Simultaneous determination of ordering and amplifications of cascaded subsystems

Letn linear, time-invariant subsystems with transfer functionsT ₁(s),T ₂(s),...,T _n (s) be given. We are required to cascade these subsystems into a system with the transfer functionT(s)=kT ₁(s)T ₂(s) ...T _n (s), wherek>0, by determining the order of the subsystems and choosing for each subsystemi an amplification factork _i such thatk=k ₁ k ₂ ...k _n .We do this using the following optimality criterion: Given a setI, minimize the ratio of the maximum and the minimum of all the absolute values of all the intermediate outputs inI, for a flat input. This criterion is applicable in cases where the intermediate outputs should not be overdriven or underdriven. In particular, the method is applicable to the problem of determining an optimal ordering and intermediate amplifications of cascadedRC-active filters.The author is indebted to E. Lueder (Ref. 1) and J. Tow, who presented various criteria for optimal arrangements ofRC-active filters and motivated the formulation of this problem.