# Protocol for Energy-Efficiency in Networked Control Systems Based on WSN
## Preliminaries
### Client-Server Control Architecture
#### 4.2.1 Success Probability and Chain Mode Accessibility  
The total number of operation modes \(\sigma\) of system (1) also corresponds to the order of the Markov chain, such that \(\mathbb{P}\in\mathbb{R}^{\sigma\times\sigma}\) models the probability of transition between the modes. In this paper two sets of particular events are defined: _i_) The success and failure of transmission of elements of \(y(k)\), which is modeled by \(\mathbb{P}^{y}\in\mathbb{R}^{\sigma_{y}\times\sigma_{y}}\) whose set of operation modes is given by \(\ell_{y}\in\mathbb{K}^{y}=[1,\ldots,\sigma_{y}]\in\mathbb{K}\); and _ii_) the success and failure of transmission of the elements of \(u(k)\), which is modeled by \(\mathbb{P}^{u}\in\mathbb{R}^{\sigma_{u}\times\sigma_{u}}\) such that each operation mode is represented by \(\ell_{u}\in\mathbb{K}^{u}=[1,\ldots,\sigma_{u}]\in\mathbb{K}\). The general process that comprises the entire system is associated with the Kronecker product of the two transition probability matrices, that is, \(\mathbb{P}^{u}\otimes\mathbb{P}^{y}=\mathbb{P}\in\mathbb{R}^{\sigma\times\sigma}\), that governs the jumps between the modes of the global Markov process. Each element in the transition matrix \(\mathbb{P}=[p_{ij}]\) represents the probability of transition of the current state \(i\) to the state \(j\). This transition may be used as a model for the network communication scheme. In the present work, a Multi-Hop network topology that implements the Hop-by-Hop transmission scheme is used. The stochastic process that determines the success or failure of transmission uses the generalized Bernoulli distribution model, which means \(p_{ij}=p_{j}\), so the probability does not depend on current state (index \(i\)). For illustrative purposes, consider that system (1) has a vector of output measurements with 3 components (\(y(k)\in\mathds{R}^{3}\)) but only two sensors are connected to the plant (\(n_{y}=2\)), such that the first two components of \(y(k)\) are simultaneously provided by the first sensor and the last component is given by the second sensor (\(\bar{y}_{1}=\{y_{1}(k),y_{2}(k)\}\) e \(\bar{y}_{2}=\{y_{3}(k)\}\)). Neglecting the packet loss of the control signal, considering only the transmission of the measurement output, one has \(\sigma_{y}=2^{n_{g}}=2^{2}=4\) operation modes and the set of the operation modes associated with \(y(k)\) is \(\ell_{y}\in\mathds{K}^{y}=[1,\ldots,\sigma_{y}]=[1,2,3,4]\). Knowing that the process of transition among the modes \(\ell_{y}\) follows a generalized Bernoulli distribution (uncorrelated events), such that it does not depend on the initial state but only on the final state (\(p_{ij}=p_{j}\)), from the formula proposed in Lemma 4 one can compute the probability of occurrence of each event (success or failure in the transmission of \(\bar{y}_{i}\), \(i=1,2\)):  
\[P_{\bar{y}_{i}}=\Pr(\bar{y}_{i})=[1-(1-p(\bar{y}_{i}))^{L(\bar{y}_{i})}]^{N( \bar{y}_{i})}. \tag{19}\]  
The possible events associated to the success or failure in the transmission of the measurement output is shown in Table 1.  
Additionally, suppose that system (1) has one vector of control inputs with 4 components (\(u(k)\in\mathds{R}^{4}\)) and only two actuators (\(n_{a}=2\)), such that the first two components of \(u(k)\) are implemented in the same actuator (\(\bar{u}_{1}=\{u_{1}(k),u_{2}(k)\}\)) and the other two components are implemented in the second actuator (\(\bar{u}_{2}=\{u_{3}(k),u_{4}(k)\}\)). The number of operation modes associated to the transmission of the control signal is \(\sigma_{u}=2^{n_{a}}=2^{2}=4\), such that \(\ell_{u}\in\mathds{K}^{u}=[1,\ldots,\sigma_{u}]=[1,2,3,4]\).