A Convex Sum-of-Squares Approach to Analysis, State Feedback and Output Feedback Control of Parabolic PDEs

TL;DR

This paper introduces a convex sum-of-squares framework for analyzing and designing boundary control and output feedback for linear parabolic PDEs without discretization, using Lyapunov functions and LMIs.

Contribution

It develops a novel convex optimization approach for PDE control that avoids discretization and incorporates boundary actuation and output feedback.

Findings

Accurately analyzes PDE stability and control using SOS and LMIs.

Demonstrates effectiveness through extensive numerical experiments.

Shows robustness of the method across different boundary conditions.

Abstract

We present an optimization-based framework for analysis and control of linear parabolic partial differential equations (PDEs) with spatially varying coefficients without discretization or numerical approximation. For controller synthesis, we consider both full-state feedback and point observation (output feedback). The input occurs at the boundary (point actuation). We use positive matrices to parameterize positive Lyapunov functions and polynomials to parameterize controller and observer gains. We use duality and an invertible state-variable transformation to convexify the controller synthesis problem. Finally, we combine our synthesis condition with the Luenberger observer framework to express the output feedback controller synthesis problem as a set of LMI/SDP constraints. We perform an extensive set of numerical experiments to demonstrate accuracy of the conditions and to prove necessity of the Lyapunov structures chosen. We provide numerical and analytical comparisons with alternative approaches to control including Sturm Liouville theory and backstepping. Finally we use numerical tests to show that the method retains its accuracy for alternative boundary conditions.