## 2 LMI-Based Control and Filter Optimization
### Robust Output Feedback Control for Case 3  
LMI-based design approaches can be employed for the design of output feedback controllers that are restricted in their parameterization according to Case 3. Here, the system's measured outputs and selected time derivatives of these signals are fed back after a suitable low-pass filtering, parameterized according to the following subsections.  
In the case of an ideal (error-free) filtering, the closed-loop dynamics are guaranteed to be robustly stable if the controller gains \(\mathbf{K}_{\mathrm{y}}\) satisfy the following theorem representing a bilinear matrix inequality.  
**Theorem 1**: (_Sufficient stability condition for robust output feedback control_) _Robust asymptotic stability of the closed-loop control system according to Case 3 is ensured for an error-free output feedback (i.e., \(\mathbf{x}\equiv\mathbf{\hat{x}}\)) if the gain matrix \(\mathbf{K}_{\mathrm{y}}\) satisfies the bilinear matrix inequalities_  
\[\left(\mathbf{A}_{v}-\mathbf{B}_{v}\mathbf{K}_{\mathrm{y}}\mathbf{C}_{ \mathrm{y}}^{+}\mathbf{\hat{C}}\right)^{T}\mathbf{P}+\mathbf{P}\cdot\left( \mathbf{A}_{v}-\mathbf{B}_{v}\mathbf{K}_{\mathrm{y}}\mathbf{C}_{\mathrm{y}}^ {+}\mathbf{\hat{C}}\right)\prec 0\enspace, \tag{11}\]  
\(\mathbf{P}=\mathbf{P}^{T}\succ 0\)_, for all vertices \(v\in\{1,\ldots,n_{v}\}\) in (10)._  
Proof: The proof of Theorem 1 is a direct consequence of setting up sufficient stability conditions for each vertex system of a linear model with polytopic uncertainty representation. In this way, Equation (11) represents the Lyapunov inequalities to be satisfied for each vertex system according to [8; 33].  
Note, the matrix inequality (11) is bilinear due to multiplicative couplings between the yet unknown matrices \(\mathbf{K}_{\mathrm{y}}\) and \(\mathbf{P}\). The following corollary provides a possibility to transfer these stability requirements into computationally feasible LMIs including a linear equality constraint.  
Figure 2: Filter-based output feedback control structure according to Case 3.  
**Corollary 1**.: _An LMI formulation of Theorem 1 is obtained by introducing a linearizing change of variables with the positive definite, symmetric unknown matrix \(\mathbf{Q}=\mathbf{Q}^{T}=\mathbf{P}^{-1}\succ 0\), as well as the equality constraints_  
\[\mathbf{M}\mathbf{C}_{\mathrm{y}}^{+}\bar{\mathbf{C}}=\mathbf{C}_{\mathrm{y}}^ {+}\bar{\mathbf{C}}\mathbf{Q}\quad\text{and}\quad\mathbf{N}=\mathbf{K}_{ \mathrm{y}}\mathbf{M}\enspace, \tag{12}\]  
_for which \(\mathbf{C}_{\mathrm{y}}^{+}\cdot\bar{\mathbf{C}}\) was assumed to be precisely known, i.e., a point matrix, according to its definition in Equations (6) and (7). Substituting the relations (12) into (11) and multiplying the matrix inequality form the left and right by \(\mathbf{Q}\) yield the LMIs_  
\[\mathbf{A}_{\mathrm{y}}\mathbf{Q}+\mathbf{Q}\mathbf{A}_{\mathrm{y}}{}^{T}- \mathbf{B}_{\mathrm{y}}\mathbf{N}\mathbf{C}_{\mathrm{y}}^{+}\bar{\mathbf{C}}- \left(\mathbf{C}_{\mathrm{y}}^{+}\bar{\mathbf{C}}\right)^{T}\mathbf{N}^{T} \mathbf{B}_{\mathrm{y}}{}^{T}\prec 0 \tag{13}\]  
_to be jointly satisfied for each vertex system \(v\in\{1,\ldots,n_{\mathrm{y}}\}\)._  
_If the matrix \(\mathbf{C}\) has full row rank, the algebraic constraint in (12) ensures that \(\mathbf{M}\) has full rank and that it is therefore invertible. Then, the resulting controller gain is given by [33]:_  
\[\mathbf{K}_{\mathrm{y}}=\mathbf{N}\mathbf{M}^{-1}\enspace. \tag{14}\]