Geometric techniques for implicit two-dimensional systems

Abstract

Geometric tools are developed for two-dimensional (2-D) models in an implicit FornasiniMarchesini form. In particular, the structural properties of controlled and conditioned invariance are defined and studied. These properties are investigated in terms of quarter-plane causal solutions of the implicit model given compatible boundary conditions. The definitions of controlled and conditioned invariance introduced, along with the corresponding output-nulling and input-containing subspaces, are shown to be richer than the one-dimensional counterparts. The analysis carried out in this paper establishes necessary and sufficient conditions for the solvability of 2-D disturbance decoupling problems and unknown-input observation problems. The conditions obtained are expressed in terms of output-nulling and input-containing subspaces, which can be computed recursively in a finite number of steps.

Self-boundedness and self-hiddenness for implicit two-dimensional systems

“…for all ∈ {0, 1, 2}, [15], [16]. Invariant subspaces are useful tools in the investigation of the so-called compatible boundary conditions of (1), i.e., the boundary conditions…”

“…The following lemma [15], [16] shows the relation between the concept of invariance for (E; A 0 , A 1 , A 2 ) defined here and the existence of compatible solutions for Σ. Lemma 1: [15], [16]. Subspace J of X is invariant for (E; A 0 , A 1 , A 2 ) if and only if (1) has a solution {x i, j ∈ J | (i, j) ∈ B} for any J -valued boundary condition with zero input.…”

“…Therefore, it admits a maximum element, which is given by the sum of all invariant subspaces of (E; A 0 , A 1 , A 2 ). This subspace, herein denoted by J ⋆ , can be computed using the following result, see [15], [16].…”

“…Lemma 2: [15], [16]. The subspace J ⋆ can be computed as the last term of the monotonically non-increasing sequence of subspaces {J i } i∈N given by…”

H−/H∞ fault detection observer design for two-dimensional Roesser systems

“…Based on state-space models, several properties have been investigated, such as stability [5,6], controllability and observability [7,8], etc. The problems of feedback control and filtering for 2-D systems have been extensively studied, e.g., [4,[9][10][11][12][13][14][15][16][17].…”