Controlled functional differential equations: Approximate and exact asymptotic tracking with prescribed transient performance

Eugene P. Ryan, Chris J. Sangwin, Philip Townsend

Research output: Contribution to journalArticlepeer-review

Abstract

A tracking problem is considered in the context of a class S of multi-input, multi-output, nonlinear systems modelled by controlled functional differential equations. The class contains, as a prototype, all finite-dimensional, linear, ℳ-input, ℳ-output, minimum-phase systems with sign-definite "high-frequency gain". The first control objective is tracking of reference signals r by the output y of any system in S: given λ ≥ 0, construct a feedback strategy which ensures that, for every r (assumed bounded with essentially bounded derivative) and every system of class S, the tracking error e = y - r is such that, in the case λ > 0, lim supt→∞ ∥e(t)∥ < λ or, in the case λ = 0, limt→∞ ∥e(t)∥ = 0. The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel ℱ (determined by a function). For suitably chosen functions α, ν and θ, both objectives are achieved via a control structure of the form u(t) = -ν(κ(t)) θ(e(t)) with κ(t) = α((t)∥e(t)∥), whilst maintaining boundedness of the control and gain functions u and κ. In the case λ = 0, the feedback strategy may be discontinuous: to accommodate this feature, a unifying framework of differential inclusions is adopted in the analysis of the general case λ ≥ 0.

Original languageEnglish
Pages (from-to)745-762
Number of pages18
JournalESAIM: Control, Optimisation and Calculus of Variations
Volume15
Issue number4
DOIs
Publication statusPublished - Oct 2009

Keywords

  • Approximate tracking
  • Asymptotic tracking
  • Functional differential inclusions
  • Transient behaviour

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