Synthesis of controllers for arbitrary pole placement in discrete plants including unstable zeros with extensions to adaptive control

This paper presents new simultaneous pole and zero placement schemes based on the use of feedforward and feedback compensators. The main novelty is that the plant is allowed to possess unstable zeros which are not required to be included as zeros of the reference model. The control objective of model-following is achieved by including a feedforward compensator between either the reference or the control signals and the plant output which is incorporated to the usual schemes consisting in a precompensator and a feedback compensator. Basically, two control schemes are presented, but some extensions are given allowing signal feedforward injections at intermediate state variables of the plant instead of the standard injections at the output and the use of blocking filters for the significant frequencies of the output of the feedforward compensator. These schemes lead to two diophantine equations, one being used for standard pole placement while the other being basically used for the placement of zeros in prescribed positions. Several numerical simulations are given to illustrate the capability of the schemes for simultaneous pole and zero placement in the absence and in the presence of noise and their application for adaptive control.