Global asymptotic linearisation of the pole placement map: a closed-form solution for the constant output feedback problem

Generic eigenvalue assignment by memoryless real output feedback

“…The criterion is based on a linearization around a dependent compensator, an idea which appeared first in purely geometric language in [12] and was later worked out in more elementary terms independently by Leventides [8,10], Wang [13] and Ariki [1].…”

“…We believe that the example below comes reasonably close to being as simple as possible. Actually it is somewhat easier to produce an example (and also to do the polynomial construction of a dependent compensator) if one starts from kernel CAR') representations [15] and for this compare with [12,10]. For reasons explained in the introduction we choose here to stay within the (A,B,C) framework.…”

“…Only later it was independently recognized by Leventides [8,10], Wang [13] and Ariki [1] that the geometric techniques used by Wang in [12] are actually based on a linearization of the pole placement map around a so-called 'dependent' compensator. All these papers use polynomial representations.…”

Reduced Sensitivity Solutions to Global Linearisation of the Pole Assignment Map

“…The Determinantal Assignment Problem (DAP) belongs to the family of algebraic synthesis methods and has arisen as the abstract problem formulation of pole, zero assignment of linear systems (Karcanias and Giannakopoulos, 1984;Leventides and Karcanias, 1995) and thus unifies the study of the corresponding frequency assignment problems of multivariable systems, which have a determinantal character, under constant, dynamic centralised, or decentralised control structure. The DAP demonstrates the significance of exterior algebra and classical algebraic geometry for control problems.…”

“…An equivalent sufficient condition for arbitrary pole assignment is that the linearisation matrix (referred also as "Blow-up" matrix) should have full rank. The paper is organized in the following way: Section 2 summarizes the theoretical background and results for the Global Linearisation method established in (Leventides and Karcanias, 1995) and formulates the constant DAP problem as Static Output Feedback (SOF) pole placement problem. In Section 3 the computational framework is introduced and a proposed numerical algorithm based on a Predictor-Corrector iterative method is presented which guarantees the maximum distance from the degenerate compensator.…”