## 2 LMI-Based Control and Filter Optimization
### Linear Output Filtering
^{T} \tag{21}\]  
in Equation (6) is obtained by collecting all outputs from (20) that are actually relevant for the output feedback design according to  
\[\hat{\mathbf{y}}_{t,i}=\mathbf{C}_{t,i}\cdot\mathbf{y}_{t,i}+\mathbf{D}_{t,i} \cdot\left(y^{\prime}_{\text{m},i}+g_{\text{m},i}w_{\text{m},i}\right)\enspace. \tag{22}\]  
Here, \(\mathbf{C}_{t,i}\) represents the dependence of the filter outputs \(\hat{\mathbf{y}}_{t,i}\) on the filters' state variables \(\mathbf{y}_{t,i}\) and contains the coefficients of the first summand of both rows in (20). The factor \(\mathbf{D}_{t,i}\) is only non-zero if the filter has a direct measurement feedthrough (and, thus, also a noise feedthrough) because the approximate of the derivative of the order \(\xi_{i}\) is expressed in terms of the last vector component of the dynamic model (16). The asymptotic stability of the filter dynamics with purely real eigenvalues is ensured by the following theorem. **Theorem 2**: (_Asymptotically stable, non-oscillatory filter dynamics_) _The filter dynamics (20) are guaranteed to be asymptotically stable with purely real eigenvalues of the deterministic part of the stochastic differential Equation (16), if the gain vectors \(\mathbf{k}_{t,i}\) satisfy the matrix inequalities_  
\[\mathbf{D}_{0}\otimes\mathbf{P}_{t,i}+\mathbf{D}_{1}\otimes\left(\left( \mathbf{A}_{t,i}-\mathbf{b}_{t,i}\mathbf{k}^{T}_{t,i}\right)\cdot\mathbf{P} _{t,i}\right)+\mathbf{D}^{T}_{1}\otimes\left(\left(\mathbf{A}_{t,i}-\mathbf{b }_{t,i}\mathbf{k}^{T}_{t,i}\right)\cdot\mathbf{P}_{t,i}\right)^{T}\preceq 0 \tag{23}\]  
_with some \(\mathbf{P}_{t,i}=\mathbf{P}^{T}_{t,i}\succ 0\) for all \(i\in\{1,\dots,n_{y}\}\), where_  
\[\mathbf{D}_{0}= \begin{bmatrix}2\gamma&0&0&0\\ 0&-2\delta&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix} \,,\ \ \mathbf{D}_{1}= \begin{bmatrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&\sin\theta&\cos\theta\\ 0&0&-\cos\theta&\sin\theta\end{bmatrix}  \tag{24}\]  
_with \(0\leq\theta<\frac{\pi}{2}\); \(\otimes\) is the matrix Kronecker product of the respective arguments; \(\gamma>0\) and \(\delta>\gamma\) represent bounds on the real parts of the eigenvalues \(s_{i}\) so that \(-\delta\leq\Re\{s_{i}\}\leq-\gamma\) holds. To obtain purely real eigenvalues, \(\theta=0\) is chosen._  
A graphical representation of the stability domain represented by (23) with (24) is given in Figure 3.  
Proof. : Theorem 2 is a direct consequence of formulating a bounded interval \([-\delta;\ -\gamma]\) on the negative real axis of the complex \(s\) plane (with \(\bar{s}\) being the conjugate complex of s) as the desired \(\Gamma\)-stability domain  
\[\mathbf{F}_{\Gamma}(s):=\mathbf{D}_{0}+s\cdot\mathbf{D}_{1}+\bar{s}\cdot \mathbf{D}^{T}_{1}\preceq 0 \tag{25}\]according to [3; 5; 7]. For a detailed derivation of the coefficient matrices \(\mathbf{D}_{0}\) and \(\mathbf{D}_{1}\), see Appendix A. A reformulation of this \(\Gamma\)-stability domain into a gain-dependent matrix inequality according to ([20], Equation (11)) completes the proof. **Remark 1**. : _The specification of \(\Gamma\)-stability domains is analogously possible for the output feedback parameterization. For a corresponding generalized formulation, see Appendix B. From a practical point of view, enforcing purely real eigenvalues with \(\theta=0\) in the filter parameterization is often not necessary. Commonly, it is sufficient to specify large enough damping ratios, for example from the sector \(0\leq\theta<\frac{\pi}{4}\), where the upper bound of this interval would correspond to the value \(\frac{1}{2}\sqrt{2}\) for [PERSON]'s damping coefficient in a second-order differential equation._  
**Corollary 2**.