# 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
Knowing that the process of transition among the modes \(\ell_{u}\) also follows a generalized Bernoulli distribution, one can compute the probability of occurrence of each event \(\ell_{u}=i\), \(i=1,\ldots,\sigma_{u}=1,\ldots,4\), by:  
\[P_{\bar{u}_{i}}=\Pr(\bar{u}_{i})=[1-(1-p(\bar{u}_{i}))^{L(\bar{u}_{i})}]^{N( \bar{u}_{i})}. \tag{20}\]  
The possible events associated with success or failure of transmission of the control input are shown in Table 2.  
Knowing that the transition probability matrix of the general process \(\mathbb{P}\) is given by the Kronecker product of \(\mathbb{P}^{y}\) and \(\mathbb{P}^{u}\), the elements of \(\mathbb{P}=[p_{ij}]=[p_{j}]\) can be calculated by  
\[p_{j}=\Pr(j)=\Pr(\ell_{y})\times\Pr(\ell_{u}).\]  
Particularly, for the mode \(j=10\), one has  
\[\Pr(j=10)=\Pr(\ell_{y}=3)\times\Pr(\ell_{u}=2)=(1-P_{\tilde{y}_{1}})P_{\tilde{y }_{2}}\times P_{\tilde{u}_{1}}(1-P_{\tilde{u}_{2}}).\]  
Notice that the operation mode is direct associated with both sets of events (\(\ell_{y}\in\mathbf{K}^{y}\) and \(\ell_{u}\in\mathbf{K}^{u}\)). For this reason, if any of the operation modes associated separately with \(u\) or \(y\) is not accessible, it is not possible to know which packet was correctly transmitted and the only feasible option is to employ mode-independent controllers or filters. A recent approach presented in [45; 46; 47; 48] proposes a synthesis procedure where the filters/controllers no longer depend on the access to the actual operation mode \(i\) of the Markov chain, instead, they depend on the estimated operation mode \(l\) of the Markov chain. This kind of approach seems appealing, however, the estimation algorithm in the implementation process increases the complexity of the solution proposed in this paper. Another solution, more widely exploited, is the use of clusters, meaning that the Markov chain modes are organized in subsets. In the case under investigation, the information about the events \(\ell_{u}\in\mathbf{K}^{u}\) is considered not accessible. On the other hand, the events \(\ell_{y}\in\mathbf{K}^{y}\) are completely observable, that is, it is always possible to know if a set of measurements (\(\tilde{y}_{i}\), \(i=1,\ldots,\sigma_{y}\)) was correctly transmitted. For this reason, this paper proposes the use of an approach based on _clusters_, that are mutually exclusive groups whose union generates the set of states \(\mathbf{K}\). In this sense, the operation modes of system (1) are grouped in \(\sigma_{y}\) disjoint subsets, such that \(\ell\in[1,\ldots,\sigma_{y}]\). For the particular case of the illustrative example, instead of designing 16 controllers (associated with \(\sigma=16\) distinct operation modes), only 4 controllers must be designed (associated with \(\sigma_{y}=4\) distinct clusters), constituting a partially mode-dependent control technique. It is important to emphasize that dynamic output-feedback control synthesis conditions that are necessary and sufficient for the case of incomplete availability of the Markov states do not exist, meaning that only sub-optimal solutions are available in the literature for this scenario.