## 3 Novel Subdivision-Based State-Space Transformation  
As the bridging element between both transformations summarized in the previous Sections 2.3 and 2.4, a novel subdivision-based procedure is introduced. Its basic components are summarized in the following Algorithms 1-4. In a first stage, the procedure given in Section 2.3 for systems with real eigenvalues is employed (line 1 in Algorithm 1). The novel components of this algorithm are activated in the case that--after a predefined wall-clock time--no suitable transformation of the uncertain system model into an interval-valued Metzler matrix has been found, or if the resulting transformed model contains unstable realizations despite provable asymptotic stability of the original state Equation (12). To reduce the wrapping effect during the transformation of the state equations, this novel algorithm does not only try to find transformations for the complete parameter domain \(\mathcal{L}^{(*)}\).\(\bar{\mathbf{p}}\) (which denotes a storage element of the currently investigated interval box for the system parameter) as a whole. In addition, the list elements of yet undecided parameter subintervals (i.e., those intervals for which a solution of the desired transformation has not yet been found) also carry several \(\mathcal{L}^{(*)}\).\(N\) subintervals, which are all stored in \(\mathcal{L}^{(*)}\).\(\mathbf{p}_{1}\). In such a way, it becomes possible to individually investigate each of the corresponding subboxes, whether at least for some of them the algorithm of Section 2.3 produces stable realizations of the transformed system model. According to Remark 1, a stopping criterion needs to be implemented which avoids an endless search for point-valued transformations. Currently, this is done by checking whether a timeout occurred (line 1 in Algorithm 1). Suitable alternatives would be progression rates for the parameter \(\mu\) which fall below a certain threshold or a maximum number of non-successful trials. If this stopping criterion becomes active, the stable intervalstransformed into interval-valued Metzler matrices--are appended to the list \(\mathcal{L}_{\text{s}}\) (starting in line 1 of Algorithm 1, as well as Algorithm 2). ```
input :Initial parameter domain \([\mathbf{p}]\) and symbolic expression for the parameter-dependent system matrix \(\mathbf{A}(\mathbf{p})\) output :List \(\mathcal{L}_{\text{s}}\) of stable interval-valued Metzler matrices \([\hat{\mathbf{A}}]^{\langle\hat{i}\rangle}\) and corresponding transformation matrices \(\mathbf{\Theta}_{i}=\mathbf{V}_{i}\cdot\mathbf{S}_{i}\) initialize the list \(\mathcal{L}_{\text{u}}\) of undecided intervals with the element \(\mathcal{L}^{\langle 1\rangle}\) according to \(\mathcal{L}^{\langle 1\rangle}.N:=1\), \(\mathcal{L}^{\langle 1\rangle}.\mathbf{p}_{1}:=[\mathbf{p}]\), \(\mathcal{L}^{\langle 1\rangle}.\mathbf{\bar{p}}:=[\mathbf{p}]\), \(\mathcal{L}^{\langle 1\rangle}.\mathbf{A}_{1}:=\mathbf{A}([\mathbf{p}])\), \(\mathcal{L}^{\langle 1\rangle}.\bar{\mathbf{A}}:=\mathbf{A}([\mathbf{p}])\), \(\mathcal{L}^{\langle 1\rangle}.v:=\text{vol}\Big{(}\mathcal{L}^{\langle 1 \rangle}.\bar{\mathbf{p}}\Big{)}\); initialize the empty list \(\mathcal{L}_{\text{s}}\) of stable transformations; set \(i^{*}:=1\); while\(\mathcal{L}_{\text{u}}\
eq\emptyset\)do search for a transformation \(\mathbf{\Theta}_{i^{*}}=\mathbf{V}_{i^{*}}\cdot\mathbf{S}_{i^{*}}\) of \(\mathcal{L}^{\langle i^{*}\rangle}\) into a cooperative state-space representation acc. to Section 2.3; iftransformation found before timeoutthen ifat least one \(\mathbf{\Theta}_{i^{*}}^{-1}\cdot\Big{(}\mathcal{L}^{\langle i^{*}\rangle}.