## 2 Cooperativity-Enforcing Similarity Transformations
### Mixed and Conjugate-Complex Eigenvalues  
If the original system (12) contains conjugate-complex eigenvalues or if a solution with the help of the previous approach cannot be found for the complete uncertainty domain, the following procedure can be applied. The most important differences in comparison with the previous approach are:  
1. the transformation is usually time-varying [1; 3];
2. mapping the uncertainty directly into the location of the eigenvalues turns the transformation into an interval-valued expression. Hence, evaluating the system matrix \(\mathbf{A}(\mathbf{p})\) in (12) for the whole range of parameters leads to a variability of the real and imaginary parts \(\sigma_{i}\) and \(\omega_{i}\) of the eigenvalues, where  
\[[\sigma_{i}]=[\underline{\sigma}_{i}\ ;\ \overline{\sigma}_{i}]\quad\text{and} \quad[\omega_{i}]=[\underline{\omega}_{i}\ ;\ \overline{\omega}_{i}] \tag{22}\]  
are their respective interval bounds. Typically, it is necessary to determine these bounds with the help of interval techniques. Possible solution procedures (verifyeig) are included in the Matlab toolbox IntLab[40]. Assume that the system under investigation contains \(\tilde{n}\) mutually disjoint conjugate-complex eigenvalue pairs with \(n^{*}=n-2\tilde{n}\geq 0\) as the number of remaining real eigenvalues. To simplify the notation, assume further that all eigenvalues are sorted so that all complex pairs are listed first, cf. [28]. Now, an equivalent system representation in block diagonal structure with  
\[\tilde{\mathbf{A}}=\mathrm{blkdiag}\big{(}\tilde{\mathbf{A}}_{1},\ldots, \tilde{\mathbf{A}}_{\tilde{n}+n^{*}}\big{)} \tag{23}\]  
can be obtained, where pairs of conjugate-complex eigenvalues with  
\[\omega_{i}=-\omega_{i+1}\,\ i\in\{1,3,\ldots,2\tilde{n}-1\}\ \, \tag{24}\]  
lead to the blocks  
\[\tilde{\mathbf{A}}_{j}\in \begin{bmatrix}[\sigma_{j}]&[\omega_{j}]\\ -[\omega_{j}]&[\sigma_{j}]\end{bmatrix}  \tag{25}\]  
with \(j\in\{1,2,\ldots,\tilde{n}\}\), while an uncertain real eigenvalue is reflected by  
\[\tilde{\mathbf{A}}_{j}\in[\sigma_{i}]\,\quad j=i-\tilde{n}\,\ i\in\{2\tilde{n}+ 1,2\tilde{n}+2,\ldots,n\}\ . \tag{26}\]  
The respective transformation between the original system matrix \(\mathbf{A}(\mathbf{p})\) and the representation in (23) is given by the column-wise partitioned matrix  
\[\tilde{\mathbf{T}}\in\big{[}\tilde{\mathbf{T}}\big{]}=\big{[}\big{[}\tilde{ \mathbf{T}}_{1}\big{]},\ldots,\big{[}\tilde{\mathbf{T}}_{\tilde{n}+n^{*}} \big{]}\big{]} \tag{27}\]  
with  
\[\big{[}\tilde{\mathbf{T}}_{j}\big{]}=\big{[}\Re\{[\mathbf{v}_{j}]\}\quad\Im\{ \big{[}\mathbf{v}_{j}\big{]}\}\big{]},\ j\in\{1,\ldots,\tilde{n}\}\ \, \tag{28}\]  
consisting of interval enclosures for the real and imaginary parts of the eigenvectors of the interval matrix \([\mathbf{A}]\) and the eigenvectors  
\[\big{[}\tilde{\mathbf{T}}_{j}\big{]}=\big{[}\mathbf{v}_{j}\big{]}\,\quad j=i- \tilde{n}\ \, \tag{29}\]  
for each of the real eigenvalues with \(i\in\{2\tilde{n}+1,2\tilde{n}+2,\ldots,n\}\). In a second stage, a (generally) time-varying transformation is performed according to  
\[\mathbf{z}=\mathbf{S}^{-1}(t)\cdot\tilde{\mathbf{z}}\ \, \tag{30}\]  
where  
\[\mathbf{S}(t)=\Big{(}\mathbf{S}^{-1}(t)\Big{)}^{T}=\mathrm{blkdiag}(\mathbf{ S}_{1}(t),\ldots,\mathbf{S}_{\tilde{n}+n^{*}}(t)) \tag{31}\]  
consists of the orthogonal blocks  
\[\mathbf{S}_{\tilde{j}}(t)= \begin{bmatrix}\cos(\omega_{j}t)&\sin(\omega_{j}t) \\ -\sin(\omega_{j}t)&\cos(\omega_{j}t)\end{bmatrix}