Control systems analysis for the Fornasini-Marchesini 2D systems model – progress after four decades

Abstract

The development of a control systems theory for multidimensional linear systems, i.e., systems with more than one independent variable is an area where a very significant volume of research is based on representations for the underlying dynamics introduced in the 1970s. One of these is the Fornasini-Marchesini state-space model published in 1978. This paper reviews the significant developments in the succeeding four decades in both theory and application, where the second area includes examples where this model description is an enabler the analysis of other classes of systems.

Robust Interval Observer for Systems Described by the Fornasini–Marchesini Second Model

“…S INCE their introduction in the second half of the nineteen seventies, two-dimensional 1 (2D) systems have been widely studied [1]. Such systems are described by different state-space models such as the ones introduced by Roesser [2], Fornasini and Marchesini [3], [4] or Kurek [5].…”

“…Extensive studies of 2D system properties such as stability, controllability, observability, etc. have been conducted [1], [6]. Finally, several control [7], [8], [11] and estimation [7], [12]- [16] strategies have been investigated.…”

“…Rauh is with Carl von Ossietzky Universit ät Oldenburg, Department of Computing Science, Group: Distributed Control in Interconnected Systems, 26111 Oldenburg, Germany (e-mail: andreas.rauh@unioldenburg.de) T.N. Dinh and T. Raïssi are with Conservatoire National des Arts et M étiers (CNAM), Cedric-Laetitia, 292 Rue St-Martin, 75141, Paris Cedex 03, France (e-mail: ngoc-thach.dinh@lecnam.net, tarek.raissi@cnam.fr) 1 Following the majority of the literature on the subject, the name "twodimensional system" is used here to refer to double-indexed systems. contain the state vector when a system is subject to bounded disturbances and measurement noise [17]- [19].…”

Derivation and reduction of the singular Fornasini–Marchesini state‐space model for a class of multidimensional systems

“…Singular multidimensional (n-D) systems appear naturally in real systems such as thermal processes [1], spatially interconnected systems [2] and linear fractional representations for robust control applications [3]. In support of the dynamic analysis and control synthesis, we often desire to derive a certain kind of n-D statespace models [4][5][6][7][8][9][10], typically the Roesser model or the Fornasini-Marchesini (F-M) (second) model, from a given transfer function matrix. A wide range of results are available for the realisation of standard n-D systems, see e.g.…”

Distributed Set-Membership Fusion Filtering for Nonlinear 2-D Systems Over Sensor Networks: An Encoding–Decoding Scheme

“…, 6) are shown in Fig.12, from which we can see that the Tr(Q(f ) i,j )is the smallest of all. Thus, Figs [4][5][6][7][8][9][10][11][12]. show the effectiveness of the proposed encoding-decoding-based distributed SMFF algorithm.…”