## 2 Cooperativity-Enforcing Similarity Transformations  
### Special Case: Linear and Quasi-Linear Systems with Bounded Parameters  
As a special case of the general system model introduced in (1), assume the parameter-dependent linear ODEs  
\[\dot{\mathbf{x}}(t)=\mathbf{A}(\mathbf{p})\cdot\mathbf{x}(t)\enspace. \tag{12}\]  
Here, the parameter vector \(\mathbf{p}\) can either be uncertain but constant or time-varying within its interval bounds. For a scenario where the latter case was accounted for during a robust control synthesis, the reader is referred to [33]. For a compact notation of the system model (12), which contains all possible parameter-dependent realizations of \(\mathbf{A}(\mathbf{p})\), we introduce an interval matrix \([\mathbf{A}]\) that is formed of the bounds \(\big{[}A_{i,j}\big{]}\) for each entry \(i,j\in\{1,\ldots,n\}\). Those bounds are obtained according to the generally not minimal interval extension [30; 31]  
\[ \begin{cases}\mathbf{p}\in[\mathbf{p}]\\ \mathcal{A}_{i,j}=A_{i,j}(\mathbf{p})\end{cases} \quad\implies\mathcal{A}_{i,j }\in\big{[}A_{i,j}\big{]}=\big{[}\underline{A}_{i,j}\;;\ \overline{A}_{i,j}\big{]}=\big{[}A_{i,j}\big{]}([\mathbf{p}])\enspace. \tag{13}\]  
In this paper, it is foremost desired to find a time-invariant change of coordinates for the case that the inequalities  
\[\underline{A}_{i,j}\geq 0 \tag{14}\]  
are not satisfied for at least one \(i,j\in\{1,\ldots,n\}\), \(i\
eq j\), i.e., that the uncertain linear system model is not proven to be cooperative. The change of coordinates should then lead to a set of state equations  
\[\dot{\mathbf{z}}(t)=\hat{\mathbf{A}}\cdot\mathbf{z}(t)\quad\text{with}\quad \mathbf{z}(t):=\boldsymbol{\Theta}^{-1}\cdot\mathbf{x}(t) \tag{15}\]  
with an interval-valued system matrix \(\hat{\mathbf{A}}\in[\hat{\mathbf{A}}]\) with non-negative off-diagonal elements \(\underline{\hat{A}}_{i,j}\geq 0\) for each \(i\
eq j\in\{1,\ldots,n\}\). Unfortunately, this change of coordinates may lead to the case that--despite the original system model (12) was asymptotically stable for all possible \(\mathbf{p}\in[\mathbf{p}]\)--the transformed matrix \(\big{[}\hat{\mathbf{A}}\big{]}\) contains unstable realizations due to the wrapping effect. Especially in the area of control engineering, dynamic system models are often not purely autonomous as stated in (12). They typically contain additive input terms \(\mathbf{B}\cdot\mathbf{u}(t)\). Due to the fact that these terms can be included additively in Equation (12), they can also be included additively in the transformed model (15) by the term  
\[\boldsymbol{\Theta}^{-1}\cdot\mathbf{B}\cdot\mathbf{u}(t)\enspace, \tag{16}\]  
where uncertainty needs to be taken into account by the same \(\inf(\cdot)\) and \(\sup(\cdot)\) operators that are included subsequently in Equation (17). As described in the introduction, cooperativity (together with stability) allows for simplifying the simulation of the transformed system model (15). Guaranteed bounds \(\mathbf{z}(t)\in[\mathbf{z}](t)=[\mathbf{v}(t)\;;\mathbf{w}(t)]\) of all reachable states in the new coordinate frame \(\mathbf{z}\) are obtained by an evaluation of the following coupled set of state equations (which are a direct consequence of [35])  
\[ \begin{split}\dot{v}_{i}(t)&=\inf([\hat{A}_{i,i}]\cdot v _{i}(t))+\inf\left(\sum_{\begin{subarray}{c}j=1\\ i\
eq i\end{subarray}}^{n}[\hat{A}_{i,j}]\cdot[z_{j}](t)\right)\\ \dot{w}_{i}(t)&=\sup\bigl{(}[\hat{A}_{i,j}]\cdot w_{i}(t) \bigr{)}+\sup\left(\sum_{\begin{subarray}{c}j=1\\ j\
eq i\end{subarray}}^{n}[\hat{A}_{i,j}]\cdot[z_{j}](t)\right)\end{split}  \tag{17}\]  
with the resulting state bounds  
\[[z_{i}](t)=\left[v_{i}(t)\ ;\ w_{i}(t)\right]\enspace, \tag{18}\]  
where couplings between the vectors \(\mathbf{v}(t)\) and \(\mathbf{w}(t)\) can be ignored if the system is positive, i.e., \(v_{i}(t)\geq 0\) holds for all \(t\geq 0\) as discussed in (9)-(11).