## 3 Methodological Background: LMI-Based Control and Observer Design  
In this paper, dynamic system models are considered, which are given by the stochastic differential equations  
\[\mathrm{d}\mathbf{x}(t)=\mathbf{A}(\mathbf{p})\cdot\mathbf{x}(t)\mathrm{d}t+ \mathbf{B}(\mathbf{p})\cdot(\mathbf{u}(t)\mathrm{d}t+\mathbf{G}_{\mathbf{u}}( \mathbf{p})\cdot\mathrm{d}\mathbf{w}_{\mathbf{u}})+\mathbf{G}_{\mathrm{p}}( \mathbf{p})\cdot\mathrm{d}\mathbf{w}_{\mathbf{p}} \tag{8}\]  
with the state vector \(\mathbf{x}:=\mathbf{x}(t)\in\mathbb{R}^{n_{\mathrm{s}}}\) and the input vector \(\mathbf{u}:=\mathbf{u}(t)\in\mathbb{R}^{n_{\mathrm{u}}}\); \(\mathbf{A}(\mathbf{p})\) and \(\mathbf{B}(\mathbf{p})\) are the system and input matrices, where \(\mathbf{p}\in\mathbb{R}^{n_{\mathrm{p}}}\) is a vector of either constant or time-varying bounded parameters. Alternatively, this vector may denote the dependence of all system matrices on the state variables \(\mathbf{x}\), cf. [9]. For the sake of a compact notation, all entries of \(\mathbf{p}\) are treated as mutually independent with an affine dependence of \(\mathbf{A}\) and \(\mathbf{B}\) on these quantities. Moreover, \(\mathbf{w}_{\mathbf{u}}\in\mathbb{R}^{m_{\mathrm{u}}}\) and \(\mathbf{w}_{\mathbf{p}}\in\mathbb{R}^{m_{\mathrm{p}}}\) are stochastically independent standard normally distributed Brownian motions of actuator and process noise. In this sense, \(\mathbf{G}_{\mathbf{u}}(\mathbf{p})\) and \(\mathbf{G}_{\mathrm{p}}(\mathbf{p})\) represent the (element-wise non-negative) disturbance input matrices containing the corresponding standard deviations. The measured system output is given by  
\[\mathbf{y}_{\mathrm{m}}(t)=\mathbf{C}_{\mathrm{m}}\mathbf{x}(t)+\mathbf{G}_{ \mathrm{m}}\mathbf{w}_{\mathrm{m}}, \tag{9}\]  
where the output matrix \(\mathbf{C}_{\mathrm{m}}\in\mathbb{R}^{n_{\mathrm{y}}\times n_{\mathrm{s}}}\) is assumed to be exactly known; \(\mathbf{w}_{\mathrm{m}}\) is the standard normally distributed sensor noise, while \(\mathbf{G}_{\mathrm{m}}\) is the weighting matrix representing the actual standard deviation of the output disturbance. In the following, we aim at designing either an observer-based state feedback (Case 1) or an observer-based output feedback control strategy (Case 2). In the second case, the measured output does not necessarily coincide with the system output to be controlled. A linear filter-based output feedback (Case 3), in which the filter is designed in a model-free manner, is only mentioned for the sake of completeness and has been treated by the authors in a separate publication [21]. The reason for this separate investigation is the fact that the design criteria do not fully coincide with the requirements derived for the Cases 1 and 2 in this paper:  
**Case 1:**: The control signal is defined as  
\[\mathbf{u}=\mathbf{u}_{\mathrm{ff}}-\mathbf{K}\cdot\hat{\mathbf{x}}, \tag{10}\]  
Figure 1: Stability analysis of the observer-based closed-loop control system (7) for different values of \(a_{\mathrm{nom}}\) and \(a\in[1\;;\;3]\): The choice of \(a_{\mathrm{nom}}\
eq\overline{n}\) requires a significant increase of both \(k\) and \(h\) above the thresholds \(k>3\) and \(h>3\). where (without loss of generality) \(\mathbf{u}_{\text{ff}}\) is a constant feedforward signal and \(\hat{\mathbf{x}}\) is the state estimate determined by the robust observer  
\[ \begin{split}\hat{\mathbf{x}}&=\mathbf{A}_{\text{nom} }\cdot\hat{\mathbf{x}}+\mathbf{B}_{\text{nom}}\cdot\mathbf{u}+\mathbf{H}\cdot( \mathbf{y}_{\text{m}}-\hat{\mathbf{y}})\\ \hat{\mathbf{y}}&=\mathbf{C}_{\text{m}}\cdot\hat{ \mathbf{x}},\end{split}  \tag{11}\]  
which makes use of the nominal system and input matrices \(\mathbf{A}_{\text{nom}}\) and \(\mathbf{B}_{\text{nom}}\), see Section 3.4. For the following stabilization and performance optimization, \(\mathbf{u}_{\text{ff}}=\mathbf{0}\) is assumed. Desired operating points outside the origin \(\mathbf{x}=\mathbf{0}\) are easily achievable by adding suitable nonzero offset terms.