Non-asymptotic State Estimation of Linear Reaction Diffusion Equation using Modulating Functions

Lilia Ghaffour,Matti Noack,Taous-Meriem Laleg-Kirati,Johann Reger

doi:10.1016/j.ifacol.2020.12.2570

Abstract

Abstract In this paper, we propose a non-asymptotic state estimation method for the linear reaction diffusion equation with general boundary conditions. The method is based on the modulating function approach utilizing a modulation functional in time and space. This results in a signal model control problem for a system of auxiliary PDEs in order to determine the modulation kernels. First, the algorithm is mathematically derived and then numerical simulations are presented for illustrating the good performance of the proposed approach and demonstrating the efficient implementation scheme.

Highlights

An extension to distributed systems can be obtained by defining the kernel function in the time and spatial domain (Fischer et al, 2018), (Fischer and Deutscher, 2016): Definition 2. (Modulation Functional)
The state modulation functional is defined by t
The contributions of this paper is the development of a generalizable methodology for the modulating function based state estimation applied to distributed systems and an expansion of the framework to a new problem class

Summary

PRELIMINARIES

We present the definition of the modulating functions and some of its useful properties. The modulating function for ODEs is defined in the following way (Aldoghaither et al, 2015): Definition 1. A function φ ∈ Ck([a, b], R) is called a modulating function of order k with k ∈ IN∗ if and only if φ(i)(a) = φ(i)(b) = 0, i = 0, 1, ..., k − 1. An extension to distributed systems can be obtained by defining the kernel function in the time and spatial domain (Fischer et al, 2018), (Fischer and Deutscher, 2016): Definition 2. The state modulation functional is defined by t. T−T 0 where h : [0, L] × R+0 → R and m : [0, L] × [0, T ] → R is the modulating function to be constructed. If the integration only concerns the temporal or spatial variable, m, h I and m, h Ω are used

Problem Statement

MODULATING FUNCTIONAL METHOD FOR STATE ESTIMATION

SOLVABILITY OF THE MODULATING FUNCTION SYSTEM

Time transformation

Implementation

CONCLUSION