A geometric approach to 2-D implicit systems

Abstract

We consider the existence of solutions with certain geometric properties for the implicit Roesser model (IRM) and the second implicit Fornasini-Marchesini model. We define invariant subspaces and some geometric notions which do not have one-dimensional counterparts. Using these notions, we analyze a Fornasini-Marchesini-like model which encompasses two alternative representations of the IRM. For the IRM we consider boundary conditions on any side of the boundary to investigate existence of the solutions.We provide recursions to compute relevant subspaces for each model. We relate the 2-D output nulling (A,E,B)-invariant subspaces of a dual system to observer design. We construct an asymptotic state-space observer for each model by using geometric techniques, and include an illustrative example.

Conditioned invariance and unknown-input observation for two-dimensional Fornasini-Marchesini models

“…Over the last twenty years, several extensions of important geometric concepts, such as controlled invariance, have been proposed for 2-D latent variable models such as the Fornasini-Marchesini and Roesser state-space models, [5], [6], [12], [13]. While definitions of controlled invariance and output-nulling subspaces are not difficult to establish for FM-I, FM-II or Kurek models, a definition for conditioned invariant subspaces is less natural, since duality cannot be exploited as in the 1-D case.…”

“…In fact, Fornasini-Marchesini models are such that their dual is not in the form described by (1), or the class corresponding to the FM-I or FM-II models. To overcome this difficulty, definitions of conditioned invariance were proposed in [12] for two particular classes of models, motivated by the the search for duality properties similar to the 1-D case. The first is yet another subclass of (1) with A 0 = 0, B 1 = B 2 = 0, see also [8].…”

A geometric approach with stability for two-dimensional systems

“…Consider a basis adapted to J . By (8) it turns out that system (6-7) is asymptotically stable if and only if the two pairs…”

“…In the last two decades, many valuable results have been achieved in the attempt to develop a geometric theory for 2-D systems, [3], [8], [9], [12]. In particular, in [3] a definition of controlled invariance was proposed for Fornasini-Marchesini (FM) models.…”

Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems

“…The geometric approach to 2D linear systems was introduced by Conte and Perdon (1988), Conte et al (1991), Kaczorek (1992), Karmanciolu and Lewis (1990;1992). The problem of internally and externally stabilizing controlled and output-nulling subspaces for 2D FornasiniMarchesini models using state-feedbacks was investigated in (Ntogramatzis, 2010).…”