Analysis of delay-throughput-reliability tradeoff in a multihop wireless channel for the control of unstable systems A. Chiuso, N. Laurenti, L. Schenato, A. Zanella Abstract—In recent years, a growing interest has been devoted to the performance analysis of control systems in presence of realistic feedback channels, which are constrained in terms of transmission rate, delay and reliability. In this paper we leverage on some recent findings both in the control and information the- ory domains to deeply analyze the effect of communication non- idealities on the control and performance of unstable stochastic scalar systems. More specifically, we consider a multihop wireless feedback channel for the control of an unstable system and investigate how the delay-throughput tradeoff impacts on the controllability of the system. Results show that control channels with multiple short hops, hence having longer delay but higher throughput, are preferable when the direct link has a low SNR or a high bandwidth is avialable. Anyway, even in a very simple scenario, it is possible to identify an optimal number of hops that allows to stabilize the largest class of plants. I. I NTRODUCTION The progress made by wireless communication technologies in terms of transmit rate and reliability are promoting their adoption in scenarios that were traditionally prerogative of wired transmission technologies, such as process control, automotive, and factory automation [1], [2], [3]. Moreover, wireless feedback channels are implicitly used in environ- mental monitoring and control applications based on wireless sensor networks, where data collected by remote sensors are reported to the control station by means of wireless multihop channels. The recent years have then witnessed a growing interest on the performance analysis of control systems with wireless feedback channels [4], [5]. The problem is of particular interest because, traditionally, control theory and communication theory have been developed mostly independently. In particular, the primary objective of control theory was to develop tools to stabilize unstable plants and to optimize some performance metrics in closed loop, under the assumption that the communication channel between sensors and controller and between the controller and the plant were ideal, i.e. without signal distortion, packet loss or delay. These assumptions, however, do not hold with feedback This work is supported by the PRIN grant nr. 20085FFJ2Z “New Algo- rithms and Applications of System Identification and Adaptive Control” by the Progetto di Ateneo CPDA090135/09 funded by the University of Padova, by the European Community’s Seventh Framework Programme [FP7/2007- 2013] under agreement nr. FP7-ICT-223866-FeedNetBack and under grant agreement nr. 257462 HYCON2 Network of Excellence All authors are with the Department of Information Engineer- ing, University of Padova, Via Gradenigo 6/b, 35131 Padova, Italy name.lastname@dei.unipd.it channels realized by means of practical communication tech- nologies, such as Internet connections or multihop wireless links. The analysis of control systems wherein the control loops are closed through a real-time network and feedback signals are exchanged in the form of data packets has given rise to a new branch of research, called Networked Control System (NCS). Recent results in this area have revealed the exis- tence of a strict connection between the performance of the controlled plant and the Shannon capacity of the feedback channel. However, this is not sufficient to completely charac- terize the communication channel from a control perspective [6], [7]. For instance, it has been proved that to stabilize an unstable plant through a control loop, the signal-to-noise ratio (SNR) of the feedback channel must be larger than some threshold depending on the unstable eigenvalues of the plant [8], [9], [10]. Another line of research addressed the problem of stabilizing an unstable plant in presence of a feedback channel prone to random packet losses [11], [12], [13], [14], or rate-limited [15], [16], [17]. A subsequent step has been made to include multiple channel limitations into the model, such as packet loss and quantization [18], [19], which however result in complex optimization problems. The impact of realistic feedback channels on the solution of an LQG (Linear-Quadratic-gaussian) control problem, which consists in finding the control signal of a Linear time-invariant (LTI) plant that minimizes a Quadratic cost function of the system state, when both the system state and the output signal are affected by Gaussian noise, has been recently investigated in [20], [21]. The analysis led to a stability condition that depends on the packet loss probability, signal to quantization noise ratio (SQNR), and end-to-end delay of feedback channel, which are however considered as independent properties of the feedback channel. These parameters, however, are clearly interrelated, as, for instance, reducing the erasure probability may require increasing the delay or reducing the information rate across the channel, which is to day, decreasing the maximum achievable SQNR. In this paper, we investigate the effect of such inter- dependencies on the performance of a closed loop LTI control system. More specifically, we consider a discrete-time unstable plant, which is controlled by means of a feedback signal generated by a suitable controller, on the basis of the output signal from the plant. The output signal is transmitted to the controller through a wireless communication channel, which can either be direct or multihop, exploiting in the last case