## 2 LMI-Based Control and Filter Optimization
### Optimal Output Feedback Control  
Under the consideration of the structure of the control law of Case 3, the stochastic differential Equation (1) for the controlled polytopic system model turns into  
\[\mathrm{d}\mathbf{x}_{v}=\Big{(}\Big{(}\mathbf{A}_{v}-\mathbf{B}_{v}\mathbf{ K}_{y}\mathbf{C}_{\overline{y}}^{+}\mathbf{C}\Big{)}\mathbf{x}_{v}+\mathbf{B}_{v} \mathbf{K}_{y}\mathbf{C}_{y}\mathbf{e}_{t}\Big{)}\mathrm{d}t+\big{[}\mathbf{ B}_{v}\mathbf{G}_{u,v}\quad\mathbf{G}_{p,v}\big{]}\cdot \begin{bmatrix}\mathrm{d} \mathbf{w}_{u}\\ \mathrm{d}\mathbf{w}_{p}\end{bmatrix} \enspace. \tag{28}\]  
In addition, the ideal filtering process (assuming a noise-free setting, where the following equation turns exactly into a disturbance-free ordinary differential equation representation in which \(\bar{\mathbf{y}}_{t,i}\) represents the state vector after removing the noise term from (16)) is described by  
\[\mathrm{d}\bar{\mathbf{y}}_{t,i}=\Big{(}\Big{(}\mathbf{A}_{t,i}-\mathbf{b}_{ t,i}\mathbf{k}_{t,i}^{T}\Big{)}\cdot\bar{\mathbf{y}}_{t,i}+\Big{(}\mathbf{b}_{ t,i}\mathbf{e}_{1}^{T}\mathbf{k}_{t,i}\Big{)}\cdot y_{m,i}^{\prime}\Big{)}\; \mathrm{d}t\enspace. \tag{29}\]  
After introducing the vectors of output estimation errors  
\[\mathbf{e}_{t,i}=\bar{\mathbf{y}}_{t,i}-\mathbf{y}_{t,i}\enspace,\enspace i \in\{1,\ldots,n_{y}\}\enspace, \tag{30}\]  
and a stacked vector notation \(\mathbf{e}_{t}\) according to (21), the error dynamics of the linear filters are given by  
Figure 3: Domain of the eigenvalues compatible with the constraints (23) and (24), where \(\theta=0\) is desired to guarantee non-oscillatory dynamics. \[\mathrm{d}\mathbf{e}_{t,i}=\left(\left(\mathbf{A}_{t,i}-\mathbf{b}_{t,i}\mathbf{k} _{t,i}^{T}\right)\cdot\mathbf{e}_{t,i}\right)\mathrm{d}t-\left(\left(\mathbf{b} _{t,i}\mathbf{e}_{1}^{T}\mathbf{k}_{t,i}\right)\cdot g_{\mathrm{m},i}\right) \mathrm{d}w_{\mathrm{m},i}\enspace. \tag{31}\]  
Now, introduce the stacked vector  
\[\mathbf{z}_{v}=\left[\mathbf{x}_{v}^{T}\quad\mathbf{e}_{t,i}^{T}\quad\cdots \quad\mathbf{e}_{t,n_{y}}^{T}\right]^{T} \tag{32}\]  
consisting of system states and noise-induced filter errors. The stochastic differential equations corresponding to (32) are given by  
\[\mathrm{d}\mathbf{z}_{v}=\boldsymbol{\mathcal{A}}_{v}\cdot\mathbf{z}_{v} \,\mathrm{d}t+\boldsymbol{\mathcal{G}}_{v}\cdot \begin{bmatrix}\mathrm{d} \mathbf{w}_{\mathrm{u}}\\ \mathrm{d}\mathbf{w}_{\mathrm{p}}\\ \mathrm{d}\mathbf{w}_{\mathrm{m}}\end{bmatrix}  \tag{33}\]  
with the system matrix  
\[\boldsymbol{\mathcal{A}}_{v}= \begin{bmatrix}\mathbf{A}_{v}-\mathbf{B}_{v} \mathbf{K}_{y}\mathbf{C}_{y}^{+}\bar{\mathbf{C}}\quad\mathbf{B}_{v}\mathbf{K }_{y}\mathbf{C}_{y}\\ \mathbf{0}\qquad\boldsymbol{\mathcal{A}}_{v,2,2}\end{bmatrix} \enspace, \tag{34}\]  
in which its lower right sub-block has the block diagonal structure  
\[\boldsymbol{\mathcal{A}}_{v,2,2}=\mathrm{blkdiag}\Big{(}\left(\mathbf{A}_{t,1 }-\mathbf{b}_{t,1}\mathbf{k}_{t,1}^{T}\right),\ldots,\left(\mathbf{A}_{t,n_{y }}-\mathbf{b}_{t,n_{y}}\mathbf{k}_{t,n_{y}}^{T}\right)\Big{)} \tag{35}\]  
and the matrix of standard deviations  
\[\boldsymbol{\mathcal{G}}_{v}= \begin{bmatrix}\mathbf{B}_{v}\mathbf{G}_{\mathrm{ u},v}&\mathbf{G}_{\mathrm{p},v}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\boldsymbol{\mathcal{G}}_{v,2,3}\end{bmatrix} \enspace, \tag{36}\]  
with the block diagonal sub-matrix  
\[\boldsymbol{\mathcal{G}}_{v,2,3}=-\mathrm{blkdiag}\Big{(}\Big{(}\mathbf{b}_{ t,1}\mathbf{e}_{1}^{T}\mathbf{k}_{t,i}\Big{)}\cdot g_{\mathrm{m},1},\ldots, \left(\mathbf{b}_{t,n_{y}}\mathbf{e}_{1}^{T}\mathbf{k}_{t,n_{y}}\right)\cdot g _{\mathrm{m},n_{y}}\Big{)}\enspace. \tag{37}\]  
**Theorem 3**.