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Description
Model truncation is one of the oldest MOR methods for linear time invariant systems
\( E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad y(t)=Cx(t)+Du(t) \quad \quad (1) \)
The main idea is to construct the projection matrices as \(V=[x_1,\ldots,x_r], W=[y_1,\ldots,y_r]\) where the \(x_i, y_i\) are right and left eigenvectors corresponding to certain eigenvalues \(\lambda_i\) of \((A,E)\). The eigentriples \((\lambda_i,x_i,y_i)\) satisfy.
\( Ax_i=\lambda_iEx_i,\quad A^Hy_i=\overline{\lambda_i}E^Hy_i,\quad i=1,\ldots,r. \)