Sampled-data finite-dimensional boundary control of 1D parabolic PDEs under point measurement via a novel ISS Halanay's inequality

Rami Katz, Emilia Fridman

Research output: Contribution to journalArticlepeer-review

Abstract

Recently, finite-dimensional observer-based controllers were introduced for 1D parabolic PDEs via the modal decomposition method. In the present paper we suggest a sampled-data implementation of a finite-dimensional boundary controller for 1D parabolic PDEs under discrete-time point measurement. We consider the heat equation under boundary actuation and point (either in-domain or boundary) measurement. In order to manage with point measurement, we employ dynamic extension and prove H1-stability. Due to dynamic extension, which leads to proportional–integral controller, we suggest a sampled-data implementation of the controller via a generalized hold device. We take into account the quantization effect that leads to a disturbed closed-loop system and input-to-state stability (ISS) analysis. We use Wirtinger-based piecewise continuous in time Lyapunov functionals which compensate sampling in the finite-dimensional state and lead to the simplest efficient stability conditions for ODEs. To compensate sampling in the infinite-dimensional tail, we introduce a novel form of Halanay's inequality for ISS, which is appropriate for functions with jump discontinuities that do not grow in the jumps. Numerical examples demonstrate the efficiency of our method.

Original languageEnglish
Article number109966
JournalAutomatica
Volume135
DOIs
StatePublished - Jan 2022

Keywords

  • Distributed parameter systems
  • Halanay's inequality
  • ISS
  • Observer-based control
  • Sampled-data control

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