## 3 Methodological Background: LMI-Based Control and Observer Design
### Optimality of the Noise Insensitive Observer-Based Control Design
The non-provable stability domain is, hence, characterized by the interior of an ellipsoid that is described by the boundary \(L(V)=0\) according to  
\[\mathbf{z}_{v}^{T}\boldsymbol{\mathcal{M}}_{v}^{-1}\mathbf{z}_{v}-1=0, \tag{49}\]  
where  
\[\boldsymbol{\mathcal{M}}_{v}^{-1}=\bigg{(}\frac{-\boldsymbol{\mathcal{A}}_{v}} {\mathrm{trace}\big{\{}\boldsymbol{\mathcal{B}}_{v}^{T}\boldsymbol{\mathcal{B}} \boldsymbol{\mathcal{B}}_{v}\big{\}}}\bigg{)} \tag{50}\]  
and  
\[\boldsymbol{\mathcal{A}}_{v}:=\boldsymbol{\mathcal{A}}_{v}^{T}\boldsymbol{ \mathcal{B}}+\boldsymbol{\mathcal{P}}\boldsymbol{\mathcal{A}}_{v}\prec 0. \tag{51}\]  
The volume of this ellipsoid is proportional to  
\[\sqrt{\mathrm{det}(\boldsymbol{\mathcal{M}}_{v})}=\sqrt{\frac{\mathrm{trace} \big{\{}\boldsymbol{\mathcal{S}}_{v}^{T}\boldsymbol{\mathcal{P}}\boldsymbol{ \mathcal{B}}_{v}\big{\}}}{\mathrm{det}(-\boldsymbol{\mathcal{A}}_{v})}}. \tag{52}\]Following the same steps as in ([14], Equations (17)-(20)), it is desired to minimize the quality criterion  
\[\mathcal{F}_{v}=\frac{\operatorname{trace}\{\mathbf{\mathscr{F}}_{v}^{T}\mathbf{\mathscr{ P}}\mathbf{\mathscr{G}}_{v}\}}{\det(-\mathbf{\mathscr{A}}_{v})}\cdot\frac{\det(\mathbf{P}_{ \text{O}}^{\prime})}{\det(\mathbf{Q})} \tag{53}\]  
by tuning the control and observer gains, where the second factor helps to maximize the domains for which the linear feedback in the closed-loop system is bounded by some positive constant [28]. According to [28], multiplicative couplings between \(\mathbf{\mathscr{P}}\) and the gains included in \(\mathbf{\mathscr{G}}_{v}\) are removed by an iterative LMI formulation of the optimization task. For that purpose, the term \(\mathbf{\mathscr{F}}_{v}^{T}\mathbf{\mathscr{P}}\mathbf{\mathscr{G}}_{v}\) is relaxed into the matrix inequality  
\[\mathcal{M}_{v}\succ\mathbf{\mathscr{F}}_{v}^{T}\mathbf{\mathscr{P}}\mathbf{\mathscr{G}}_ {v}\ \text{ corresponding to }\ \mathcal{N}_{v}-\mathbf{\mathscr{F}}_{v}^{T}\mathbf{\mathscr{P}}\mathbf{\mathscr{G}}_{v}\succ 0 \tag{54}\]  
with \(\mathcal{M}_{v}=\mathcal{M}_{v}^{T}\succ 0\). The inequality (54) is finally re-written by applying the [PERSON] complement formula according to  
\[ \begin{bmatrix}\mathcal{M}_{v}&\mathbf{\mathscr{G}}_{v}^{T}\\ \mathbf{\mathscr{G}}_{v}&\begin{bmatrix}\mathbf{Q}&\mathbf{0}\\ \mathbf{0}&\left(\mathbf{P}_{\text{O}}^{\prime}\right)^{-1}\end{bmatrix} \end{bmatrix} \succ 0. \tag{55}\]  
The proof is completed by approximating \(\left(\mathbf{P}_{\text{O}}^{\prime}\right)^{-1}\) with the solution of the inequality (45) and by replacing all nonlinearities related to the gains and parameterizations of \(V(\mathbf{z}_{v})\) in (46) by the results of the preceding iteration step. **Remark 5**: _The pure block diagonal structure of this matrix \(\mathbf{\mathscr{P}}\) in (47) with independent parameters for the state and error dynamics, may introduce some degree of conservativeness. However, the positive definiteness of this matrix always guarantees closed-loop stability (see the discussion of Equation (48)). Moreover, the simulations in the following section have shown that the resulting control signals outperform classical design approaches in terms of a reduction of the control effort. In addition, notice that the obtained solution can be conservative due to the use of a common quadratic Lyapunov function for the whole polytope. The use of parameter-dependent Lyapunov functions, such as those discussed in [29] for discrete-time processes with time-varying parameters, will be a subject of future research._