Solving Infinite-Dimensional Harmonic Lyapunov and Riccati equations

TL;DR

This paper develops a numerical method for solving infinite-dimensional harmonic Lyapunov and Riccati equations, crucial for analyzing and controlling periodic systems, and demonstrates its application in harmonic LQ control design.

Contribution

It introduces a closed-form Floquet factorization, studies spectral properties, and proposes an efficient algorithm for solving these equations with arbitrary precision.

Findings

The algorithm effectively solves infinite-dimensional harmonic Lyapunov equations.

The combined approach successfully addresses harmonic Riccati equations.

Application to harmonic LQ control demonstrates practical utility.

Abstract

In this paper, we address the problem of solving infinite-dimensional harmonic algebraic Lyapunov and Riccati equations up to an arbitrary small error. This question is of major practical importance for analysis and stabilization of periodic systems including tracking of periodic trajectories. We first give a closed form of a Floquet factorization in the general setting of L 2 matrix functions and study the spectral properties of infinite-dimensional harmonic matrices and their truncated version. This spectral study allows us to propose a generic and numerically efficient algorithm to solve infinite-dimensional harmonic algebraic Lyapunov equations up to an arbitrary small error. We combine this algorithm with the Kleinman algorithm to solve infinite-dimensional harmonic Riccati equations and we apply the proposed results to the design of a harmonic LQ control with periodic trajectory tracking.