Fast optimal \(\mathcal H_2\) model reduction algorithms based on Grassmann manifold optimization (Q2865655)

The linear dynamical system NEWLINE\[NEWLINE \frac{dx}{dt}=A x(t)+B u(t),NEWLINE\]NEWLINE NEWLINE\[NEWLINE y(t)=C x(t),NEWLINE\]NEWLINE where \(x\in \mathbb R^{n}\), \(A\in \mathbb R^{n\times n}\), \(B\in \mathbb R^{n\times p}\), \(C \in \mathbb R^ {q\times p}\), \(y\in \mathbb R^{q}\), \(x(t_{0})=x_{0}\) is reduced, via projection by a special selecting matrix \(U\) such that \( U^{T}U=I\), to a lower-order system. This problem can be formulated as a minimization problem NEWLINE\[NEWLINE \min J(U)\quad \text{ on the set of Grassmann manifold}. NEWLINE\]NEWLINE The functional \(J(U)\) is obtained using the Laplace transform of the function \( y(t)\). This method is illustrated by numerical results for two problems: the heat transfer equation and the model of a spiral inductor.