• Jingfan Zhang

Student thesis: Phd


A Lurye system is a feedback interconnection between a linear plant and a nonlinear operator that belongs to some class of nonlinear operators. The stability problem of Lurye systems has been being a core topic in control theory for the last 70 years due to the lack of necessary and sufficient conditions in general. In this thesis, we study the absolute stability of discrete-time Lurye systems where the nonlinearities are sector-bounded and slope-restricted. For completeness’ sake, we review the searches over two subclasses of Zames-Falb multipliers, infinite impulse response (IIR) and finite impulse response (FIR) Zames-Falb multipliers. The search leads to the lower bound of the maximum slope bound of the nonlinearities that preserve the absolute stability of a Lurye system. The numerical results obtained by FIR multipliers are less conservative than all existing literature. As a contribution, this thesis shows the equivalence between the second-order FIR searches and previous literature using Lyapunov functions. The main contribution of this thesis is the development of tractable duality conditions for Zames-Falb multipliers. The result allows us to find an upper bound of the maximum slope of the nonlinearities where linear time-invariant (LTI) Zames-Falb multipliers can guarantee the absolute stability. As a result, it is shown that the search over FIR Zames-Falb multipliers are very efficient as we can show that there is no suitable LTI multipliers when the FIR search fails to find a multiplier besides numerical precision. The significance of these results is two folds. First, they can be used as the stopping criterion in the iterative process in selecting the order of FIR multipliers. Second, they may be used to show unstable behaviour if the recent conjecture on the necessity of the LTI Zames-Falb multipliers for the absolute stability is shown to be true. Inspired by the sound result of FIR Zames-Falb multipliers, we propose a novel parametrisation of a class of Lyapunov-Lurye functionals (LLFs), which are the time-domain equivalence to FIR Zames-Falb multipliers. It may have some advantages to analyse Lurye systems in pure time domain, and the numerical results are less conservative than all existing Lyapunov methods. In addition to the absolute stability problem, the exponential convergence rate of stable Lurye systems has aroused the interest in recent years as the first order optimisation algorithms for convex objective functions can be modelled as Lurye systems. In this thesis, a convex search over the suitable subclass of Zames-Falb multipliers has been developed leading to less-conservative upper bound of the convergence rate of Lurye systems with slope-restricted nonlinearities. However, it remains open to analyse the conservativeness of the results.
Date of Award31 Dec 2021
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorWilliam Heath (Supervisor) & Joaquin Carrasco Gomez (Supervisor)


  • Zames-Falb multipliers
  • Absolute stability
  • Nonlinear systems

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