A Complete Approach to Time Varying Linear Systems

TL;DR

This paper introduces a unifying theory for linear second order systems that handles both time-varying and time-invariant cases through a spectrum-invariant diagonalization and a canonical fundamental matrix form.

Contribution

It provides a novel transformation and canonical form that generalize the characteristic equation for time-varying systems, unifying their analysis with time-invariant systems.

Findings

Unified approach to solving time-invariant and time-varying systems

Transformation diagonalizes arbitrary time-varying matrices spectrum-invariantly

Examples demonstrate the method's effectiveness across different system types

Abstract

This paper presents a unifying theory of Linear second order systems that allows time-varying and time invariant systems to be treated in the same way for the first time. In the process, a transformation is given that diagonalizes an arbitrary time varying state matrix in a spectrum invariant way. A canonical form for the fundamental matrix is given that depends on dynamic eigenvalues and related eigenvectors dependent upon the Riccati Characteristic Equation for the system, which intuitively generalizes the standard characteristic equation for time invariant systems. The technique is shown by examples to give a unified approach to the solutions of time invariant, time-varying, and periodic systems.