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

This letter proposes a novel robust interval observer for a two-dimensional (treated as a synonym for a double-indexed system) linear time-invariant discrete-time system described by the Fornasini-Marchesini second model. This system is subject to unknown but bounded state disturbances and measurement noise. Built on recent interval estimation strategies designed for one-dimensional systems, the proposed observer is based on the introduction of weighting matrices which provide additional degrees of freedom in comparison with the classical structure relying on a change of coordinates. Linear matrix inequality conditions for the exponential stability and peak-to-peak performance of a two-dimensional system described by the Fornasini-Marchesini second model are then proposed, and applied to the design of a robust interval observer. Numerical simulation results are provided to show the efficiency of the proposed estimation strategy.

I. Introduction

Since their introduction in the second half of the nineteen seventies, two-dimensional (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]. 2D systems can be used to represent many physical processes [6] such as image processing [2], [7], repetitive industrial processes [8], spatio-temporal systems of which the behavior is governed by hyperbolic partial differential equations [9] or the task of iterative learning control synthesis [10]. 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.¹

Following the majority of the literature on the subject, the name “two-dimensional system” is used here to refer to double-indexed systems.