Issue 
Natl Sci Open
Volume 3, Number 1, 2024



Article Number  20230036  
Number of page(s)  3  
Section  Information Sciences  
DOI  https://doi.org/10.1360/nso/20230036  
Published online  06 November 2023 
PERSPECTIVE
Network system capacity: Towards integrating sensing, communication and control
^{1}
Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
^{2}
Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China
^{3}
Shanghai Engineering Research Center of Intelligent Control and Management, Shanghai 200240, China
^{*} Corresponding author (email: xpguan@sjtu.edu.cn)
Received:
21
June
2023
Revised:
28
July
2023
Accepted:
1
August
2023
Network systems refer to a new generation of systems with integrated information perception, transmission and utilization capabilities through communication networks, which are adopted to achieve desirable objectives under physical and information related uncertainties and/or adversary conditions. In the informationrich era [1, 2], one of the fundamental issues is to exploit the limit of feedback control on dissipating such uncertainties in the scenarios of networked sensing and communication [3]. The existing information and control theories are difficult to reconcile due to different mathematical basis and conceptual paradigms from their aspects. It is challenging to effectively reveal the intrinsic triadic relation among networked sensing, communication, and feedback control. The concept of “network system capacity" is proposed to give a unified analytical expression for the integration of sensing, communication and control.
A typical architecture of network systems is shown in Figure 1, which consists of the controlled plant, sensor network, communication network, and feedback controllers. The state information of the controlled plant is collected through the sensor network, and transmitted to the estimator for achieving system state estimation. The estimation is passed to the controller to generate control commands, which are sent to the actuator through the communication network to realize the closedloop control. The dynamics of the plant is x[k+1]=a(x[k]+B[k]u^{*}[k])+w[k], where x[k], u^{*}[k], and w[k] denote the state, control input vector, and the noise of the plant, respectively, the scalar a is the system gain. The term u^{*}[k]=γ[k]u[k], where $u\left[k\right]=K\widehat{x}\left[k\right]$ is the designed control law with the feedback gain K and the estimate $\widehat{x}\left[k\right]$ of x[k], and the probability of successful transmission through the noisy channel γ[k]~p_{γ}, is of independent identical distribution (i.i.d.). The mean and variance of γ[k] are denoted as μ_{b} and ${\sigma}_{b}^{2}$. The sensing equation is y[k]=C[k]G[k]x[k]+v[k], where G[k] and C[k] are the observation vector and indicator matrix of the activated sensors, respectively. The stochastic variables w[k]~p_{w} and v[k]~p_{v} are the white noises. The mean and variance of v[k] are denoted as μ_{v} and ${\sigma}_{v}^{2}$, respectively. Based on the aforementioned network system model, the “network system capacity" is derived from the following three aspects.
Figure 1 Network system architecture. (a) “Boomerang” like relationship between channel capacity and networked sensing capacity. (b) Relationship between the gain of network system capacity and channel capacity. (c) Relationship between the gain of network system capacity and networked sensing capacity. 
Firstly, communication networks offer competitive advantages in collaboration of the sensing and control processes, such as flexibility and scalability. To improve the collaborative sensing and control performance, it is necessary to further reveal the analytical relationship of channel capacity and sensing/control performance. The channel capacity is defined to be the maximum rate at which information can be transmitted through a channel [4], i.e., ${\mathcal{C}}_{c}=W{{\displaystyle \mathrm{log}}}_{2}\left(1+{\mu}_{b}^{2}/{\sigma}_{b}^{2}\right)$, where W is the bandwidth. It portrays the theoretical limit of the amount of sensing data G[k]x[k] and control data u[k] that can be transmitted over the noisy channels. The channel capacity thus constrains the sensing and control processes in the loop of the system.
Secondly, networked sensing enables the collaboration and data interaction of multiple sensors to achieve continuous and efficient monitoring of the system. It is a fundamental process for information acquisition and fusion through sensor networks, and plays an important role in state/parameter estimation and control of network systems. However, it is corrupted by the inevitable uncertainties such as ambient noise, interference and multipath fading [5, 6]. Thus, the new concept called networked sensing capacity ${\mathcal{C}}_{s}=1/{\sigma}_{v}^{2}$ is defined to measure the estimation performance and sensor measurements under a given network topology.
Finally, feedback control is the process of returning the output information of the system to the input side, and using the deviation of the output and input information to steer the state of the controlled plant to a predetermined stable or equilibrium trajectory [7]. To describe the capability of controllers to stabilize a system, the concept of control capacity is introduced by [8, 9]. However, in the network system as in Figure 1, the networked sensing capacity and channel capacity limit the accuracy of the state estimation and successful transmission rate of the control command, respectively. Those uncertainties from sensing and communication processes induce challenges in capturing the fundamental limits of the control performance. Inspired by [8], the network system capacity is expressed based on the secondmoment stability in onestep from the perspective of feedback control with the considered uncertainties, i.e., ${\mathcal{C}}_{N}={{\displaystyle \mathrm{max}}}_{u\left[k\right]}\frac{1}{2}\mathrm{log}\mathbb{E}\left[{\left(\frac{x\left[k\right]}{x\left[k1\right]}\right)}^{2}\right]$ for all $k\in {\mathbb{R}}^{+}$, where the maximum can be obtained by the proper design of feedback gain K similarly as [8]. Thus, the network system capacity is expressed as ${\mathcal{C}}_{N}=\frac{1}{2}\mathrm{log}\left[1+\frac{{\mathcal{C}}_{s}}{{\alpha}_{0}\cdot {2}^{{\mathcal{C}}_{c}/W}+\left({2}^{{\mathcal{C}}_{c}/W}1\right)\cdot {\mathcal{C}}_{s}}\right]$ where α_{0} is a constant determined by the initial state x[0], and C_{N} characterizes the capability of the controller to stabilize the plant under sparse networked sensing and lossy communication, i.e., the uncertainty dissipation ability under the optimal system feedback gain.
Figure 1 reveals the gain of network system capacity composed of $\frac{\partial {\mathcal{C}}_{N}}{\partial {\mathcal{C}}_{c}}$ and $\frac{\partial {\mathcal{C}}_{N}}{\partial {\mathcal{C}}_{s}}$ under the influence of C_{s} and C_{c}. Inset (a) shows the “boomerang” like relationship of C_{s} and C_{c}. It reveals that C_{s} decreases with respect to C_{c} for a given gain of network system capacity. Insets (b) and (c) reflect that the gain of network system capacity varies with the incremental change of C_{s} and C_{c}. The top point corresponds to the optimal gain of network system capacity. The optimal values of channel capacity C_{c} and networked sensing capacity C_{s} can also be easily found, respectively. When C_{c} and C_{s} are smaller than their optimums, the gain of network system capacity increases very fast. However, when C_{c} and C_{s} are larger than their optimums, the gain of network system capacity drops greatly. It means the network system capacity is becoming saturated.
The network system capacity is tightly related to networked sensing capacity and channel capacity and the intrinsic triadic relationship is exploited from the perspective of control. When considering the general linear systems with multiple state variables, an expression of network system capacity can be obtained in a similar way. When considering multiple controlled plants, there exist complex communication and sensing resource competitions among different loops. It then yields more difficulties in giving a unified analytical expression for the integration of sensing, communication and control. Of course, the relationship can also be explored from the perspective of networked sensing capacity and/or channel capacity. From the perspective of sensing, the control force makes the network system obtain richer measurements and more accurate estimation. The channel capacity then limits the ability of interacting among sensors, the controller and actuator. From the perspective of communication, efficient sensing of the communication environment is required to accurately identify interference to achieve optimized network capacity [10]. Besides, control theory can also be developed for congestion avoidance and quality of service (QoS) improvements. Based on different perspectives, the intrinsic coupling among sensing, communication and control can be explored comprehensively so that the overall system performance could be optimized by integrating appropriate sensing mechanisms, communication protocols and control laws.
Funding
This work was supported by the National Natural Science Foundation of China (92167205).
Author contributions
The author thanks Dr. Ling Lyu, Mr. Haifan Su, and Mr. Zhiduo Ji for the simulation and illustration, and thanks Mr. Jinglong Zhang and Mr. Lei Xu for the constructive discussion.
Conflict of interest
The author declares no conflict of interest.
References
 Murray RM, Astrom KJ, Boyd SP, et al. Future directions in control in an informationrich world. IEEE Control Syst Maga 2003; 23: 20–33. [Google Scholar]
 LamnabhiLagarrigue F, Annaswamy A, Engell S, et al. Systems & control for the future of humanity, research agenda: Current and future roles, impact and grand challenges. Annu Rev Control 2017; 43: 1–64. [Article] [CrossRef] [Google Scholar]
 Guan X, Yang B, Chen C, et al. A comprehensive overview of cyberphysical systems: From perspective of feedback system. IEEE/CAA J Autom Sinica 2016; 3: 1–14. [Google Scholar]
 Shannon CE. Claude Elwood Shannon: Collected Papers. New York: IEEE Press, 1993. [Google Scholar]
 Rachlin Y, Negi R, Khosla PK. The sensing capacity of sensor networks. IEEE Trans Inform Theor 2011; 57: 1675–1691. [Article] [Google Scholar]
 Tzoumas V, Carlone L, Pappas GJ, et al. LQG control and sensing codesign. IEEE Trans Automat Contr 2021; 66: 1468–1483. [Article] [Google Scholar]
 Wiener N. Cybernetics. Bull Am Acad Arts Sci 1950; 3: 2–4. [Google Scholar]
 Ranade G, Sahai A. Control capacity. IEEE Trans Inform Theor 2019; 65: 235–254. [Article] [CrossRef] [MathSciNet] [Google Scholar]
 Fang S, Chen J, Hideaki I. Towards integrating control and information theories. In: Allgower F, Morari M, eds. Lecture Notes in Control and Information Sciences. Switzerland: Springer International Publishing, 2017. [CrossRef] [Google Scholar]
 Gupta P, Kumar PR. The capacity of wireless networks. IEEE Trans Inform Theor 2000; 46: 388–404. [Article] [Google Scholar]
© The Author(s) 2023. Published by Science Press and EDP Sciences
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
All Figures
Figure 1 Network system architecture. (a) “Boomerang” like relationship between channel capacity and networked sensing capacity. (b) Relationship between the gain of network system capacity and channel capacity. (c) Relationship between the gain of network system capacity and networked sensing capacity. 

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