## 2 LMI-Based Control and Filter Optimization
### Optimal Output Feedback Control
: \((\)_Optimal control and filter gains\()\) The controller and filter gains from Corollary 1 in Section 2.2 and Corollary 2 in Section 2.3 are jointly optimal if they are chosen so that the cost function_  
\[J=\sum_{v=1}^{n_{y}}\left(\frac{\mathrm{trace}\{\boldsymbol{\mathcal{N}}_{v} \}}{\mathrm{det}\Big{(}-\bar{\boldsymbol{\mathcal{A}}}_{v}\Big{)}}\cdot\frac{ 1}{\mathrm{det}\Big{(}\bar{\boldsymbol{\mathcal{Q}}}\Big{)}\cdot\prod\limits_ {i=1}^{n_{y}}\mathrm{det}\big{(}\bar{\boldsymbol{\mathcal{Q}}}_{t,i}\big{)}}\right) \tag{38}\]  
_is minimized, where the abbreviation \(\bar{\boldsymbol{\mathcal{A}}}_{v}=\bar{\boldsymbol{\mathcal{A}}}_{v}^{T} \boldsymbol{\mathcal{P}}+\boldsymbol{\mathcal{P}}\bar{\boldsymbol{\mathcal{A}}} _{v}\prec 0\) is defined and \(\boldsymbol{\mathcal{N}}_{v}=\boldsymbol{\mathcal{N}}_{v}^{T}\succ 0\) is a free matrix variable. Here, the matrices \(\bar{\boldsymbol{\mathcal{A}}}_{v}\) are defined for the vertices of the polytope (10) according to_  
\[\bar{\boldsymbol{\mathcal{A}}}_{v}= \begin{bmatrix}\mathbf{A}_{v}-\mathbf{B}_{v }\bar{\mathbf{K}}_{y}\mathbf{C}_{y}^{+}\bar{\mathbf{C}}&\mathbf{B}_{v}\bar{ \mathbf{K}}_{y}\mathbf{C}_{y}\\ \mathbf{0}&\bar{\boldsymbol{\mathcal{A}}}_{v,2,2}\end{bmatrix}  \tag{39}\]  
_with_  
\[\bar{\boldsymbol{\mathcal{A}}}_{v,2,2}=\mathrm{blkdiag}\Big{(}\Big{(}\mathbf{ A}_{t,1}-\mathbf{b}_{t,1}\mathbf{k}_{t,1}^{T}\Big{)},\ldots,\Big{(}\mathbf{A}_{t,n_{y }}-\mathbf{b}_{t,n_{y}}\mathbf{k}_{t,n_{y}}^{T}\Big{)}\Big{)}\enspace. \tag{40}\]  
_In addition, the definiteness constraint_  
\[ \begin{bmatrix}\boldsymbol{\mathcal{N}}_{v}&\bar{\boldsymbol{\mathcal{G}}}_{v} ^{T}\\ \tilde{\boldsymbol{\mathcal{G}}}_{v}&\begin{bmatrix}\mathbf{Q}&\mathbf{0}\\ \mathbf{0}&\mathrm{blkdiag}\Big{(}\mathbf{Q}_{t,1},\ldots,\mathbf{Q}_{t,n_{y }}\Big{)}\end{bmatrix}\end{bmatrix} \succ 0 \tag{41}\]  
_with_  
\[\tilde{\boldsymbol{\mathcal{G}}}_{v}= \begin{bmatrix}\mathbf{B}_{v}\mathbf{G}_{ \mathrm{u},v}&\mathbf{G}_{\mathrm{p},v}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\tilde{\boldsymbol{\mathcal{G}}}_{v,2,3}\end{bmatrix}  \tag{42}\]_and_  
\[\mathcal{G}_{v,2,3}=-\mathrm{blkdiag}\Big{(}\Big{(}\mathbf{b}_{l,1}\mathbf{e}_{1 }^{T}\mathbf{\bar{k}}_{l,i}\Big{)}\cdot g_{m,1},\ldots,\Big{(}\mathbf{b}_{l,l_{y }}\mathbf{e}_{1}^{T}\mathbf{\bar{k}}_{l,n_{y}}\Big{)}\cdot g_{m,n_{y}}\Big{)} \tag{43}\]  
_must be satisfied; \((\cdot)\) symbols indicate an iterative evaluation, where all such values are replaced by the outcome of the previous iteration stage._  
Proof. : Define a positive definite Lyapunov function candidate  
\[V(\mathbf{z}_{v})=\frac{1}{2}\Big{(}\mathbf{z}_{v}^{T}\mathbf{\mathcal{P}}\mathbf{ z}_{v}\Big{)} \tag{44}\]  
with the block diagonal matrix  
\[\mathbf{\mathcal{P}}=\mathrm{blkdiag}\Big{(}\mathbf{P},\mathbf{P}_{l,1},\ldots, \mathbf{P}_{l,n_{y}}\Big{)}\enspace. \tag{45}\]  
By applying the Ito differential operator [27], its time derivative is obtained as  
\[L(V)=\frac{1}{2}\Big{(}\mathbf{z}_{v}^{T}\Big{(}\mathbf{\mathcal{A}}_{v}^{T}\mathbf{ \mathcal{P}}+\mathbf{\mathcal{P}}\mathbf{\mathcal{A}}_{v}\Big{)}\mathbf{z}_{v}+ \mathrm{trace}\Big{\{}\mathbf{\mathcal{G}}_{v}^{T}\mathbf{\mathcal{P}}\mathbf{\mathcal{G }}_{v}\Big{\}}\Big{)}\enspace. \tag{46}\]  
Following the reasoning in [26], the interior of the ellipsoid  
\[\mathbf{z}_{v}^{T}\mathbf{\mathcal{M}}_{v}^{-1}\mathbf{z}_{v}-1=0\enspace, \tag{47}\]  
where  
\[\mathbf{\mathcal{M}}_{v}^{-1}=\left(\frac{-\mathbf{\mathcal{A}}_{v}}{\mathrm{trace} \Big{\{}\mathbf{\mathcal{G}}_{v}^{T}\mathbf{\mathcal{P}}\mathbf{\mathcal{G}}_{v}\Big{\}}}\right) \tag{48}\]  
and  
\[\mathbf{\mathcal{A}}_{v}:=\mathbf{\mathcal{A}}_{v}^{T}\mathbf{\mathcal{P}}+\mathbf{\mathcal{ P}}\mathbf{\mathcal{A}}_{v}\prec 0 \tag{49}\]  
hold, is the domain for which no stability properties can be verified.