Fading-memory feedback systems and robust stability

Jeff S. Shamma, Rongze Zhao

Research output: Contribution to journalArticlepeer-review


This paper considers fading memory for nonlinear time-varying systems and associated problems of robust stability. We define two notions of fading memory for stable dynamical systems: uniform and pointwise. We then provide conditions under which stable linear or nonlinear systems exhibit uniform or pointwise fading memory. In particular, we show that (1) all stable discrete-time linear time-varying (LTV) systems have uniform fading-memory, (2) all stable continuous-time LTV systems have pointwise fading-memory, and (3) stable finite-dimensional continuous-time LTV systems have uniform fading-memory. We then show that a version of the small gain theorem which employs the asymptotic gain of a fading-memory system is necessary for the stable invertibility of certain feedback operators. These results are presented for both continuous-time and discrete-time systems using general ℓp or Lp notions of input/output stability and generalize existing results for ℓ2 stability. We further investigate fading memory in a closed-loop context. For linear plants, we parametrize all nonlinear controllers which lead to closed-loop pointwise fading memory.

Original languageEnglish (US)
Pages (from-to)191-200
Number of pages10
Issue number1
StatePublished - Jan 1993
Externally publishedYes


  • Nonlinear systems
  • robust control
  • robustness
  • system theory
  • time-varying systems

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Electrical and Electronic Engineering

Cite this