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Parameter identification for an uncertain reaction-diffusion equation via setpoint regulation

Gildas Besançon, Andrea Cristofaro, Francesco Ferrante

Abstract

The problem of estimating the reaction coefficient of a system governed by a reaction-diffusion partial differential equation is tackled. An estimator relying on boundary measurements only is proposed. The estimator is based upon a setpoint regulation strategy and leads to an asymptotically converging estimate of the unknown reaction coefficient. The proposed estimator is combined with a state observer and shown to provide an asymptotic estimate of the actual system state. A numerical example supports and illustrates the theoretical results.

Parameter identification for an uncertain reaction-diffusion equation via setpoint regulation

Abstract

The problem of estimating the reaction coefficient of a system governed by a reaction-diffusion partial differential equation is tackled. An estimator relying on boundary measurements only is proposed. The estimator is based upon a setpoint regulation strategy and leads to an asymptotically converging estimate of the unknown reaction coefficient. The proposed estimator is combined with a state observer and shown to provide an asymptotic estimate of the actual system state. A numerical example supports and illustrates the theoretical results.
Paper Structure (12 sections, 6 theorems, 71 equations, 1 figure)

This paper contains 12 sections, 6 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Problem setting
  5. Main results
  6. Outline of the proposed estimator
  7. Well-posedness
  8. Convergence analysis
  9. Recovery of the unknown parameter
  10. Joint state and parameter estimation
  11. Numerical example and simulations
  12. Conclusions and extensions

Key Result

Proposition 1

The unbounded operator $A_e$ defined in eq:Ae_op generates a strongly continuous semigroup on the Hilbert space $\mathcal{X}$ equipped with the following inner product

Figures (1)

  • Figure 1: Estimation results

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • Corollary 1
  • proof
  • Proposition 3
  • ...and 2 more