## 3 Methodological Background: LMI-Based Control and Observer Design
### Observer-Based Feedback Control in the Presence of Noise  
To ensure stability of the control laws (10) or (12) despite the general invalidity of the separation principle according to Section 2 and to simultaneously optimize the controller and observer gain matrices in Cases 1 and 2, the closed-loop control systems' dynamics as well as the point-valued observers' error dynamics \(\mathbf{e}_{v}=\mathbf{x}_{v}-\hat{\mathbf{x}}_{v}\) are described by using a combined set of stochastic differential equation models. In the following, all system models are derived for each of the worst-case realizations \(v\in\{1,\ldots,n_{v}\}\), to simultaneously account for bounded parameter uncertainty and stochastic noise. For that purpose, the deterministic system models from the previous section (i.e., the noise-free, however, parameter-dependent terms) are extended by the terms for stochastic process and sensor noise as well as by accounting for the control signals  
\[\mathbf{u}_{v}=-\mathbf{K}_{\mathrm{o}}\mathbf{C}\hat{\mathbf{x}}_{v}=- \mathbf{K}_{\mathrm{o}}\mathbf{C}\cdot(\mathbf{x}_{v}-\mathbf{e}_{v}). \tag{31}\]  
Hence, each vertex realization for the closed-loop control system is represented by a stochastic differential equation  
\[\mathrm{d}\mathbf{x}_{v}=\Big{(}(\mathbf{A}_{v}-\mathbf{B}_{v}\mathbf{K}_{ \mathrm{o}}\mathbf{C})\mathbf{x}_{v}+\mathbf{B}_{v}\mathbf{K}_{\mathrm{o}} \mathbf{C}\mathbf{e}_{v}\Big{)}\mathrm{d}t+\big{[}\mathbf{B}_{v}\mathbf{G}_{ \mathrm{u},v}\quad\mathbf{G}_{\mathrm{p},v}\big{]}\cdot \begin{bmatrix} \mathrm{d}\mathbf{w}_{\mathrm{u}}\\ \mathrm{d}\mathbf{w}_{\mathrm{p}}\end{bmatrix} . \tag{32}\]  
Here and in the following, \(\mathbf{C}\equiv\mathbf{I}\) and \(\mathbf{K}_{\mathrm{o}}\equiv\mathbf{K}\) hold for Case 1. Similarly, the stochastic differential equation for the observer dynamics at each vertex \(v\) is obtained as  
\[\mathrm{d}\hat{\mathbf{x}}_{v}=\Big{(}(\mathbf{A}_{\mathrm{nom}}-\mathbf{B}_ {\mathrm{nom}}\mathbf{K}_{\mathrm{o}}\mathbf{C})\hat{\mathbf{x}}_{v}+\mathbf{ HC}_{\mathrm{m}}\mathbf{e}_{v}\Big{)}\mathrm{d}t+\mathbf{HG}_{\mathrm{m}} \,\mathrm{d}\mathbf{w}_{\mathrm{m}}. \tag{33}\]  
The latter leads to the associated error dynamics  
\[ \begin{split}\mathrm{d}\mathbf{e}_{v}&=\Big{(}(( \mathbf{A}_{v}-\mathbf{A}_{\mathrm{nom}})-(\mathbf{B}_{v}-\mathbf{B}_{ \mathrm{nom}})\mathbf{K}_{\mathrm{o}}\mathbf{C})\mathbf{x}_{v}\Big{)}\mathrm{d }t\\ &\quad+\Big{(}((\mathbf{A}_{\mathrm{nom}}-\mathbf{HC}_{\mathrm{m}}) -(\mathbf{B}_{\mathrm{nom}}-\mathbf{B}_{v})\mathbf{K}_{\mathrm{o}}\mathbf{C}) \mathbf{e}_{v}\Big{)}\mathrm{d}t\\ &\quad+\big{[}\mathbf{B}_{v}\mathbf{G}_{\mathrm{u},v}\quad\mathbf{ G}_{\mathrm{p},v}\quad-\mathbf{HG}_{\mathrm{m}}\big{]}\cdot\begin{bmatrix} \mathrm{d}\mathbf{w}_{\mathrm{u}}\\ \mathrm{d}\mathbf{w}_{\mathrm{p}}\\ \mathrm{d}\mathbf{w}_{\mathrm{m}}\end{bmatrix}.\end{split}  \tag{34}\]  
As discussed in [14], it may be impossible to verify stability in the close vicinity of the desired stationary operating point in the presence of stochastic noise. This is true even if a robust stabilization by simultaneously designing the controller and observer for the noise-free polytopic uncertainty model is successful. To design a state observer minimizing the sensitivity against noise, our previous work [14] suggests estimating the volume of an ellipsoid centered at the desired operating point for which stability properties cannot be proven. For the case of exactly known system models, an LMI-based optimization approach was introduced in [14] to reduce the corresponding ellipsoid volume. This approach is now extended by the following two aspects: (i) simultaneously computing the controller and observer gains in a jointly stabilizing sense in the presence of noise, as well as (ii) additionally considering the goal of achieving insensitivity against bounded parameter uncertainty in the system matrices by the minimization of the aforementioned ellipsoid volume. For that purpose, define an augmented vector  
\[\mathbf{z}_{v}=