Design of causal reversed-frame-normalizing controllers using bicausal expansions

Reverse-frame-normalizing controllers overcome the sensitivity problems of commutative controllers and balance the tolerance of multivariable feedback systems to input and output multiplicative unstructured uncertainty. They are, however, based on the singular value decomposition of transfer function matrices and lead to implementation difficulties on account of the irrational nature of the decomposition. Realizable approximations can be derived through the use of least-squares frequency response fit algorithms, but these are either nonlinear or require an a priori definition ofthe controller poles. In this paper we deploy a bicausal sequence representation for the singular value decomposition and derive conditions that overcome anticausality difficulties. This treatment leads to a characterization of the whole class of controllers and proves that frequency response targets for the generalized Nyquist diagrams cannot be defined arbitrarily. Finally we propose an algorithm for the systematic trade-off between the objective of achieving normality and that of reaching specific frequency response targets.