Frequency Assignment Problems in Linear Multivariable Systems: Exterior Algebra and Algebraic Geometry Methods

Abstract

The determinantal assignment problem (DAP) is defined as a generalisation of the pole, zero assignment problems of linear multivariable theory. The multilinear nature of DAP is reduced to a linear problem of zero assignment of polynomial combinants and a standard problem of decomposability of multivectors. The characterisation of the system problems by decomposable polynomial multivectors leads to the definition of the various system Plücker matrices. Necessary conditions for the solvability of frequency assignment problems are given in terms of the new system invariants, the Plücker matrices. The multilinear problem of decomposability is characterised by a minimal set of algebraically independent quadratics, the Reduced Quadratic Plücker Relations (RQPR). The set of RQPRs is used for the study of linearising compensators (feedbacks), and a linearising family of feedbacks superior to that of dyadic feedbacks is defined. A new proof to the pole assignment theorem by state feedback is given. The approach unifies the various problems of frequency assignment, provides a common algebrogeometric framework for their study, and establishes the basis for the development of a common algorithmic procedure for the computation of solution.