Finite-dimensional control of the heat equation: Dirichlet actuation and point measurement

TL;DR

This paper develops finite-dimensional observer-based controllers for the 1D heat equation with unbounded Dirichlet actuation and point measurement, using modal decomposition, Lyapunov methods, and LMIs, including sampled-data implementation.

Contribution

First to design finite-dimensional controllers for the 1D heat equation with both unbounded control and observation operators, using a novel Lyapunov approach and LMIs.

Findings

Controllers achieve exponential decay rates.

Sampled-data implementation is effective.

Numerical examples confirm method efficiency.

Abstract

Recently finite-dimensional observer-based controllers were introduced for the 1D heat equation, where at least one of the observation or control operators is bounded. In this paper, for the first time, we manage with such controllers for the 1D heat equation with both operators being unbounded. We consider Dirichlet actuation and point measurement and use a modal decomposition approach via dynamic extension. We suggest a direct Lyapunov approach to the full-order closed-loop system, where the finite-dimensional state is coupled with the infinite-dimensional tail of the state Fourier expansion, and provide LMIs for finding the controller dimension and the resulting exponential decay rate. We further study sampled-data implementation of the controller under sampled-data measurement. We use Wirtinger-based, discontinuous in time, Lyapunov functionals which compensate sampling in the finite-dimensional state. To compensate sampling in the infinite-dimensional tail, we use a novel form of Halanay's inequality, which is appropriate for Lyapunov functions with jump discontinuities that do not grow in the jumps. Numerical examples demonstrate the efficiency of the method.