# Protocol for Energy-Efficiency in Networked Control Systems Based on WSN
## Preliminaries
### Client-Server Control Architecture  
A control design scheme where the connection among the elements is made through digital networks can be summarized as a Client-Server control architecture [3], as depicted in Figure 1. The client is the controller (Algorithm), the server is the plant (Dynamic System), and the control loop is denoted as a multi-hop network that, in the case investigated in this paper, is represented by a Wireless Sensor Network.  
The packets are transported via a topology chain using hops between intermediary terminals (router nodes) from the Source to the Sink.  
The classical control schemes are designed assuming that the communication between all the elements is made via an ideal channel, meaning that the packet loss rate must be null. However, in real network applications, more specifically in a WSN (presented in Figure 1), to perform a network communication without packet loss is a troublesome task. Thus, to reach such ideal conditions,  
Figure 1: Cliente-Server for NCS.  
some packet flow control mechanisms are implemented to get as close as possible a full-reliable communication. These mechanisms usually perform re-transmissions and use acknowledgment signals. As examples, one can mention the transport layer End-to-End and Hop-by-Hop schemes [44]. To accomplish such level of reliability in a WSN generally it is required a high energy cost, because each unit must retransmit the packet by an unknown number of times until the packet arrives to the proper destination and the acknowledge signal (ACK) is properly received by the sink.  
Operation Modes Associated with Output-Feedback Control of A Dynamical System through A Semi-Reliable Network  
Using the client-server scheme shown in Figure 1, it is proposed a standard formulation to devise a transition probability matrix of the Markov model, supposing the existence of multiple sensors and actuators. In this scheme, the source (in this case, the set of sensors) sends a set of measurements (elements composing vector \(y(k)\)) to the sink (in this case, the controller). Subsequently, the controller (now acting as the source) sends the control signal \(u(k)\), which is received by the plant (sink). In the classical control, the number of independent sensors that generate the signal \((y(k))\) and the number of actuators that receive the signal \((u(k))\) are irrelevant due to the hypotheses that they are accessible in every time instant \(k\). When considering the packet loss, one can model the measurement and control vectors as a composition of some subsets extracted from the original vectors if the signals composing those vectors are emitted by a set of different sources.  
The classical control case assumes that all elements composing the vectors of measurement and control signals are correctly transmitted. However, in a plant subject to communication failure, there are \(\sigma\) operation modes representing different combinations of the original vectors, associated with the success and failure in the packet transmission, that is, the total number of modes can be obtained using  
\[\sigma=\sigma_{y}\sigma_{u}=2^{n_{g}+n_{a}},\;\;n_{g}\leq n_{y},\;\;n_{a}\leq n _{u}, \tag{18}\]  
since \(\sigma_{y}=2^{n_{g}}\) and \(\sigma_{u}=2^{n_{a}}\), where \(n_{g}\) and \(n_{a}\) represent the number of independent sensors and independent actuators, such that those values can not surpass the size of vectors \(y(k)\) (\(n_{y}\)) and \(u(k)\) (\(n_{u}\)). Such limitation is motivated by the fact that a single sensor (or actuator) may send (or receive) a packet with different types of information. As an example, one can mention an encoder that provides both the velocity and the position at the same time. Furthermore \(n_{g}\) and \(n_{a}\) also denote the number of elements of the vectors of packets of measurement signals (\(\bar{g}=[g_{1},\ldots,g_{n_{g}}]\)) and control input signals (\(\bar{u}=[\bar{u}_{1},\ldots,\bar{u}_{n_{2}}]\)).