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IMA Journal of Mathematical Control and Information Advance Access originally published online on November 20, 2006
IMA Journal of Mathematical Control and Information 2007 24(3):411-423; doi:10.1093/imamci/dnl032
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© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

On the geometry of stability regions of Smith predictors subject to delay uncertainty

Constantin-Irinel Morarescu{dagger}

HeuDiaSyC (UMR CNRS 6599), Université de Technologie de Compiègne, Centre de Recherche de Royallieu, BP 20529, 60205 Compiègne, France and Department of Mathematics, University ‘Politehnica’ of Bucharest, Romania

Silviu-Iulian Niculescu{ddagger}

Laboratoire de Signaux et Systèmes (L2S), Supélec, 3, rue Joliot Curie, 91190 Gif-sur-Yvette, France

Keqin Gu§

Department of Mechanical and Industrial Engineering, Southern Illinois University at Edwardsville, Edwardsville, IL 62026-1805, USA

{dagger} Email: constantin.morarescu{at}math.pub.ro

{ddagger} Corresponding author. Email: Silviu.Niculescu{at}lss.supelec.fr. On leave from HeuDiaSyC (UMR CNRS 6599), Université de Technologie de Compiègne, Centre de Recherche de Royallieu, BP 20529, 60205, Compiègne, France

§ Email: kgu{at}siue.edu

Received on May 4, 2006; Accepted on September 13, 2006

In this paper, we present a geometric method for describing the effects of the ‘delay-induced uncertainty’ on the stability of a standard Smith predictor control scheme. The method consists of deriving the ‘stability crossing curves’ in the parameter space defined by the ‘nominal delay’ and ‘delay uncertainty’, respectively. More precisely, we start by computing the ‘crossing set’, which consists of all frequencies corresponding to all points on the stability crossing curve, and next we give their ‘complete classification’, including also the explicit characterization of the ‘directions’ in which the zeros cross the imaginary axis. This approach complements existing algebraic stability tests, and it allows some new insights in the stability analysis of such control schemes. Several illustrative examples are also included.

Keywords: delay stability; robustness; Smith predictor.


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