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Multidimensional Systems and Signal Processing

, Volume 23, Issue 3, pp 381–411 | Cite as

\({{\mathcal H}_{\infty}}\) control of linear multidimensional discrete systems

  • Zhi-Yong Feng
  • Qinghe Wu
  • Li Xu
Article

Abstract

This paper presents a comprehensive investigation on the \({{\mathcal H}_{\infty}}\) control problem of linear multidimensional (nD) discrete systems described by the nD Roesser (local) state-space model. A Bounded Real Lemma consisting of a series of conditions is first established for general nD systems. The proposed nD conditions directly reduce to their 1D counterparts when n = 1, and besides several sufficient conditions which include the existing 2D results as special cases, some necessary and sufficient conditions are also shown to explore further insights to the considered problem. By applying a linear matrix inequality (LMI) condition of the nD Bounded Real Lemma, the nD \({{\mathcal H}_{\infty}}\) control problem is then considered for three kinds of control laws, namely, static state feedback (SSF) control, dynamic output feedback (DOF) control and static output feedback (SOF) control, respectively. The nD \({{\mathcal H}_{\infty}}\) SSF and DOF control problems are formulated in terms of an LMI and LMIs, respectively, and thus tractable by using any available LMI solvers. In contrast, the solution condition of the nD \({{\mathcal H}_{\infty}}\) SOF controller is not strictly in terms of LMIs, therefore an iterative algorithm is proposed to solve this nonconvex problem. Finally, numerical examples are presented to demonstrate the application of these different kinds of nD \({{\mathcal H}_{\infty}}\) control solutions to practical nD processes as well as the effectiveness of the proposed methods.

Keywords

Multidimensional discrete systems Roesser model Stability Bounded Real Lemma \({{\mathcal H}_{\infty}}\) control Linear matrix inequality 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Electronics and Information SystemsAkita Prefectural UniversityYuri-HonjoJapan
  2. 2.Department of Automatic ControlBeijing Institute of TechnologyBeijingChina

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