Structure assignment problems in linear systems: Algebraic and geometric methods

Abstract

The Determinantal Assignment Problem (DAP) is a family of synthesis methods that has emerged as the abstract formulation of pole, zero assignment of linear systems. This unifies the study of frequency assignment problems of multivariable systems under constant, dynamic centralized, or decentralized control structure. The DAP approach is relying on exterior algebra and introduces new system invariants of rational vector spaces, the Grassmann vectors and Plücker matrices. The approach can handle both generic and non-generic cases, provides solvability conditions, enables the structuring of decentralisation schemes using structural indicators and leads to a novel computational framework based on the technique of Global Linearisation. DAP introduces a new approach for the computation of exact solutions, as well as approximate solutions, when exact solutions do not exist using new results for the solution of exterior equations. The paper provides a review of the tools, concepts and results of the DAP framework and a research agenda based on open problems.

Partially Fixed Structure Determinantal Assignment Problems

“…Therefore, more sophisticated methods have to be employed. This can be either the Global Linearization Method [9], [22], which can be applied when mp > ρ (this is the case here) or the methodology of the present technical note. This system has transfer function the following 3 × 3 rational matrix:…”