## 2 LMI-Based Control and Filter Optimization
### Linear Output Filtering  
As shown in [26], a linear low-pass output filtering, as well as the derivative estimation of the scalar measured variables \(y_{\mathrm{m},i}\in\{1,\ldots,n_{\mathrm{y}}\}\), can be described in terms of the input-output representation  
\[\sum_{k=0}^{\xi_{i}}\alpha_{k,i}\cdot\frac{\mathrm{d}^{k}y_{t,i}}{\mathrm{d}t ^{k}}=\alpha_{0,i}\cdot y_{\mathrm{m},i}\enspace. \tag{15}\]  
The linear differential Equation (15) has the order \(\xi_{i}\) and contains the \(k\)-th order time derivatives \(y_{t,i}^{(k)}:=\frac{\mathrm{d}^{k}y_{t,i}}{\mathrm{d}t^{k}}\) that represent the filtered quantities that can be utilized in the controller according to Case 3, Equation (6). In this subsection, we present an LMI-based design of these filters as a systematic generalization of the pole (respectively, time constant) assignment that was performed in [26]. When additionally accounting for the influence of stochastic noise with quasi-continuous measurements, Equation (15) turns into the state-space representation  
\[\mathrm{d}\mathbf{y}_{t,i}=\left(\left(\mathbf{A}_{t,i}-\mathbf{b}_{t,i} \mathbf{k}_{t,i}^{T}\right)\cdot\mathbf{y}_{t,i}+\left(\mathbf{b}_{t,i} \mathbf{e}_{1}^{T}\mathbf{k}_{t,i}\right)\cdot y_{\mathrm{m},i}^{\prime} \right)\mathrm{d}t+\left(\left(\mathbf{b}_{t,i}\mathbf{e}_{1}^{T}\mathbf{k}_{ t,i}\right)\cdot g_{\mathrm{m},i}\right)\mathrm{d}w_{\mathrm{m},i} \tag{16}\]  
of a stochastic differential equation with the state vector  
\[\mathbf{y}_{t,i}=\left[y_{t,i}\quad\ldots\quad y_{t,i}^{(\xi_{i}-1)}\right]^{ T}\enspace, \tag{17}\]  
in which the superscript index denotes the corresponding temporal derivative order, the coefficient matrices  
\[\mathbf{A}_{t,i}= \begin{bmatrix}0&1&0&\ldots&0\\ 0&0&1&\ldots&0\\ \vdots&&&\ddots&\vdots\\ 0&0&0&\ldots&1\\ 0&0&0&\ldots&0\end{bmatrix} \in\mathbb{R}^{\xi_{i}\times\xi_{i}}\;,\;\mathbf{b }_{t,i}= \begin{bmatrix}0\\ 0\\ \vdots\\ 1\end{bmatrix} \in\mathbb{R}^{\xi_{i}}\enspace, \tag{18}\]  
the first unit vector \(\mathbf{e}_{1}= \begin{bmatrix}1&0&\ldots&0\end{bmatrix} ^{T}\in\mathbb{R}^{ \xi_{i}}\), and the yet unknown filter gain vector  
\[\mathbf{k}_{t,i}= \begin{bmatrix}\alpha_{0,i}&\ldots&\alpha_{\xi_{i}-1,i} \end{bmatrix} ^{T} \tag{19}\]  
with \(\alpha_{\xi_{i},i}\equiv 1\). This simplification results from a normalization of both sides of (15) under the restriction of steady-state accuracy due to which the derivatives of order zero on both sides of (15) have identical coefficients. For the sake of compactness, it is assumed that thematrix \(\mathbf{G}_{y}\) in (2) is purely diagonal. This corresponds to vanishing correlations between the noise of all scalar measurements in (2) with \(y_{\text{m},i}=y^{\prime}_{\text{m},i}+g_{\text{m},i}w_{\text{m},i}\). Hence, the low-pass filtered derivative of the order \(j\), \(j\in\{0,\dots,\xi_{i}\}\), for the \(i\)-th measured output is related to the state vector \(\mathbf{y}_{t,i}\) of the stochastic differential equation model (16) by  
\[\hat{y}^{(j)}_{t,i}= \begin{cases}\mathbf{e}^{T}_{j+1}\mathbf{y}_{t,i}+0\cdot y _{\text{m},i}&\text{for }j\in\{0,\dots,\xi_{i}-1\}\\ \mathbf{e}^{T}_{\xi_{i}}\mathbf{A}_{t}\mathbf{y}_{t,i}+\alpha_{0,i}\cdot\left( y^{\prime}_{\text{m},i}+g_{\text{m},i}w_{\text{m},i}\right)&\text{for }j=\xi_{i}\end{cases}  \tag{20}\]  
with \(\mathbf{e}_{j}\in\mathbb{R}^{\xi_{i}}\) denoting the \(j\)-th unit vector. In the equations above, the subscript m denotes the measured data, the prime symbol \((\cdot)^{t}\) the ideal noise-free outputs, the subscript f the filtered data, and \((\cdot)\) the estimates used by the controller. A compact notation of the filtered output vector  
\[\hat{\mathbf{y}}_{t}= \begin{bmatrix}\hat{\mathbf{y}}^{T}_{t,1}&\dots&\hat{ \mathbf{y}}^{T}_{t,n_{y}}\end{bmatrix}