# Protocol for Energy-Efficiency in Networked Control Systems Based on WSN
## 5 Energy Saving Protocol  
Three distinct heuristics of the energy saving protocol are investigated in this paper.  
* **Trade-off approach:** The first technique consists in the use of a semi-reliable network with a single and finite value for the maximum number of transmissions allowed (\(L<\infty\)) in the global network. Consequently, due to the limitation of \(L\), the probability of successful communication for each packet is less than one. Despite this disadvantage, the limitation of \(L\) provides energy-savings in the use of network resources. This protocol was proposed in [6; 8; 13].
* **Energy for node approach:** The second method is based on image transmission in WSN [15], where the probability of successful communication per packet between nodes is directly associated with the energy level of the transmission for a particular node. The energy level of the nodes starts at the maximum, which means that, initially, the probability of successful transmission of each node is equal to one (without packet loss, corresponding to the classical control theory, \(L=\infty\)). After, according to the use of each node, its energy consumption increases, the value of \(L\) becomes finite and it is gradually reduced, implying a reduction in the successful transmission probability.
* **Mixed approach:** The third method is a mix between the previous approaches, that is, the probability of successful transmissions depends on the energy consumption of each network node, but the maximum number of transmissions allowed starts at a finite value. The idea is to obtain a performance as close as possible to classical control but maintaining desirable energy efficiency characteristics (\(L<\infty\)). As a consequence, it is possible to achieve a better trade-off between performance and the use of network resources.  
In order to perform a proper comparison between the classical control \(P_{S}=1\) and the energy saving protocol proposed in this paper, some performance measurements variables are proposed:  
* \(\mathrm{Y}_{H}=\frac{\|\mathcal{G}_{d}\|_{\infty}(L<\infty)}{\|\mathcal{G}_{d}\|_ {\infty}(L\to\infty)}\) denotes the rate between the norm obtained with packet loss and the one using the classical approach (without loss).
* \(\mathrm{Y}_{M}=\frac{\mathcal{E}[M|L<\infty]}{\mathcal{E}[M|L\to\infty]}\) represents the ratio between the mathematical expectation of the global number of transmissions considering a limited value for \(L\) and the situation without any boundaries.
* \(\mathrm{Y}_{E}=\frac{\mathcal{E}(E_{T}(p,L<\infty,N))}{\mathcal{E}(E_{T}(p,L =\infty,N))}\) represents the ratio between the mathematical expectation of the global energy consumption using semi-reliable network and the energy consumption for full-reliable network (see (17)).
* \(\Omega_{H}=\mathrm{Y}_{H}-1\) is the normalized norm degradation arising from the use of a bounded \(L\).
* \(\Omega_{M}=1-\mathrm{Y}_{M}\) is the normalized decrease of the mathematical expectation of the global number of transmissions due to the limitation of \(L\).
* \(\Omega_{E}=1-\mathrm{Y}_{E}\) is the normalized decrease of the global energy consumption due to the limitation of \(L\).
* \(\Theta=\Omega_{M}-\Omega_{H}\) is the percentage difference between the normalized norm degradation and the normalized decrease of the mathematical expectation of the global number of transmissions due to the limitation of \(L\).  
The variable \(\Theta\) was conceived with the task of verifying if there is any advantage in limiting the maximum number of transmissions per packet. Thus, if \(\Theta>0\), the decrease in the average number of global transmissions is proportionally greater than the increase in the \(\mathcal{H}_{\infty}\) norm, whereas \(\Theta<0\) represents the opposite. Therefore, when \(\Theta>0\), one assumes that the limitation of \(L\) is advantageous (positive trade-off), because it this implies a greater saving in energy resources than performance loss.