# Protocol for Energy-Efficiency in Networked Control Systems Based on WSN
## 5 Energy Saving Protocol
### Energy Saving Scheme by Energy for Node Approach  
In WSN, the energy level of the autonomous units changes according to time (usually nonlinearly) and it can be estimated or measured by the battery voltage output [15]. When the battery life span is at the end, it is necessary to switch it to guarantee a good communication link, routes and control signal, otherwise the system may become unstable. Clearly, the best performance is attained by the classical digital controller, however, the energy consumption is maximum. Using the Trade-off approach (Lemma 5), the obtained \(\mathcal{H}_{\infty}\) performance is worse, but the energy consumption is improved.  
Considering that initially all the network nodes have a good energy level, it would be convenient for the network to operate in a full-reliable working mode (no packet loss, \(L=\infty\)), since this situation does not require energy savings. When a given node is near the end of its life span, it is appropriate to change its operation to energy savings working mode, extending its life span until a battery replacement. In this case, it is sufficient to limit the maximum number of retransmission (\(L=\epsilon\)). However, besides the two working modes described above, it is possible to segment the battery life span in several working modes: starting from the full-reliable (\(L=\infty\)) and going through a set of saving work modes, such that \(L\in\{L_{1},\ldots,L_{x},\ldots,L_{m}\}\) with \(L_{1}=\infty\) and \(L_{x}\in\mathbb{N}<\infty\), \(\kappa\in\{1,2,\ldots,m\}\), where \(m\) represents the total number of the network working modes.  
Since the switch of the working modes per node is associated with an external event (battery level), the nodes can work in different modes (\(L_{x}\)), simultaneously. Each possible combination of \(L_{x}\) in the set of nodes \(\Psi\) corresponds to a different transition probability matrix \(\mathbb{P}\). The total number \(\xi\) of transition probability matrices is equal to the possible settings of the network, that is associated to the number of the working modes that each node has. For example, consider a network (illustrated in Figure 2) with 4 nodes (\(U=\{1,2,3,4\}\)) with a single route \(\mathcal{R}=1\), transmitting a single signal packet \(\mathcal{S}=\{\bar{u}_{1}(k)\}\) such that the source and sink nodes (\(\mathcal{N}_{1}^{[1,a_{1}]}\), \(\mathcal{N}_{4}^{[1,a_{1}]}\)) have unlimited energy, implying that the corresponding number of transmissions does not depend on the energy, i.e., \(L_{1}=L_{4}=\infty\).  
The nodes whose maximum number of transmissions \(L\) is limited correspond to \(\mathcal{N}_{2}^{[1,a_{1}]}\) and \(\mathcal{N}_{3}^{[1,a_{1}]}\), with three work modes (\(m=3\)) given by \(L_{1}=\infty\), \(L_{2}=10\) and \(L_{3}=2\), generating \(\xi=9\) network distinct settings:  
\[\xi=1: \quad\Psi=\{\mathcal{N}_{1}^{[1,a_{1}]}(L=\infty),\mathcal{N}_{2}^ {[1,a_{1}]}(L=L_{1}),\mathcal{N}_{3}^{[1,a_{1}]}(L=L_{1}),\mathcal{N}_{4}^{[1,a _{1}]}(L=\infty)\}\] \[\xi=2: \quad\Psi=\{\mathcal{N}_{1}^{[1,a_{1}]}(L=\infty),\mathcal{N}_{2}^ {[1,a_{1}]}(L=L_{1}),\mathcal{N}_{3}^{[1,a_{1}]}(L=L_{2}),\mathcal{N}_{4}^{[1,a _{1}]}(L=\infty)\}\] \[\xi=3: \quad\Psi=\{\mathcal{N}_{1}^{[1,a_{1}]}(L=\infty),\mathcal{N}_{2}^ {[1,a_{1}]}(L=L_{1}),\mathcal{N}_{3}^{[1,a_{1}]}(L=L_{3}),\mathcal{N}_{4}^{[1,a _{1}]}(L=\infty)\}\] \[\xi=4: \quad\Psi=\{\mathcal{N}_{1}^{[1,a_{1}]}(L=\infty),\mathcal{N}_{2}^ {[1,a_{1}]}(L=L_{2}),\mathcal{N}_{3}^{[1,a_{1}]}(L=L_{1}),\mathcal{N}_{4}^{[1,a _{1}]}(L=\infty)\}\] \[\vdots \vdots \vdots\] \[\xi=9: \quad\Psi=\{\mathcal{N}_{1}^{[1,a_{1}]}(L=\infty),\mathcal{N}_{2}^ {[1,a_{1}]}(L=L_{3}),\mathcal{N}_{3}^{[1,a_{1}]}(L=L_{3}),\mathcal{N}_{4}^{[1,a _{1}]}(L=\infty)\}\]  
Since each network setting is associated with a distinct transition probability matrix (\(\mathbb{P}(\xi)\)) computed according to the procedures explained in Section 4.2.1, in this particular case, \(\xi=9\) distinct sets of partially mode-dependent stabilizing controllers must be designed.