# Protocol for Energy-Efficiency in Networked Control Systems Based on WSN
## 1 Introduction
The first one, called _Trade-off_ in the context of dynamic systems, is an extension of the results originally proposed in [6] (which treat the design of filters in communication networks) to cope with control in semi-reliable networks. The solution proposed in [6] consists in choosing a single and finite number of retransmissions for all the nodes of the network, providing resources saving while, at the same time, assuring an \(\mathcal{H}_{\infty}\) performance suitable to the application. By limiting the maximum number of retransmissions, the expected value of global transmissions is reduced when compared with a full-reliable communication. As a consequence, it is possible to obtain a reduction in the network congestion, the average delay time of packet transmission, among other meaningful parameters. Differently from [6], that handles problems associated with the transmission of a single packet in a single path, this paper generalizes the approach to deal with the transmission of an arbitrary number of packets in several paths. The second heuristic is an adaptation of the methodology presented in [15] (that deals with another research area: image transmission in WSN) to cope with control of dynamical systems through communication networks. In [15], the authors proposed to modify the network reliability (the maximum number of retransmissions) by considering the status (particularly the battery charge) of the intermediate nodes in the path over a Hop-by-Hop sensor network. In the NCS context, subject of investigation of this paper, this method is adapted to start from a full-reliable communication (unlimited maximum number of retransmissions) and, gradually, to achieve a semi-reliable network by limiting the number of retransmissions of each intermediate node. Finally, a third heuristic combining some features of the previous two is an original proposition of this paper. Differently from the second, in this new heuristic the network always operates in a semi-reliable communication working mode (limited number of retransmissions). Nevertheless, distinct from the first, the node energy status is taken into account, such that, when the nodes have high energy level (requiring no energy savings), the network tends to operate in a working mode close to the full-reliable communication (limited but large number of retransmissions allowed). Then, gradually, the nodes switch to a saving resources working mode, such that all the network nodes converge to a single and finite number of retransmissions by packet (status similar to the first heuristic). Observe that, regardless the adopted heuristic, by using the proposed energy saving protocol the number of retransmissions per packet in the Hop-by-Hop scheme is always limited and it is possible to achieve substantially better energy consumption indexes in the communication link than using full-reliable network (compatible with classical control theory). To quantify the energy savings of the proposed protocol, it is used a stochastic model for the multi-hop network, which provides average expected energy consumption for the entire network. Another important parameter that is investigated is the energy consumed by each individual node, because it allows to determine the node with lowest path lifetime and also identify the network nodes that are more susceptible to failure due to energy consumption issues. The proposed resource saving protocol is formulated in a generalist structure, that is, it can be applied to any NCS problem with multiple information sources, such as filtering, state-feedback, static output-feedback or dynamic output-feedback control. Nevertheless, to show the viability of employment and main features of the protocol, the problem of output-feedback control via WSN is particularly investigated in this paper. To deal with the packet loss in the communication channel, the controlled plant is modeled as a discrete-time Markov Jump Linear System (MJLS), whose jumps between the operation modes can represent the possible transmission failures [16; 17].