# Protocol for Energy-Efficiency in Networked Control Systems Based on WSN
## 6 Example of Application of the Energy-Efficiency Protocol
### Results
#### 6.3.2 Energy Saving Protoc--Second Method: \(L\) Variable  
In the first method, the value of \(L\) is fixed and the resulting performance, for instance the robustness against exogenous inputs, is always worse than the one provided by the classic case. In the second method, the value (per node) of \(L\in\{L_{1},\ L_{2}\}\) depends exclusively on the battery energy levels, which are time-varying. In the network presented in Figure 4, the nodes that change their maximum number of transmissions \(L\) over time are \(U\in\{1,2,\ldots,16\}\), while nodes \(U\in\{17,18,\ldots,24\}\) always operate in \(L_{1}=\infty\) because they (nodes connected to the sensors and actuators) are assumed to have unlimited energy. The proposed method starts with a full-reliable communication (\(L=\infty\)), providing the best performance in terms of \(\mathcal{H}_{\infty}\) norm. The vector \(L=\{L_{1}=\infty,L_{2}=5\}\) starts with a norm value of \(\mathcal{H}_{\infty}=63.2076\) and an global energy consumption for the entire network equal to 16.2177 J and gradually declines to the value of \(L=5\) for all nodes with a performance of \(\mathcal{H}_{\infty}=88.2573\) and a energy consumption 11.5137 J/h. Table 6 exhibits the probabilities of successful transmission between the source and sink (\(\text{Pr}(\bar{y}_{1})\), \(\text{Pr}(\bar{y}_{2})\), \(\text{Pr}(\bar{u}_{1})\), and \(\text{Pr}(\bar{u}_{2})\)) associated with each measurement or control input signal (\((\bar{y}_{1})\), \((\bar{y}_{2})\), \((\bar{u}_{1})\), and \((\bar{u}_{2})\)) for each new setting of the network. More precisely, Table 6 shows the sequence where the nodes change from \(L_{1}=\infty\) to \(L_{2}=5\), such that column \(U\) indicates the node that is changing in the current time interval and column \(I_{L_{m}}\) indicates the order of each setting in terms of \(\mathbb{P}(\xi)\). By imposing a one-way exchange sequence (from \(L_{1}=\infty\) to \(L_{2}=5\) and never the opposite), the network under investigation can operate only in \(\xi=17\) distinct settings, such that the change in the value of \(L\) is made via an alarm signal sent to the controller. Observe that the exchange sequence by node is supposedly done according to the battery consumption of each node (The selection of the \begin{table}
\begin{tabular}{c c c c c c} \hline \(I_{L_{m}}\) & \(U\) & \(\text{Pr}(\bar{y}_{1})\) & \(\text{Pr}(\bar{y}_{2})\) & \(\text{Pr}(\bar{u}_{1})\) & \(\text{Pr}(\bar{u}_{2})\) \\ \hline
1 & – & 1 & 1 & 1 & 1 \\
2 & 13 & 1.0000 & 1.0000 & 1.0000 & 0.9222 \\
3 & 9 & 1.0000 & 1.0000 & 0.9497 & 0.9222 \\
4 & 14 & 1.0000 & 1.0000 & 0.9497 & 0.8505 \\
5 & 10 & 1.0000 & 1.0000 & 0.9019 & 0.8505 \\
6 & 15 & 1.0000 & 1.0000 & 0.9019 & 0.7844 \\
7 & 11 & 1.0000 & 1.0000 & 0.8565 & 0.7844 \\
8 & 1 & 0.9688 & 1.0000 & 0.8565 & 0.7844 \\
9 & 6 & 0.9688 & 0.9688 & 0.8565 & 0.7844 \\
10 & 16 & 0.9688 & 0.9688 & 0.8565 & 0.7234 \\
11 & 12 & 0.9688 & 0.9688 & 0.8134 & 0.7234 \\
12 & 2 & 0.9385 & 0.9688 & 0.8134 & 0.7234 \\
13 & 3 & 0.9091 & 0.9688 & 0.8134 & 0.7234 \\
14 & 4 & 0.8807 & 0.9688 & 0.8134 & 0.7234 \\
15 & 7 & 0.8807 & 0.9385 & 0.8134 & 0.7234 \\
16 & 5 & 0.8532 & 0.9385 & 0.8134 & 0.7234 \\
17 & 8 & 0.8532 & 0.9091 & 0.8134 & 0.7234 \\ \hline \end{tabular}
\end{table} 
Table 6: Probabilities of successful transmission for each new setting of the network such that the nodes \(U\) exchange \(L\in\{\infty,5\}\) following the sequence \(I_{Lm}\) (Energy for node approach). switching sequence is merely illustrative. To determine an optimized switching sequence for a real network is outside the scope of investigation of the present work). Figure 6a shows the \(\mathcal{H}_{\infty}\) guaranteed costs of the closed-loop system computed for each transition probability matrix \(\mathbb{P}(\xi)\) associated to a new network setting \(\xi=[1,\ 17]\) respecting the exchange order indicated by \(I_{L_{m}}\) in Table 6. Note that the guaranteed costs do not increase monotonically.