nent. Hence, it is natural to decompose H(s) as H(s) = HLTI(s) AH(s) (3.77) = (s)1(s)) (3.78) Here, HLTI (s) is a diagonal HTM corresponding to the diagonal (LTI subsection) of H(s) while -AH(s) contains the off-diagonal elements. H(s) is dominated by HLTI(s) if the major part of the output signal's energy is generated by this LTI component, or (3.79) (See appendix A for a discussion on HTM norms.) Assuming the presence of a dominant LTI component, i.e. with (3.79) being satisfied, the inverse of H(s) can be approximated by means of the series expansion (3.80) With the inverse of the diagonal HTM HLTI(s) being straightforward to compute, this expansion requires only matrix multiplications and additions. Therefore, (3.80) can be evaluated using both numerical and symbolic techniques. For practical purposes, the expansion is truncated after a finite number of terms R. Note that, especially for symbolic computations, use of (3.80) is only practical if R can be kept low, i.e. if (3.80) converges sufficiently fast. Note that the theoretical justification of the expansion (3.80) builds on the Banach fixed-point theorem [Cor91, Huts80, Vana02c]. 3.4.3.3 Rank-reduced approximation Another strategy assumes that the HTM H(s) that is to be inverted can be approximated H(s)>Z(s)+U(s)V(s)7 (3.81) i.e. the sum of an HTM Z(s) whose inverse can be determined at low computational cost and a low-rank update U(s)V(s)T(4). Here, U(s) and V(s) are column matrices with a finite number M of columns. Formally, U(s), V(s) E CooxM For the feedback loop in Fig. 3.14, the approximation (3.81) is possible if, for example, the open-loop gain G(s) = is nearly rank-deficient. In such a case Z(s) = I. Often, the such an approximation only holds for S belonging to a bounded subset of the complex plane. 4For this approximation to be valid, the largest singular value of the approximation error matrix should be well below the smallest singular value of (3.81), i.e. Omax (H(s) -Z(s)+U(s)V(s)T) << Omin(Z(s) - + U(s)V(s)T). 3.4.3 Feedback systems and H