Lyapunov and Riccati Equations from a Positive System Perspective

TL;DR

This paper offers a novel perspective on Lyapunov and Riccati equations by framing them within positive system theory, enabling new insights into their solutions and convergence properties.

Contribution

It introduces a positive system approach to analyze Lyapunov and Riccati equations, establishing existence, uniqueness, and convergence results from positive system theory.

Findings

Constructed positive systems related to Lyapunov equations show exponential convergence.

Homogeneous positive systems for Riccati equations exhibit complex dynamics.

Existence and uniqueness of solutions are proven using positive system properties.

Abstract

This paper presents a new interpretation of the Lyapunov and Riccati equations from the perspective of positive system theory. We show it is possible to construct positive systems related to these equations, and then certain conclusions -- such as the existence and uniqueness of solutions -- can be drawn from positive systems theory. Specifically, under standard observability assumptions, a strictly positive linear system can be constructed for Lyapunov equations, leading to exponential convergence in Hilbert metric to the Perron-Frobenius vector -- closely related to the solution of the Lyapunov equation. For algebraic Riccati equations, homogeneous strictly positive systems can be constructed, which exhibit more complex dynamical behaviors. While the existence and uniqueness of the solution can still be proven, only asymptotic convergence can be obtained.