Multiplicative approximation of linear multivariable systems with L∞ error bounds

It is shown that a p^xm transfer function G(s) with p⩾m can be decomposed as G = (I-ν_NΔ_N)^-1...(I-ν_r+2Δ_r+2)^-1 (I-ν_r=1Δ_r=1)^-1G_O where Δ_i are stable all-pass transfer functions, 1=ν₁..=ν_r≫ν_r=1..≫ν_N≫0 are the Hankel-singular-values of GW*^-1 where G*G=WW* with W stable and minimum phase. Results on the McMillan degree of G^Λ:=(I-ν_iΔ_i)..(I-ν_NΔ_N)G then show that G^Λ gives a good low order approximation to G in the sense of relative error.