# Protocol for Energy-Efficiency in Networked Control Systems Based on WSN
## Preliminaries
### Markovian Model
#### 3.2.2 Full Order Dynamic Output-Feedback Controller for MJLS  
A full-order dynamic output-feedback controller for system (1) with the best (optimal) \(\mathcal{H}_{\infty}\) performance can be obtained with the design method given in [42] if all Markov operation modes (\(i\in\mathbf{K}\)) are accessible. Since this assumption is unrealistic in many applications, leading to only suboptimal solutions when employing the technique of [42], in this paper a different output-feedback approach is pursued, based on the separated design of a state-observer filter and a state-feedback controller. Although in the WSN-based NCS literature there are plenty of works dealing with the design and implementation of filters and controllers, the proposed approach is described in the MJLS framework associated with a Markov chain with Bernoulli distribution. In this particular case, even when the operation modes are not available, both the obtained filter [20] and controller [21] provide optimal solutions individually. Concerning the design of the state-observer, consider a Luenberger filter \(\mathcal{F}\) (internal model-based filter), given by  
\[\mathcal{F}: \begin{cases}x_{f}(k+1)=A_{i}x_{f}(k)\;+G_{f_{i}}\left(y(k)-C_{yi} x_{f}(k)\right),\\ z_{f}(k)=C_{zi}x_{f}(k)+K_{f_{i}}\left(y(k)-C_{yi}x_{f}(k)\right),\end{cases}  \tag{4}\]  
where \(x_{f}(k)\in\mathbb{R}^{n_{x}}\), \(z_{f}(k)\in\mathbb{R}^{n_{z}}\) and \(y(k)\in\mathbb{R}^{n_{y}}\), are, respectively, the filter state vector, the estimated output and the measured signal output. Additionally, the filter matrices are given by \(A_{f_{i}}=\{A_{i}-G_{f_{i}}C_{yi}\}\), \(B_{f}=G_{f_{i}}\), \(C_{f_{i}}=\{C_{zi}-K_{f_{i}}E_{yi}\}\) and \(D_{fi}=K_{f}\), where \(G_{f_{i}}\) and \(K_{f_{i}}\) are the matrices to be designed. The estimation error is defined as \(e(k)=z(k)-z_{f}(k)\), and its dynamics is represented by the set of equations given in (5), connecting the filter (4) to the MJLS (1) with \(u(k)=0\)  
\[\mathcal{G}_{o}: \begin{cases}\tilde{x}(k+1)=\left(A_{i}-G_{f_{i}}C_{yi}\right) \tilde{x}(k)+\left(E_{i}-G_{f_{i}}E_{yi}\right)w(k),\\ e(k)=\left(C_{zi}-K_{f_{i}}C_{yi}\right)\tilde{x}(k)+\left(E_{zi}-K_{f_{i}}E_{ yi}\right)w(k),\end{cases}  \tag{5}\]  
where \(\tilde{x}(k)=x(k)-x_{f}(k)\). As mentioned before, the assumption of complete Markov mode availability is not realistic. Thus, in some applications, it may be more adequate to consider cluster availability. Therefore, consider the set \(\mathbb{L}=1,2,\ldots,\sigma_{c}\) with \(\sigma_{c}\leq\sigma\) and define the set of Markov chain states \(\mathbf{K}\) as the union of \(\sigma_{c}\) disjoint sets, or clusters, that is, \(\mathbf{K}=\bigcup_{l\in\mathbb{L}}\mathbb{U}_{l}\) such that \(\mathbb{U}_{i}\cap\mathbb{U}_{j}=\emptyset\), \(\forall i=j\in L\). The matrices of filter (4) designed in the case \(z(k)=\tilde{x}(k)\) (state-observer filter) can be synthesized by solving a set of LMI conditions presented in [21] and reproduced in the sequence for convenience. **Lemma 2**. : _There exists a filter as in Equation (4), such that \(\|\mathcal{G}_{\theta}\|_{\infty}^{2}<\gamma\), under the assumption of \(p_{ij}\ =\ p_{j}\ \forall i,j\in\mathbb{K}\), if and only if there exist symmetric matrices \(H_{i}\), \(X\), and matrices \(F_{I}\), \(K_{I}\) with compatible dimensions fulfilling the LMIs_  
\[ \begin{bmatrix}H_{i}&\bullet&\bullet&\bullet\\ 0&\gamma I&\bullet&\bullet\\ XA_{i}+F_{I}C_{yi}&XE_{i}+F_{I}E_{yi}&X&\bullet\\ C_{zi}-K_{I}C_{yi}&E_{zi}-K_{I}E_{yi}&0&I\end{bmatrix} >0,\forall i\in\mathbf{K },\forall l\in\mathbb{L}, \tag{6}\]  
_and_  
\[H_{p}-X<0, \tag{7}\]  
_where \(H_{p}=\sum_{j=1}^{\sigma}p_{j}H_{j}\). If a feasible solution is achieved, the filter gains are given by_  
\[G_{f_{I}}=-X^{-1}F_{I}\quad\text{and}\quad K_{f_{I}}=K_{I}\ \ \forall l\in\mathbb{L}. \tag{8}\]  
On the other hand, consider the general problem of designing a state-feedback control law  
\[u(k)=K_{i}x(k), \tag{9}\]  
for system (1) where \(K_{i}\) are state-feedback gains to be designed.