36
reduced model
ɪ = Afeξ(i) + Bfeu(t), y(t) = Cfeξ(i), (2.39)
where
Afc = (WfVfc)-1WfcAVfe, Bfc = (WfVfc)-1WfB, Cfe = CVfe. (2.40)
The reduced-order system is computed by finding V∕c and W⅛ so that the L2-n0rm
of the error between the transfer functions of the original and reduced systems along
the imaginary axis is minimized, i.e., we solve the optimization problem
min
Vb Wk
IlCτ(iωl - A)-1B - Cf(zωl - Afc)-1Bfc∣∣2dω.
One strategy for solving this is to interpolate the full transfer function, to first order,
at the negative of each of its poles. Since these poles are not generally known a
priori and may be hard to compute, we make an initial guess and then iterate until
convergence, indicating that we have arrived at the reduced system. This is achieved
in a computationally efficient manner via the Iterative Rational Krylov Algorithm
(IRKA), whose implementation details are found in (Gugercin et al., 2008).
I give the IRKA algorithm here for the multiple-input multiple-output (MIMO)
case, which is exactly what we have in the quasi-active system.