\(\mathcal{H}_\infty\) control of linear multidimensional discrete systems. (English) Zbl 1308.93074
Multidimensional Syst. Signal Process. 23, No. 3, 381-411 (2012); erratum ibid. 23, No. 3, 413 (2012).
Summary: This paper presents a comprehensive investigation on the \(\mathcal{H}_\infty\) control problem of linear multidimensional (\(n\)D) discrete systems described by the \(n\)D Roesser (local) state-space model. A Bounded Real Lemma consisting of a series of conditions is first established for general \(n\)D systems. The proposed \(n\)D 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 \(n\)D Bounded Real Lemma, the \(n\)D \(\mathcal{H}_\infty\) 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 \(n\)D \(\mathcal{H}_\infty\) 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 \(n\)D \(\mathcal{H}_\infty\) 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 \(n\)D \(\mathcal{H}_\infty\) control solutions to practical \(n\)D processes as well as the effectiveness of the proposed methods.
MSC:
| 93B36 | \(H^\infty\)-control |
| 93C55 | Discrete-time control/observation systems |
| 93C05 | Linear systems in control theory |
| 93C35 | Multivariable systems, multidimensional control systems |
Keywords:
multidimensional discrete systems; Roesser model; stability; bounded real lemma; \(\mathcal{H}_\infty\) control; linear matrix inequalityReferences:
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