Convex Constrained Controller Synthesis for Evolution Equations

TL;DR

This paper introduces a convex controller synthesis method for infinite-dimensional linear systems, enabling optimization before discretization and handling complex constraints, demonstrated on a linear Boltzmann equation.

Contribution

It develops a convex synthesis framework for infinite-dimensional systems, incorporating structural constraints while maintaining convexity, extending existing methods to PDEs and integral equations.

Findings

Framework successfully applied to a linear Boltzmann equation.

Allows optimization-then-discretize approach for complex systems.

Handles delays and locality constraints within convex optimization.

Abstract

We propose a convex controller synthesis framework for a large class of constrained linear systems, including those described by (deterministic and stochastic) partial differential equations and integral equations, commonly used in fluid dynamics, thermo-mechanical systems, quantum control, or transportation networks. Most existing control techniques rely on a (finite-dimensional) discrete description of the system, via ordinary differential equations. Here, we work instead with more general (infinite-dimensional) Hilbert spaces. This enables the discretization to be applied after the optimization (optimize-then-discretize). Using output-feedback SLS, we formulate the controller synthesis as a convex optimization problem. Structural constraints like sensor and communication delays, and locality constraints, are incorporated while preserving convexity, allowing parallel implementation and extending key SLS properties to infinite dimensions. The proposed approach and its benefits are demonstrated on a linear Boltzmann equation.