# Protocol for Energy-Efficiency in Networked Control Systems Based on WSN
## 6 Example of Application of the Energy-Efficiency Protocol
### Results
#### 6.3.3 Energy Saving Protocol--Third Method: Mixed Problem  
Although associated with the optimal performance in terms of the \(\mathcal{H}_{\infty}\) norm, the implementation of a full-reliable communication is unfavorable with respect the energy consumption. Depending on the network and the dynamic system to be controlled, to allow a small packet loss percentage may provide a significant decrease in the use of network resources and a proportionally smaller decrease in the performance. A similar problem was discussed in [6; 7; 8; 13], concerning filtering design through semi-reliable networks. The main goal in those works is to keep a positive trade-off between the norm degradation and the energy savings, denoted by \(\Theta\) and defined in Section 5.1 of this paper. To employ the mixed energy saving protocol approach, first it is necessary to determine the initial finite value of \(L\). By setting the maximum number of transmissions of all the network nodes to the same fixed and finite value of \(L\), it is possible to compute the resulting value of \(\Theta\) (based on the results of Section 6.3.1, see Figure 7). Since the mixed approach does not include the case \(L_{1}=\infty\) and aiming to use values associated with positive trade-off, two distinct finite values of \(L\) were chosen: \(L_{2}=8\) and \(L_{3}=5\), that is \(L\in\{L_{2},L_{3}\}=\{8,5\}\). The initial value was chosen \(L=8\) since it results in the maximum trade-off \(\Theta\), and the energy savings work mode was fixed in \(L=5\) to allow a better comparison with the second  
Figure 6: (a) \(\mathcal{H}_{\infty}\) norm of the closed-loop system (**b**) Expected number of global transmissions by route (\(\mathcal{E}(M)\)) (**c**) Global energy consumption of the network (\(E_{GE}\)) computed for each setting \(\xi\) of the network associated with the exchange sequence \(I_{Lm}\). approach, where the finite maximum number of transmissions was also 5. To compute the number of distinct network settings (in this case, the same as the previous method: \(\xi=17\), as shown in Table 7), first it is necessary to identify which nodes change the maximum number of transmissions \(L\) over time (\(U\in\{1,2,\ldots,16\}\)), and which nodes always operate in fixed \(L=8\) (\(U\in\{17,18,\ldots,24\}\)) because they are connected to the sensors and actuators with unlimited energy. Figure 8 shows the results of the third method in terms of the \(\mathcal{H}_{\infty}\) performance and the global energy consumption \(E_{GE}\), and also a comparison with the results obtained by the second approach. Observe, for instance, that for \(I_{L_{m}}=1\), all the nodes of the network are using \(L=8\), corresponding to the performance presented in Table 5 for column \(L=8\). Similarly to the second method, Figure 8a shows that the \(\mathcal{H}_{\infty}\) norm does not increase monotonically. Figure 8b illustrates the percentage augmentation of the \(\mathcal{H}_{\infty}\) norm for each network setting when compared with the value obtained by the second method:  
\[\mathcal{H}_{\infty}\%=\frac{\mathcal{H}_{\infty}^{A3}(I_{L_{m}})-\mathcal{H }_{\infty}^{A2}(I_{L_{m}})}{\mathcal{H}_{\infty}^{A2}(I_{L_{m}})}\times 100,\]