Lyapunov and Riccati Equations from a Positive System Perspective

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.