33
described by (2.33) and (2.34). We now compose
T = Σ-1∕2T*L*, and T~1 = UZ^-1/2,
and note that the transformed gramians
P = TPT* and Q = T~*QT~1
are balanced and diagonal in the sense that
P= Q = Σ.
Moreover, they are the gramians of the transformed state, z = Tz, which itself is
governed by the transformed dynamical system
= Âz(t) + Bu(Z), y (Z) = Cz(Z),
(2.35)
where A = TAT-1, B = TB, C = CT-1. Based on the decay of the singular values
in Σ, we can construct a reduced model by using only the к largest singular values.
This corresponds to approximating (2.35) with
ɪ = A11ξ(Z) + Biu(Z), y(Z) = C1ξ(Z),
(2.36)