AbstractWe consider a stochastic system where the communication between the controller and the actuatoris triggered by a threshold-based rule. The communication is performed across an unreliable link thatstochastically erases transmitted packets. To decrease the communication burden, and as a partial pro-tection against dropped packets, the controller sends a sequence of control commands to the actuator ineach packet. These commands are stored in a buffer and applied sequentially until the next control packetarrives. In this context, we study dead-beat control laws and compute the expected linear-quadratic loss ofthe closed-loop system for any given event-threshold. Furthermore, we provide analytical expressions thatquantify the trade-off between the communication cost and the control performance of event-triggeredcontrol systems. Numerical examples demonstrate the effectiveness of the proposed framework.Index TermsEvent-triggering algorithms; Linear systems; Communication networks; Packet loss; Networkedcontrol
The analysis becomes even more involved when the network is unreliable. Blind and Allg " o wer @cite_4 extended the work of @cite_27 to the case where transmissions from the sensor to the controller take place over an unreliable link. They analytically derived the control cost and the expected inter-event times for different packet loss rates. Rabi and Johansson @cite_36 designed the optimal impulse control and the level triggering mechanism under packet losses with multiple loops sharing a common network. From a different perspective, @cite_6 analyzed the stability of an event-triggered implementation of a controller in the presence of packet losses and limited processing resources. However, in all these works, the unreliable channel appears between sensor and controller, while the controller-actuator communication is lossless.
@cite_36 @cite_27 @cite_4 @cite_6
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In this paper, we consider a robust network control problem. We consider linear unstable and uncertain discrete time plants with a network between the sensor and controller and the controller and plant. We investigate the effect of data drop out in the form of packet losses. Four distinct control schemes are explored and sufficient conditions to ensure almost sure stability of the closed loop system are derived for each of them in terms of minimum packet arrival rate and the maximum uncertainty. I. INTRODUCTION In the past decade, networked control systems (NCS) have gained much attention from both the control com- munity and the network and communication community. When compared with classical feedback control system, networked control systems have several advantages. For example, they can reduce the system wiring, make the system easy to operate and maintain and later diagnose in case of malfunctioning, and increase system agility (20). Although NCS have advantages, inserting a network in between the plant and the controller introduces many problems as well. For instance, zero-delayed sensing and actuation, perfect information and synchronization are no longer guaranteed in the new system architecture as only finite bandwidth is available and data packet drops and delays may occur due to network traffic conditions. These must be revisited and analyzed before networked control systems become prevalent. Recently, many researchers have spent effort on these issues and some significant results were obtained and many are in progress. Many of the aforementioned issues are studied separately. Tatikonda (19) and Sahai (13) have presented some interesting results in the area of control under communication constraints. Specifically, Tatikonda gave a necessary and sufficient condition on the channel data rate such that a noiseless LTI system in the closed loop is asymptotically stable. He also gave rate results for stabilizing a noisy LTI system over a digital channel. Sahai proposed the notion of anytime capacity to deal with real time estimation and control for a networked control system. In our paper (17), the authors have considered various rate issues under finite bandwidth, packet drops and finite controls. An optimal bit allocation scheme is given in (16) under the networked setting. The effect of pacekt drops on state estimation was studied by Sinopoli, et. al. in (3). It has further been investigated by many researchers including the present authors in (15) and (6). Distributed processing through ad hoc and sensor networks is having a major impact on scale and applications of computing. The creation of new cyber-physical services based on wireless sensor devices relies heavily on how well communication protocols can be adapted and optimized to meet quality constraints under limited energy resources. The IEEE 802.15.4 medium access control protocol for wireless sensor networks can support energy efficient, reliable, and timely packet transmission by a parallel and distributed tuning of the medium access control parameters. Such a tuning is difficult, because simple and accurate models of the influence of these parameters on the probability of successful packet transmission, packet delay, and energy consumption are not available. Moreover, it is not clear how to adapt the parameters to the changes of the network and traffic regimes by algorithms that can run on resource-constrained devices. In this paper, a Markov chain is proposed to model these relations by simple expressions without giving up the accuracy. In contrast to previous work, the presence of limited number of retransmissions, acknowledgments, unsaturated traffic, packet size, and packet copying delay due to hardware limitations is accounted for. The model is then used to derive a distributed adaptive algorithm for minimizing the power consumption while guaranteeing a given successful packet reception probability and delay constraints in the packet transmission. The algorithm does not require any modification of the IEEE 802.15.4 medium access control and can be easily implemented on network devices. The algorithm has been experimentally implemented and evaluated on a testbed with off-the-shelf wireless sensor devices. Experimental results show that the analysis is accurate, that the proposed algorithm satisfies reliability and delay constraints, and that the approach reduces the energy consumption of the network under both stationary and transient conditions. Specifically, even if the number of devices and traffic configuration change sharply, the proposed parallel and distributed algorithm allows the system to operate close to its optimal state by estimating the busy channel and channel access probabilities. Furthermore, results indicate that the protocol reacts promptly to errors in the estimation of the number of devices and in the traffic load that can appear due to device mobility. It is also shown that the effect of imperfect channel and carrier sensing on system performance heavily depends on the traffic load and limited range of the protocol parameters. Motivated by limited computational resources in sensor nodes, the impact of complexity constraints on the communication efficiency of sensor networks is studied. A single-parameter characterization of processing limitation of nodes in sensor networks is invoked. Specifically, the relaying nodes are assumed to "donate" only a small part of their total processor time to relay other nodes information. The amount of donated processor time is modelled by the node's ability to decode a channel code reliably at given rate R. Focusing on a four node network, with two relays, prior work for a complexity constrained single relay network is built upon. In the proposed coding scheme, the transmitter sends a broadcast code such that the relays decode only the "coarse" information, and assist the receiver in removing ambiguity only in that information. Via numerical examples, the impact of different power constraints in the system, ranging from per node power bound to network wide power constraint is explored. As the complexity bound R increases, the proposed scheme becomes identical to the recently proposed achievable rate by Gupta & Kumar (2003). Both discrete memoryless and Gaussian channels are considered. Accurate analytical expressions of delay and packet reception probabilities, and energy consumption of duty-cycled wireless sensor networks with random medium access control (MAC) are instrumental for the efficient design and optimization of these resource-constrained networks. Given a clustered network topology with unslotted IEEE 802.15.4 and preamble sampling MAC, a novel approach to the modeling of the delay, reliability, and energy consumption is proposed. The challenging part in such a modeling is the random MAC and sleep policy of the receivers, which prevents to establish the exact time of data packet transmission. The analysis gives expressions as function of sleep time, listening time, traffic rate and MAC parameters. The analytical results are then used to optimize the duty cycle of the nodes and MAC protocol parameters. The approach provides a significant reduction of the energy consumption compared to existing solutions in the literature. Monte Carlo simulations by ns2 assess the validity of the analysis.