Characterising discrete-time linear systems with the "mixed" positive real and bounded real property

Matheus Souza, Wynita M. Griggs, Rodrigo H. Ordóñez-Hurtado, S. Shravan K. Sajja, Alexander Lanzon, Robert N. Shorten

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3 Citations (Scopus)


In this paper, we present characterisations of linear, shift-invariant, discrete-time systems that exhibit mixtures of small gain-type properties and positive real-type behaviours in a certain manner. These "mixed" systems are already fairly well characterised in the continuous-time domain, but the widespread adoption of digital controllers makes it necessary to verify whether commonly used discretisation procedures preserve the characteristic of "mixedness". First, we analyse the effects of classical discretisation methods on the "mixed" property using Nyquist methods. A frequency domain feedback stability result is then presented. Finally, we develop a spectral-based characterisation of "mixed" discrete-time systems which provides a practical computational test that can also be applied to the MIMO case. Several examples validate the developed theory.

Original languageEnglish
Pages (from-to)259-268
Number of pages10
JournalEuropean Journal of Control
Issue number5
Publication statusPublished - Sep 2014
Externally publishedYes


  • Discrete-time system
  • Finite frequency bounded realness
  • Finite frequency positive realness
  • Linear system
  • Stability

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