Journal of Control Theory and Applications

, Volume 6, Issue 1, pp 37–44 | Cite as

Periodicity in Delta-modulated feedback control

  • Xiaohua Xia
  • Guanrong Chen
  • Rudong Gai
  • Alan S. I. Zinober
Article
First Online:
Received:

Abstract

The Delta-modulated feedback control of a linear system introduces nonlinearity into the system through switchings between two input values. It has been found that Delta-modulation gives rise to periodic orbits. The existence of periodic points of all orders of Sigma-Delta modulation with “leaky” integration is completely characterized by some interesting groups of polynomials with “sign” coefficients. The results are naturally generalized to Sigma-Delta modulations with multiple delays, Delta-modulations in the “downlink”, unbalanced Delta-modulations and systems with two-level quantized feedback. Further extensions relate to the existence of periodic points arising from Delta-modulated feedback control of a stable linear system in an arbitrary direction, for which some necessary and sufficient conditions are given.

Keywords

Switching Periodic orbit Delta-modulation Sign polynomial Nonlinear control 
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Copyright information

© Editorial Board of Control Theory and 2008

Authors and Affiliations

  • Xiaohua Xia
    • 1
  • Guanrong Chen
    • 2
  • Rudong Gai
    • 3
  • Alan S. I. Zinober
    • 4
  1. 1.Department of Electrical, Electronic and Computer EngineeringUniversity of PretoriaPretoriaSouth Africa
  2. 2.Department of Electronic EngineeringCity University of Hong KongHong KongChina
  3. 3.Department of Computation ScienceHuaiyin Institute of TechnologyHuai’an AnhuiChina
  4. 4.Department of Applied MathematicsThe University of SheffieldSheffieldUK

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