Positive real control of two-dimensional systems: Roesser models and linear repetitive processes

Abstract

This paper considers the problem of positive real control for two-dimensional (2-D) discrete systems described by the Roesser model and also discrete linear repetitive processes, which are another distinct sub-class of 2-D linear systems of both systems theoretic and applications interest. The purpose of this paper is to design a dynamic output feedback controller such that the resulting closed-loop system is asymptotically stable and the closed-loop system transfer function from the disturbance to the controlled output is extended strictly positive real. We first establish a version of positive realness for 2-D discrete systems described by the Roesser state space model, then a sufficient condition for the existence of the desired output feedback controllers is obtained in terms of four LMIs. When these LMIs are feasible, an explicit parameterization of the desired output feedback controllers is given. We then apply a similar approach to discrete linear repetitive processes represented in their equivalent 1-D state-space form. Finally, we provide numerical examples to demonstrate the applicability of the approach.

On the Kalman–Yakubovich–Popov lemma and the multidimensional models

“…No design technique is proposed. However, the authors would like to stress the link between some existing synthesis LMI conditions, such as those in Gałkowski et al (2003a,b,c), Xu et al (2003) for example, and the framework presented here. Actually, control conditions provided in the above references are based upon analytical conditions that can be subsumed to the present version of n-D KYP lemma.…”

“…For multidimensional linear models, few contributions exist. However, the 2-D discrete Roesser models are considered from the viewpoint of positive realness in Xu et al (2003). Hence, we make an attempt to extend those results to investigate positive realness of hybrid Roesser models.…”

“…Recently, much effort has been dedicated to establishing KYP lemma for n-D systems. Although few special instances of the KYP lemma have been implicitly addressed for 2-D models (Xu et al 2003(Xu et al , 2005, no real general result has been provided yet. This paper aims at filling this gap.…”

Model approximation for two-dimensional Markovian jump systems with state-delays and imperfect mode information

“…The studies of 2-D systems are motivated by their extensive applications in many modern engineering fields, such as process control (thermal processes, gas absorption, water stream heating), multi-dimensional digital filtering, and image processing (enhancement, deblurring, seismographic data processing) Kaczorek (1985); Lu and Antoniou (1992); Lin (1999); Lin et al (2001) and so on. Correspondingly, various issues on 2-D systems, including stability analysis, stabilization, and filtering design, have been investigated, and a large volume of open literature has been available Xu et al (2002Xu et al ( , 2003; Xie et al (2002); Lam et al (2004); Paszke et al (2004); Hoang et al (2005); Peng and Guan (2009).…”

ℋ∞Control for Two-Dimensional Markovian Jump Systems with State-Delays and Defective Mode Information

“…[3]. In particular, since the well-known Roesser statespace model and the Fornasini-Marchesini local state-space (FMLSS) model were proposed, theory on 2D systems has progressed greatly [4][5][6][7][8][9][10][11].…”