Dual Representations and \(H_{\infty}\)-Optimal Control of Partial Differential Equations

We consider \(H_{\infty}\)-optimal state-feedback control of the class of linear Partial Differential Equations (PDEs) class, which admit a Partial Integral Equation (PIE) representation. While linear matrix inequalities are commonly used for optimal control of Ordinary Differential Equations (ODEs)...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2024-09
Main Authors: Shivakumar, Sachin, Das, Amritam, Peet, Matthew
Format: Article
Language:English
Subjects:
Computational geometry
Controllers
Convexity
Criteria
Integral equations
Operators
Optimal control
Optimization
Ordinary differential equations
Partial differential equations
Stability tests
Synthesis
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We consider \(H_{\infty}\)-optimal state-feedback control of the class of linear Partial Differential Equations (PDEs) class, which admit a Partial Integral Equation (PIE) representation. While linear matrix inequalities are commonly used for optimal control of Ordinary Differential Equations (ODEs), the absence of a universal state-space representation and suitable dual form prevents such methods from being applied to optimal control of PDEs. Specifically, for ODEs, the controller synthesis problem is defined in state-space, and duality is used to resolve the bilinearity of that synthesis problem. Recently, the PIE representation was proposed as a universal state-space representation for linear PDE systems. In this paper, we show that any PDE system represented by a PIE admits a dual PIE with identical stability and I/O properties. This result allows us to reformulate the stabilizing and optimal state-feedback control problems as convex optimization over the cone of positive Partial Integral (PI) operators. Operator inversion formulae then allow us to construct feedback gains for the original PDE system. The results are verified through application to several canonical problems in optimal control of PDEs.
ISSN:2331-8422