Discrete-time Zames-Falb Multipliers: Complete Search and Phase Properties

  • Shuai Wang

Student thesis: Phd


The stability of a feedback interconnection between an LTI system and a memoryless nonlinearity has aroused the interests of academic researchers. This thesis focuses on absolute stability analysis in the discrete-time domain. To date, Zames-Falb multipliers are considered as the widest class of multipliers that ensure the absolute stability of the Lur’e system with a slope restricted nonlinearity. To ensure the stability of the system, a search within the class of Zames-Falb multipliers is needed. A complete and convex search for discrete-time Finite Impulse Response (FIR) Zames- Falb multipliers is proposed. Specifically we search for noncausal multipliers with FIR structure of arbitrary order. The subclass is shown to be phase-equivalent to the class of discrete-time rational noncausal Zames-Falb multipliers. The search can be expressed as LMIs whose number of parameters increases quadratically, and whose number of linear constraints increases with model order. The search may be used for the case with either slope-restricted or odd slope-restricted nonlinearity. Favourable results are reported with respect to those in the literature. We derive phase limitations of the discrete-time Zames-Falb multipliers. In some cases the phase limitations may be used to show there exists no appropriate Zames-Falb multiplier for a Lur’e system with a given LTI plant and a given class of slope-restricted nonlinearities. We illustrate the results with some numerical examples and discuss the implications for benchmarking of stability tests. The phase limitation is applied to show that there is no direct counterpart to the discrete-time off-axis circle criterion. Specifically, no Zames-Falb multiplier can be found to keep the stability if the phase limitation is reached. This is confirmed by numerical counterexample. Motivated by phase properties of multipliers, the RL and RC multipliers are considered. In the continuous time, the RL and RC multipliers preserve the positivity of memoryless and monotone nonlinearities. In this thesis, the terms, multipliers with real poles and zeros, are used to describe this specific class of multipliers. We classify their discrete-time counterparts and analyse their phase properties. The classification of the discrete-time multipliers is richer than that of their continuous-time counterparts; their phase properties are less flexible.
Date of Award31 Dec 2017
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorWilliam Heath (Supervisor) & Joaquin Carrasco Gomez (Supervisor)


  • Absolute stability
  • Nonlinear system
  • Zames-Falb multipliers

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