## 6 Virtual Sensor Design
While its dynamics is governed by:  
\[\mathbf{e}_{d,k+1} =\mathbf{d}_{k+1}-\mathbf{d}_{k}+\mathbf{d}_{k}-\mathbf{\hat{d}}_{k}-\mathbf{LE}_{k} \mathbf{e}_{d,k}-\mathbf{LWw}_{k} \tag{45}\] \[=\left(\mathbf{I}-\mathbf{LE}_{k}\right)\mathbf{e}_{d,k}+\left[\mathbf{I}\quad- \mathbf{LW}\right]\bar{w}_{k},\]  
where \(\mathbf{e}_{k}=\mathbf{d}_{k+1}-\mathbf{d}_{k}\) and \(\bar{w}_{k}= \begin{bmatrix}\mathbf{\varepsilon}_{k}\\ \mathbf{w}_{k}\end{bmatrix} \). Finally, Equation (45) is transformed into a compact form:  
\[\mathbf{e}_{d,k+1}=\mathbf{X}_{k}\mathbf{e}_{d,k}+\mathbf{Z}\bar{w}_{k}, \tag{46}\]  
with \(\mathbf{X}_{k}=\mathbf{I}-\mathbf{LE}_{k}\) and \(\mathbf{Z}= \begin{bmatrix}\mathbf{I}-\mathbf{LW}\end{bmatrix} \). To make further deliberations tractable, it is assumed that \(\bar{w}_{k}\) is bounded within an ellipsoid  
\[\bar{w}_{k}\in\mathcal{E}_{w},\mathcal{E}_{w}=\{\bar{w}:\bar{w}^{T}\mathbf{Q}_{w} \bar{w}\leq 1\}. \tag{47}\]  
This allows formulating the following theorem, which constitutes the main result of this section. Theorem 1: _The system (46) is strictly quadratically bounded for all \(\mathbf{E}_{k}\) and all allowable \(\bar{w}_{k}\in\mathcal{E}_{w}\) if there exist \(\mathbf{N},\mathbf{P}\succ\mathbf{0}\) and \(0<\alpha<1\), such the following conditions are satisfied_  
\[ \begin{bmatrix}-\mathbf{P}+\alpha\mathbf{P}&\mathbf{0}&\mathbf{P}-\mathbf{E}_{k}^{T}\mathbf{N}^{T}\\ \mathbf{0}&-\alpha\mathbf{Q}_{w}&\mathbf{R}^{T}\\ \mathbf{P}-\mathbf{NE}_{k}&\mathbf{R}&-\mathbf{P}\end{bmatrix} \prec\mathbf{0},\quad k=0,1,\ldots \tag{48}\]  
_with \(\mathbf{R}= \begin{bmatrix}\mathbf{P}-\mathbf{NW}\end{bmatrix} \)._  
Proof: Using Definition 1 and the fact that \(\bar{w}_{k}^{T}\mathbf{Q}\bar{w}_{k}\leq 1\) (cf. (47)), it can be concluded that  
\[\bar{w}_{k}^{T}\mathbf{Q}\bar{w}_{k}<\mathbf{e}_{d,k}^{T}\mathbf{P}\mathbf{e}_{d,k}\succeq\bm {e}_{d,k+1}^{T}\mathbf{P}\mathbf{e}_{d,k+1}-\mathbf{e}_{d,k}^{T}\mathbf{P}\mathbf{e}_{d,k}<0, \tag{49}\]  
where \(V_{k}=\mathbf{e}_{d,k}^{T}\mathbf{P}\mathbf{e}_{d,k}\) is the Lyapunov candidate function. Consequently, using Equation (46) and defining \(\mathbf{v}_{k}= \begin{bmatrix}\mathbf{e}_{d,k}\\ \bar{w}_{k}\end{bmatrix} \), it can be shown that  
\[\mathbf{v}_{k}^{T} \begin{bmatrix}\mathbf{X}^{T}\mathbf{P}\mathbf{X}&\mathbf{X}^{T}\mathbf{P}\mathbf{X}\\ \mathbf{Z}^{T}\mathbf{P}\mathbf{X}&\mathbf{Z}^{T}\mathbf{P}\mathbf{X}\end{bmatrix} \mathbf{v}_{k}<0. \tag{50}\]  
From (49), it is evident that, for any \(\alpha>0\),  
\[\alpha\mathbf{v}_{k}^{T} \begin{bmatrix}-\mathbf{P}&\mathbf{0}\\ \mathbf{0}&\mathbf{Q}_{w}\end{bmatrix} \mathbf{v}_{k}<0. \tag{51}\]  
Thus, applying the S-procedure to (50) and (51) gives  
\[\mathbf{v}_{k}^{T} \begin{bmatrix}\mathbf{X}_{k}^{T}\mathbf{P}\mathbf{X}_{k}-\mathbf{P}+\alpha\mathbf{ P}&\mathbf{X}_{k}^{T}\mathbf{P}\mathbf{Z}\\ \mathbf{Z}^{T}\mathbf{P}\mathbf{X}_{k}&\mathbf{Z}^{T}\mathbf{P}\mathbf{Z}-\alpha\mathbf{Q}_{w}\end{bmatrix}  \mathbf{v}_{k}<0, \tag{52}\]  
which by implementing the Schur complement yields  
\[ \begin{bmatrix}-\mathbf{P}+\alpha\mathbf{P}&\mathbf{0}&\mathbf{X}_{k}^{T}\mathbf{P}\\ \mathbf{0}&-\alpha\mathbf{Q}_{w}&\mathbf{Z}^{T}\mathbf{P}\\ \mathbf{P}\mathbf{X}_{k}&\mathbf{P}\mathbf{Z}&-\mathbf{P}\end{bmatrix} \prec\mathbf{0}. \tag{53}\]Finally, substituting  
\[P\mathbf{X}_{k} =\mathbf{P}\left(\mathbf{I}-\mathbf{L}\mathbf{E}_{k}\right)=\mathbf{P}-\mathbf{P}\mathbf{L}\mathbf{E} _{k}=\mathbf{P}-\mathbf{N}\mathbf{E}_{k}, \tag{54}\] \[\mathbf{P}\mathbf{Z} =\mathbf{P}\left[\mathbf{I}\ -\mathbf{L}\mathbf{W}\right]=\left[\mathbf{P}-\mathbf{P}\mathbf{L} \mathbf{B}\right]=\left[\mathbf{P}-\mathbf{N}\mathbf{W}\right], \tag{55}\]  
into (53) gives (48), which completes the proof. In spite of the incontestable appeal of the approach summarized by Theorem 1, it is impossible to use in order to obtain a solution of (48), which will be feasible for all \(k=0,1,\ldots\).