## 2 LMI-Based Control and Filter Optimization  
In this paper, dynamic system models are considered, which are given by the stochastic differential equations  
\[\mathrm{d}\mathbf{x}(t)=\mathbf{A}(\mathbf{p})\cdot\mathbf{x}(t)\mathrm{d}t+ \mathbf{B}(\mathbf{p})\cdot(\mathbf{u}(t)\mathrm{d}t+\mathbf{G}_{\mathrm{u}}( \mathbf{p})\cdot\mathrm{d}\mathbf{w}_{\mathrm{u}})+\mathbf{G}_{\mathrm{p}}( \mathbf{p})\cdot\mathrm{d}\mathbf{w}_{\mathrm{p}} \tag{1}\]  
with the state vector \(\mathbf{x}:=\mathbf{x}(t)\in\mathbb{R}^{n_{\mathrm{x}}}\) 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 \(\mathbf{p}\in\mathbb{R}^{n_{\mathrm{fp}}}\) is a vector of either constant or time-varying bounded parameters. Alternatively, this vector represents the dependencies of all system matrices on the state variables \(\mathbf{x}\); cf. [20]. For the sake of compactness, we assume that all entries of \(\mathbf{p}\) are mutually independent and that they influence the matrices \(\mathbf{A}\) and \(\mathbf{B}\) in an affine manner. Moreover, \(\mathbf{w}_{\mathrm{u}}\in\mathbb{R}^{m_{\mathrm{u}}}\) and \(\mathbf{w}_{\mathrm{p}}\in\mathbb{R}^{m_{\mathrm{p}}}\) are stochastically independent standard normally distributed Brownian motions of the actuator and process noise, so that \(\mathbf{G}_{\mathrm{u}}(\mathbf{p})\) and \(\mathbf{G}_{\mathrm{p}}(\mathbf{p})\) define the respective noise standard deviations in terms of element-wise non-negative matrix entries.  
In addition, the measured system output is given by  
\[\mathbf{y}_{\mathrm{m}}(t)=\mathbf{C}_{\mathrm{m}}\mathbf{x}(t)+\mathbf{G}_{ \mathrm{m}}\mathbf{w}_{\mathrm{m}}\enspace, \tag{2}\]  
where the output matrix \(\mathbf{C}_{\mathrm{m}}\in\mathbb{R}^{n_{\mathrm{fp}}\times n_{\mathrm{x}}}\) is assumed to be exactly known; \(\  
The introduction of this constraint allows us to formulate stability requirements for an output feedback control--that is based on an ideal filtering--(i.e., the algebraic relation (7) holds) in terms of matrix inequalities, which can be cast into LMIs by a suitable change of coordinates; see Section 2.2. Errors, which inevitably result from the non-negligible filter dynamics, are later on taken into consideration in Sections 2.3 and 2.4, especially in Equations (28) and (29).  
If the matrix \(\mathbf{C}_{y}\) (which extracts certain state variables or their linear combinations from the linear filter's state vectors) has full row rank, (7) can be reformulated according to  
\[\mathbf{\hat{y}}_{\mathrm{f}}=\mathbf{C}\cdot\mathbf{\hat{x}}\quad\text{with} \quad\mathbf{C}=\mathbf{C}_{\mathrm{y}}^{+}\cdot\mathbf{\hat{C}}\enspace, \tag{8}\]  
where  
\[\mathbf{C}_{\mathrm{y}}^{+}=\left(\mathbf{C}_{\mathrm{y}}^{T}\mathbf{C}_{ \mathrm{y}}\right)^{-1}\mathbf{C}_{\mathrm{y}}^{T} \tag{9}\]  
is the matrix pseudo inverse. Under the assumption of the aforementioned stationary, i.e., purely algebraic, relation, the matrix \(\mathbf{C}\) provides the possibility to express the filter outputs \(\mathbf{\hat{y}}_{\mathrm{f}}\) in terms of the internal states of the plant (1).  
For further details concerning the structured, LMI-based output feedback control design in Case 2, the reader is also referred to [31]. Figures 1 and 2 give a summary of the three different types of control structures described above, where the last one is the focus of this paper.  
To guarantee the solvability of the control design task, it is assumed that the system (1) is stabilizable using either of the inputs (3), (5), or (6). In addition, the pair \((\mathbf{A}(\mathbf{p}),\mathbf{C})\) needs to be robustly observable (or at least detectable) in Cases 1 and 2; cf. [29].  
Figure 1: Observer-based state and output feedback control structure according to Case 1 with the gain matrix \(\mathbf{K}\) and Case 2 with the structured gain \(\mathbf{K}_{\mathrm{o}}\mathbf{C}\).