A Convex Approach to Output Feedback Control of Parabolic PDEs Using Sum-of-Squares

TL;DR

This paper introduces a convex optimization framework using sum-of-squares and semidefinite programming to design output-feedback controllers for one-dimensional parabolic PDEs, accommodating various measurement and actuation configurations.

Contribution

It develops a novel convex approach for PDE output-feedback control leveraging Lyapunov operators, duality, and Luenberger observers, formulated as Linear-Operator-Inequalities.

Findings

Feasibility of control synthesis tested via SDP and SOS methods.

Framework applicable to distributed and boundary measurements and actuation.

Provides a systematic, optimization-based method for PDE control design.

Abstract

In this paper we use optimization-based methods to design output-feedback controllers for a class of one-dimensional parabolic partial differential equations. The output may be distributed or point-measurements. The input may be distributed or boundary actuation. We use Lyapunov operators, duality, and the Luenberger observer framework to reformulate the synthesis problem as a convex optimization problem expressed as a set of Linear-Operator-Inequalities (LOIs). We then show how feasibility of these LOIs may be tested using Semidefinite Programming (SDP) and the Sum-of-Squares methodology.