Network system capacity: Towards integrating sensing, communication and control

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 information-rich 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 di ﬃ cult to reconcile due to di ﬀ erent mathematical basis and conceptual paradigms from their aspects. It is challenging to e ﬀ ectively 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

inevitable uncertainties such as ambient noise, interference and multi-path fading [5,6].Thus, the new concept called networked sensing capacity C s = 1/σ 2 v 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 second-moment stability in one-step from the perspective of feedback control with the considered uncertainties, i.e., for all k ∈ 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 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 ∂C N ∂Cc and ∂C N ∂Cs 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.

Figure 1
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.