Impulse-to-Peak-Output Norm Optimal State-Feedback Control of Linear PDEs

TL;DR

This paper extends impulse-to-peak response analysis to linear PDEs using a novel PIE framework, enabling convex optimization-based bounds and optimal control design.

Contribution

It introduces a new method for I2P analysis and control of PDEs via PIE representation and Lyapunov techniques, filling a gap in existing PDE control methods.

Findings

Derived convex bounds for I2P-norms of linear PDEs.

Developed a constructive I2P optimal state-feedback control method.

Validated the approach on multiple PDE examples.

Abstract

Impulse-to-peak response (I2P) analysis for state-space ordinary differential equation (ODE) systems is a well-studied classical problem. However, the techniques employed for I2P optimal control of ODEs have not been extended to partial differential equation (PDE) systems due to the lack of a universal transfer function and state-space representation. Recently, however, partial integral equation (PIE) representation was proposed as the desired state-space representation of a PDE, and Lyapunov stability theory was used to solve various control problems, such as stability and optimal ${H}_\infty$ control. In this work, we utilize this PIE framework, and associated Lyapunov techniques, to formulate the I2P response analysis problem as a solvable convex optimization and obtain provable bounds for the I2P-norm of linear PDEs. Moreover, by establishing strong duality between primal and dual formulations of the optimization problem, we develop a constructive method for I2P optimal state-feedback control of PDEs and demonstrate the effectiveness of the method on various examples.