Analytic Methods for Differential Algebraic Equations

TL;DR

This paper presents new analytic methods for solving and characterizing differential-algebraic equations (DAEs), enabling insights into system behavior and parameter tuning beyond numerical simulation capabilities.

Contribution

It introduces a direct formulation for linear DAEs and develops automated symbolic methods for deriving full solutions and governing equations, applicable to large systems.

Findings

Analytic solutions for constant coefficient DAEs derived

Governing equations for dependent variables obtained

Methods applicable to systems of any size with automation

Abstract

We introduce methods for deriving analytic solutions from differential-algebraic systems of equations (DAEs), as well as methods for deriving governing equations for analytic characterization which is currently limited to very small systems as it is carried out by hand. Analytic solutions to the system and analytic characterization through governing equations provide insights into the behaviors of DAEs as well as the parametric regions of operation for each potential behavior. For each system (DAEs), and choice of dependent variable, there is a corresponding governing equation which is univariate ODE or PDE that is typically higher order than the constitutive equations of the system. We first introduce a direct formulation for representing systems of linear DAEs. Unlike state space formulations, our formulation follows very directly from the system of constitutive equations without the need for introducing state variables or singular matrices. Using this formulation for the system of constitutive equations (DAEs), we develop methods for deriving analytic expressions for the full solution (complementary and particular) for all dependent variables of systems that consist of constant coefficient ordinary-DAEs and special cases of partial-DAEs. We also develop methods for deriving the governing equation for a chosen dependent variable for the constant coefficient ordinary-DAEs and partial-DAEs as well as special cases of variable coefficient DAEs. The methods can be automated with symbolic coding environments thereby allowing for dealing with systems of any size while retaining analytic nature. This is relevant for interpretable modeling, analytic characterization and estimation, and engineering design in which the objective is to tune parameter values to achieve specific behavior. Such insights cannot directly be obtained using numerical simulations.