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Mathematical Problems in Engineering
Volume 2005 (2005), Issue 6, Pages 599-616

Several dissipativity and passivity implications in the linear discrete-time setting

Programa de Investigación en Matemáticas Aplicadas y Computación, Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152 AP. 14-805, Colonia San Bartolo Atepehuacan, México, DF 07730, Mexico

Received 22 September 2004; Revised 27 March 2005

Copyright © 2005 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Some important features and implications of dissipativity and passivity properties in the discrete-time setting are collected in this paper. These properties are mainly referred to as the stability analysis (feedback stability systems and study of the zero dynamics), the relative degree, the feedback passivity property, and the preservation of passivity under feedback and parallel interconnections. Frequency-domain characteristics are exploited to show some of these properties. The main contribution is the proposal of necessary and sufficient conditions in order to render a multiple-input multiple-output linear discrete-time invariant system passive by means of a static-state feedback and using the properties of the relative degree and zero dynamics of the system. A discrete-time model for the DC-to-DC buck converter is used as an example to illustrate the passivation scheme proposed. In addition, dissipativity frequency-domain properties are related to some feedback stability criteria.