Theory of LTR for non-minimum phase systems, recoverable target loops, and recovery in a subspace Part 1. Analysis

Abstract

A complete analysis of loop transfer recovery problem using full order observer based controllers for general not necessarily left invertible and not necessarily minimum phase systems is considered. The analysis here, while showing that neither exact nor asymptotic loop transfer recovery is in general possible, focuses on three fundamental issues. The first issue is concerned with what can and what canot be achieved for a given system and for an arbitrarily specified target loop transfer function, while the second issue is concerned with the development of necessary and/or sufficient conditions a target loop has to satisfy so that it can be either exactly or asymptotically be recovered for a given system. The third issue deals with the development of method(s)to test whether recovery is possible in a given subspace of the control space or not, i.e. to test whether projections of target and achievable sensitivity and complimentary sensitivity functions onto a given subspace match each other or not. Such an analysis pinpoints the limitations of the given system for the recovery of arbitrarily specified target loops via observer based controllers. These limitations are the consequences of the structural properties (i.e. finite and infinite zero structure, and invertibility) of the given system. Furthermore, the analysis discovers a multitude of ways in which freedom exists to shape the loops in a desired way as close as possible to the target shapes. Also, possible pole zero cancellations between the eigenvalues of the controller and the input and/or output decoupling zeros of the plant are characterized.