## 2 Cooperativity-Enforcing Similarity Transformations
### Purely Real Eigenvalues  
In this case, the transformation is based on the work of [PERSON] et al. in [2]. There, a general method to transform uncertain systems with purely real eigenvalues into a cooperative form (15) was presented. To extend its applicability towards larger uncertainty bounds, an extension was derived in [29], where the original approach was redesigned into an LMI-constrained optimization task [37] to make it more generally applicable and to simplify its practical use.  
We assume that the uncertain system matrix \([\mathbf{A}]\) can be enclosed by the element-wise defined inequality  
\[\mathbf{Z}_{\mathrm{a}}-\mathbf{\Delta}\leq\mathbf{Z}:=\mathbf{V}^{-1}\cdot \mathbf{A}\cdot\mathbf{V}\leq\mathbf{Z}_{\mathrm{a}}+\mathbf{\Delta} \tag{21}\]  
for all \(\mathbf{A}\in[\mathbf{A}]\) defined in (13), where \(\mathbf{Z}_{\mathrm{a}}=\mathbf{Z}_{\mathrm{a}}^{T}\) in (21) is a point matrix. Typical choices for \(\mathbf{Z}_{\mathrm{a}}\) are the midpoint of \([\mathbf{A}]\) if this matrix is diagonally dominant (with \(\mathbf{V}=\mathbf{I}\)) or the midpoint of a matrix obtained after a principal axis transformation using the invertible real-valued eigenvector matrix \(\mathbf{V}\).  
Next, a Metzler matrix \(\mathbf{R}=\mu\mathbf{E}_{n}-\mathbf{\Gamma}\) is searched for, which has the same eigenvalues as \(\mathbf{Z}_{\mathrm{a}}\). For that, \(\mu\in\mathbb{R}\) is a constant, \(\mathbf{E}_{n}\in\mathbb{R}^{n\times n}\) is a matrix with all elements equal to 1, and \(\mathbf{\Gamma}\in\mathbb{R}^{n\times n}\) is a diagonal matrix. Following the procedure detailed in [2], there exists an orthogonal matrix \(\mathbf{S}\in\mathbb{R}^{n\times n}\) such that \(\mathbf{S}^{T}\mathbf{Z}\mathbf{S}\) is Metzler with \(\mathrm{eig}(\mathbf{R})=\mathrm{eig}(\mathbf{Z}_{\mathrm{a}})\) and \(\mu>n||\mathbf{\Delta}||_{\mathrm{max}}\), where \(||\mathbf{\Delta}||_{\mathrm{max}}\) is the maximum absolute value of \(\mathbf{\Delta}\). The overall transformation according to (15) becomes \(\mathbf{\Theta}:=\mathbf{V}\cdot\mathbf{S}\), which is point-valued and time-invariant. The LMI-based optimization in [29] automatizes the search for the matrix \(\mathbf{S}\). It terminates successfully, if all realizations of the interval matrix \([\hat{\mathbf{A}}]:=\mathbf{\Theta}^{-1}\cdot[\mathbf{A}]\cdot\mathbf{\Theta}\) are Metzler.  
**Remark 1**.: _The LMI-based optimization may fail to find a solution because of two reasons. Either the LMI solver (e.g., \(\mathrm{SEDUMI}\) in combination with Yalmip[38; 39]) may produce the output of being infeasible. Then, it is ensured that the problem formulation does not have a suitable solution. However, previous work has shown that this is typically not the case. Mostly, the step size control procedure, which increases the parameter \(\mu\) defined above up to the point where \(\mu>n||\mathbf{\Delta}||_{\mathrm{max}}\) is satisfied, progresses too slowly to find a solution in acceptable time. This is an indicator that the intervals are too wide to find a common point-valued transformation for the uncertain system model._  
Instabilities and the failure of the LMI-based solution are countered by the subdivision in Section 3.