## 3 Methodological Background: LMI-Based Control and Observer Design
### Optimality of the Noise Insensitive Observer-Based Control Design  
**Theorem 4** (Optimal control and observer parameterization). : _The parameterization of the observer-based controller for the augmented system (36)-(40) is optimal in the sense of a minimization of the influence of noise if the controller and observer gains are chosen to minimize the cost function_  
\[J=\sum_{v=1}^{n_{v}}\left(\frac{\mathrm{trace}\{\boldsymbol{\mathcal{N}}_{v} \}}{\mathrm{det}\Big{(}-\boldsymbol{\mathcal{A}}_{v}^{\prime}\Big{)}}\cdot \frac{\mathrm{det}\big{(}\dot{\mathbf{P}}_{\mathrm{O}}^{\prime}\big{)}}{ \mathrm{det}(\dot{\mathbf{Q}})}\right) \tag{41}\]with \(\mathcal{M}_{v}=\mathcal{M}_{v}^{T}\succ 0\) and \[\ddot{\boldsymbol{\mathcal{A}}}_{v}:=\boldsymbol{\mathcal{A}}_{v}^{T}\ddot{ \boldsymbol{\mathcal{B}}}+\dot{\boldsymbol{\mathcal{B}}}\boldsymbol{\mathcal{A}} _{v}\prec 0,\] (42) where \[ \begin{bmatrix}\mathcal{M}_{v}&\ddot{\boldsymbol{\mathcal{B}}}_{v}^{T}\\ \ddot{\boldsymbol{\mathcal{B}}}_{v}&\begin{bmatrix}\mathbf{Q}&\mathbf{0}\\ \mathbf{0}&(\dot{\mathbf{P}}_{\mathrm{O}}^{\prime})^{-1}\end{bmatrix}\end{bmatrix}  \succ 0,\] (43)  
\[\ddot{\boldsymbol{\mathcal{B}}}_{v}= \begin{bmatrix}\mathbf{B}_{v}\mathbf{G}_{ \mathrm{u},v}&\mathbf{G}_{\mathrm{p},v}&\mathbf{0}\\ \mathbf{B}_{v}\mathbf{G}_{\mathrm{u},v}&\mathbf{G}_{\mathrm{p},v}&-\ddot{ \mathbf{Q}}_{\mathrm{O}}^{-1}\mathbf{Y}_{\mathrm{O}}^{T}\mathbf{G}_{\mathrm{m }}\end{bmatrix} , \tag{44}\]  
and  
\[\left(\mathbf{A}_{v}-\dot{\mathbf{H}}\mathbf{C}_{\mathrm{m}}\right)^{T}\cdot \mathbf{P}_{\mathrm{O}}^{\prime}+\mathbf{P}_{\mathrm{O}}^{\prime}\cdot\left( \mathbf{A}_{v}-\dot{\mathbf{H}}\mathbf{C}_{\mathrm{m}}\right)\prec 0 \tag{45}\]  
with \(\ddot{\mathbf{H}}=\dot{\mathbf{Q}}_{\mathrm{O}}^{-1}\dot{\mathbf{Y}}_{\mathrm{ O}}^{T}\) hold. Here, \((\cdot)\) symbols indicate an iterative evaluation, where all such values are replaced by the outcome of the previous iteration stage, especially by inserting the previous gain values into (37) to obtain the matrices \(\boldsymbol{\mathcal{A}}_{v}\) in (42). Proof. : Define a quadratic form  
\[V(\mathbf{z}_{v})=\frac{1}{2}\mathbf{z}_{v}^{T}\cdot\boldsymbol{\mathcal{B}} \cdot\mathbf{z}_{v} \tag{46}\]  
as the candidate for a Lyapunov function with the block diagonal matrix  
\[\boldsymbol{\mathcal{B}}=\mathrm{blkdiag}\big{(}\mathbf{P},\mathbf{P}_{ \mathrm{O}}^{\prime}\big{)}= \begin{bmatrix}\mathbf{P}&\mathbf{0}\\ \mathbf{0}&\mathbf{P}_{\mathrm{O}}^{\prime}\end{bmatrix}  \tag{47}\]  
consisting of those matrices that were obtained from the Corollaries 1 or 2, as well as from Theorem 3. The corresponding time derivative of (46) is computed by applying the Ito differential operator [14; 15]. It leads to the expression  
\[L(V)=\frac{1}{2}\Big{(}\mathbf{z}_{v}^{T}\Big{(}\boldsymbol{\mathcal{A}}_{v}^ {T}\boldsymbol{\mathcal{B}}+\boldsymbol{\mathcal{P}}\boldsymbol{\mathcal{A}} _{v}\Big{)}\mathbf{z}_{v}+\mathrm{trace}\Big{\{}\boldsymbol{\mathcal{B}}_{v}^ {T}\boldsymbol{\mathcal{P}}\boldsymbol{\mathcal{B}}_{v}\Big{\}}\Big{)}. \tag{48}\]  
Despite the strict negative definiteness of \(\boldsymbol{\mathcal{A}}_{v}^{T}\boldsymbol{\mathcal{B}}+\boldsymbol{\mathcal{ P}}\boldsymbol{\mathcal{A}}_{v}\) (corresponding to the asymptotic stability of the deterministic part of the augmented system model due to the existence of a positive definite matrix \(\boldsymbol{\mathcal{B}}=\boldsymbol{\mathcal{B}}^{T}\succ 0\)), the value of \(L(V)\) may become positive in the close vicinity of \(\mathbf{z}_{v}=\mathbf{0}\) due to persistent stochastic excitation.