Delayed finite-dimensional observer-based control of 1D heat equation under Neumann actuation

TL;DR

This paper introduces an improved, computationally efficient method for finite-dimensional observer-based control of 1D heat equations with Neumann actuation, handling larger and variable delays using predictors and reduced-order LMIs.

Contribution

It presents a novel approach with reduced-order LMIs and predictors for delayed control of 1D heat equations, including cases with unknown delays and non-local measurements.

Findings

Reduced LMI dimension independent of observer order

Feasibility of LMIs for large observer dimensions

Effective handling of large and fast-varying delays

Abstract

Recently a constructive method was introduced for finite-dimensional observer-based control of 1D parabolic PDEs. In this paper we present an improved method in terms of the reduced-order LMIs (that significantly shorten the computation time) and introduce predictors to manage with larger delays. We treat the case of a 1D heat equation under Neumann actuation and non-local measurement, that has not been studied yet. We apply modal decomposition and prove $L^2$ exponential stability by a direct Lyapunov method. We provide reduced-order LMI conditions for finding the observer dimension $N$ and resulting decay rate. The LMI dimension does not grow with $N$. The LMI is always feasible for large $N$, and feasibility for $N$ implies feasibility for $N+1$. For the first time we manage with delayed implementation of the controller in the presence of fast-varying (without any constraints on the delay-derivative) input and output delays. To manage with larger delays, we construct classical observer-based predictors. For the known input delay, the LMIs dimension does not grow with $N$, whereas for unknown one the LMIs dimension grows, but it is ssentially smaller than in the existing results. A numerical example demonstrates the efficiency of our method.