Distributed Control Systems with Wired and Wireless Sensors. In Wireless Sensors in Industrial Time-Critical Environments; Springer International Publishing: Cham, Switzerland, 2014. [Google Scholar]
- Ramotsoela, D.; Abu-Mahfouz, A.; Hancke, G. A survey of anomaly detection in industrial wireless sensor networks with critical water system infrastructure as a case study. Sensors
**2018**, 18, 2491. [Google Scholar] [CrossRef] [PubMed] - Cloosterman, M.B.G.; van de Wouw, N.; Heemels, W.P.M.H.; Nijmeijer, H. Stability of networked control systems with uncertain time-varying delays. IEEE Trans. Autom. Control
**2009**, 54, 1575–1580. [Google Scholar] [CrossRef] - Hespanha, J.P.; Naghshtabrizi, P.; Xu, Y. A survey of recent results in networked control systems. Proc. IEEE
**2007**, 95, 138–162. [Google Scholar] [CrossRef] - Zhang, W.; Branicky, M.S.; Phillips, S.M. Stability of networked control systems. IEEE Control Syst.
**2001**, 21, 84–99. [Google Scholar] [Green Version] - Sinopoli, B.; Schenato, L.; Franceschetti, M.; Poolla, K.; Jordan, M.I.; Sastry, S.S. Kalman filtering with intermittent observations. IEEE Trans. Autom. Control.
**2004**, 49, 1453–1464. [Google Scholar] [CrossRef] - Smith, S.C.; Seiler, P. Estimation with lossy measurements: jump estimators for jump systems. IEEE Trans. Autom. Control
**2003**, 48, 2163–2171. [Google Scholar] [CrossRef] - Fujioka, H. Stability analysis for a class of networked/embedded control systems: A discrete-time approach. In Proceedings of the American Control Conference (ACC), Seattle, WA, USA, 11–13 June 2008; pp. 4997–5002. [Google Scholar]
- Montestruque, L.A.; Antsaklis, P. Stability of model-based net-worked control systems with time-varying transmission times. IEEE Trans. Autom. Control
**2004**, 49, 1562–1572. [Google Scholar] [CrossRef] - Hetel, L.; Daafouz, J.; Iung, C. Stabilization of arbitrary switched linear systems with unknown time-varying delays. IEEE Trans. Autom. Control
**2006**, 51, 1668–1674. [Google Scholar] [CrossRef] - Gielen, R.; Olaru, S.; Lazar, M.; Heemels, W.; van de Wouw, N.; Niculescu, S.-I. On polytopic inclusions as a modeling framework for systems with time-varying delays. Automatica
**2010**, 46, 615–619. [Google Scholar] [CrossRef] [Green Version] - Seiler, P.; Sengupta, R. An H
_{∞}approach to networked control. IEEE Trans. Autom. Control**2005**, 50, 356–364. [Google Scholar] [CrossRef] - Yue, D.; Han, Q.-L.; Peng, C. State feedback controller design of networked control systems. IEEE Trans. Circ. Syst. II Express Brief.
**2004**, 51, 640–644. [Google Scholar] [CrossRef] - Xiong, J.; Lam, J. Stabilization of linear systems over networks with bounded packet loss. Automatica
**2007**, 43, 80–87. [Google Scholar] [CrossRef] - Zhang, W.-A.; Yu, L. Modelling and control of networked control systems with both network-induced delay and packet-dropout. Automatica
**2008**, 44, 3206–3210. [Google Scholar] [CrossRef] - Heemels, W.P.M.H.; Teel, A.R.; van de Wouw, N.; Nesic, D. Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance. IEEE Trans. Autom. Control
**2010**, 55, 1781–1796. [Google Scholar] [CrossRef] - Baldi, M.; Giacomelli, R.; Marchetto, G. Time-driven access and forwarding for industrial wireless multihop networks. IEEE Trans. Ind. Inform.
**2009**, 5, 99–112. [Google Scholar] [CrossRef] - Antepli, M.A.; Uysal-Biyikoglu, E.; Erkan, H. Optimal packet scheduling on an energy harvesting broadcast link. IEEE J. Sel. Areas Commun.
**2011**, 29, 1721–1731. [Google Scholar] [CrossRef] - Demirel, B.; Zou, Z.; Soldati, P.; Johansson, M. Modular design of jointly optimal controllers and forwarding policies for wireless control. IEEE Trans. Autom. Control
**2014**, 59, 3252–3265. [Google Scholar] [CrossRef] - Ergen, S.C.; Varaiya, P. TDMA scheduling algorithms for wireless sensor networks. Wirel. Netw.
**2010**, 16, 985–997. [Google Scholar] [CrossRef] - Park, P.; Di Marco, P.; Fischione, C.; Johansson, K.H. Modeling and Optimization of the IEEE 802.15.4 Protocol for Reliable and Timely Communications. IEEE Trans. Parallel Distrib. Syst.
**2013**, 24, 550–564. [Google Scholar] [CrossRef] [Green Version] - Park, P.; Di Marco, P.; Fischione, C.; Bonivento, A.; Johansson, K.H.; Sangiovanni-Vincent, A. Breath: An Adaptive Protocol for Industrial Control Applications Using Wireless Sensor Networks. IEEE Trans. Mob. Comput.
**2011**, 10, 821–838. [Google Scholar] [CrossRef] [Green Version] - Park, P.; Araujo, J.; Johansson, K.H. Wireless Networked Control System Co-Design. In Proceedings of the IEEE International Conference on Networking, Sensing and Control (ICNSC), Delft, The Netherland, 11–13 April 2011; pp. 486–491. [Google Scholar]
- Sadi, Y.; Ergen, S.C.; Park, P. Minimum energy data transmission for wireless networked control systems. IEEE Trans. Wirel. Commun.
**2014**, 13, 2163–2175. [Google Scholar] [CrossRef] - Sadi, Y.; Ergen, S.C. Energy and delay constrained maximum adaptive schedule for wireless networked control systems. IEEE Trans. Wirel. Commun.
**2015**, 14, 3738–3751. [Google Scholar] [CrossRef] - Park, P.; Marco, P.D.; Johansson, K.H. Cross-layer optimization for industrial control applications using wireless sensor and actuator mesh networks. IEEE Trans. Ind. Electron.
**2017**, 64, 3250–3259. [Google Scholar] [CrossRef] - Ma, Y.; Gong, B.; Sugihara, R.; Gupta, R. Energy-efficient deadline scheduling for heterogeneous systems. J. Parall. Distrib. Comput.
**2012**, 72, 1725–1740. [Google Scholar] [CrossRef] - Pang, Z.; Luvisotto, M.; Dzung, D. Wireless High-Performance Communications: The Challenges and Opportunities of a New Target. IEEE Ind. Electron. Mag.
**2017**, 11, 20–25. [Google Scholar] [CrossRef] - Tramarin, F.; Vitturi, S.; Luvisotto, M. A Dynamic Rate Selection Algorithm for IEEE 802.11 Industrial Wireless LAN. IEEE Trans. Ind. Inform.
**2017**, 13, 846–855. [Google Scholar] [CrossRef] - Donkers, M.C.F.; Heemels, W.P.M.H.; van de Wouw, N.; Hetel, L. Stability analysis of networked control systems using a switched linear systems approach. IEEE Trans. Autom. Control
**2011**, 56, 2101–2115. [Google Scholar] [CrossRef] - Lofberg, J. YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), New Orleans, LA, USA, 26 April–1 May 2004; pp. 284–289. [Google Scholar]
- Sturm, J.F. Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones. Optim. Methods Softw.
**1999**, 11, 625–653. [Google Scholar] [CrossRef] - Nesic, D.; Teel, A.R. Input-output stability properties of networked control systems. IEEE Trans. Autom. Control
**2004**, 49, 1650–1667. [Google Scholar] [CrossRef] - Petersen, S.; Carlsen, S. WirelessHART versus ISA100.11a: The format war hits the factory floor. IEEE Ind. Electron. Mag.
**2011**, 5, 23–34. [Google Scholar] [CrossRef] - Rom, R.; Sidi, M. Aloha Protocols. In Multiple Access Protocols: Performance and Analysis; Springer: Berlin, Germany, 1990. [Google Scholar]
- Kelly, F.P.; Maulloo, A.K.; Tan, D.K.H. Rate control for communication networks: Shadow prices, proportional fairness and stability. J. Oper. Res. Soc.
**1998**, 49, 237–252. [Google Scholar] [CrossRef] - Srikant, R. The Mathematics of Internet Congestion Control; Boston Birkhauser: New York, NY, USA, 2004. [Google Scholar]
- Park, P. Power controlled fair access protocol for wireless networked control systems. Wirel. Netw.
**2015**, 21, 1499–1516. [Google Scholar] [CrossRef] - Neely, M. Lyapunov Optimization. In Stochastic Network Optimization with Application to Communication and Queueing Systems; Morgan & Claypool: San Rafael, CA, USA, 2010. [Google Scholar]
- Zolertia RE-Mote Revision B Internet of Things Hardware Development Platform. Available online: https://github.com/Zolertia/Resources/wiki/RE-Mote (accessed on 1 December 2018).
- Joy, W.; Ross, Y. Wireless Sensor Networking for the Industrial Internet of Things; White Paper; Linear Technology: Milpitas, CA, USA, 2015. [Google Scholar]
- Gross, D.; Harris, C.M. Advanced Markovian Queueing Models. In Fundamentals of Queueing Theory; Wiley: Hoboken, NJ, USA, 1998. [Google Scholar]

**Figure 3.**Stability region over different MATI and MAD values. The circle and rectangular marker present the stability and instability operating region of control systems for a given MATI and MAD value.

**Figure 4.**Transmission interval and delay of both TDMA and slotted Aloha schemes over the stability region: (

**a**) TDMA performance over stability region; and (

**b**) slotted Aloha performance over stability region.

**Figure 6.**CDF of slack of ideal solution, centralized random access, distributed random access, and Lyapunov-based approach with $N=8$.

**Figure 7.**Minimum slack, average TI, and outage probability of ideal solution, centralized random access, distributed random access, and Lyapunov-based approach as a function of different number of nodes $N=3,\dots ,30$: (

**a**) minimum slack vs. number of nodes; (

**b**) average TI vs. number of nodes; and (

**c**) outage probability vs. number of nodes.

**Figure 8.**MATI requirement and average TI of each node using different transmission scheduling schemes with $N=9$.

**Figure 9.**Access probability of each node using centralized random access and distributed random access schemes with $N=9$.

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**MDPI and ACS Style**

Park, B.; Nah, J.; Choi, J.-Y.; Yoon, I.-J.; Park, P.
Transmission Scheduling Schemes of Industrial Wireless Sensors for Heterogeneous Multiple Control Systems. *Sensors* **2018**, *18*, 4284.
https://doi.org/10.3390/s18124284

**AMA Style**

Park B, Nah J, Choi J-Y, Yoon I-J, Park P.
Transmission Scheduling Schemes of Industrial Wireless Sensors for Heterogeneous Multiple Control Systems. *Sensors*. 2018; 18(12):4284.
https://doi.org/10.3390/s18124284

**Chicago/Turabian Style**

Park, Bongsang, Junghyo Nah, Jang-Young Choi, Ick-Jae Yoon, and Pangun Park.
2018. "Transmission Scheduling Schemes of Industrial Wireless Sensors for Heterogeneous Multiple Control Systems" *Sensors* 18, no. 12: 4284.
https://doi.org/10.3390/s18124284