The name is absent

35

of the controllability gramian is much slower than that of the observability gramian,
which can be understood intuitively by the fact that there is only one output but
the reduced system must be able to accurate represent the action of many inputs. In
this case, these Lyapunov solvers converge slowly and are not of practical use in this
setting.

Krylov methods, on the other hand, construct approximate reduced systems of a
given dimension from the beginning. Instead of transforming the original system and
then truncating, the Krylov process iteratively projects the dynamics of the original
system onto a smaller subspace. This reduces the memory requirements significantly,
since only matrix-vector products are used, and in turn drastically speeds up the
reduction process. We now describe the main ideas behind this algorithm; for full
details see (Gugercin et al., 2008).

Consider the quasi-active system given in (2.33) and (2.34). We construct two
matrices, V_fc, W⅛ ∈ R^yvxfc such that z(t) = V_fcξ(t) for some ξ(i) ∈ R^fc and such that

WΓ (V_fcξ'(i) - AV_fcξ(i) - Bu(t)) = 0                (2.37)

and

Range(V_fc) ∩ Null(Wj∏ = {0}.                   (2.38)

From (2.38) we see that W⅛V_fc is invertible. Hence we can use (2.37) to construct a

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