The Riccati Characteristic Equation

TL;DR

This paper introduces the Riccati Characteristic Equation (RCE), a unifying framework for analyzing linear time-varying and invariant systems through solutions of the Riccati differential equation.

Contribution

It presents the first general form of solutions for the RCE, unifies existing methods, and offers new insights into Floquet theory and linear system analysis.

Findings

Solutions form a continuum based on a primitive pair.

Purely real solutions can always be found.

The general solution form encompasses all known solutions.

Abstract

The Riccati differential equation is examined in light of its connection to second order linear time varying systems. In that light it becomes the clear generalization for the characteristic equation of linear time invariant systems, and is called the Riccati Characteristic Equation (RCE). Consequently, the RCE becomes the unifying centerpiece for the study of linear systems. Its solutions are considered in complementary pairs that form a continuum based on a primitive pair. Pairs may always be found as purely real solutions, despite the fact that complex conjugate primitive solutions are shown to exist in many cases. Not only is the pairing unique, but the general form of solutions, shown here for the first time, is uniquely compact and encompasses all known solutions, while allowing for all initial conditions. Classical engineering mathematics examples are shown to conform to this approach, which provides new insights to all, especially Floquet theory.