Solutions of differential-algebraic equations as outputs of LTI systems: application to LQ control problem

TL;DR

This paper develops a geometric framework to represent all solutions of DAE-LTIs as outputs of ODE-LTIs, enabling solution synthesis and control design for complex systems.

Contribution

It introduces a unified geometric approach to represent DAE-LTI solutions as ODE-LTI outputs and provides an algorithm for their computation, advancing control theory.

Findings

All solutions of DAE-LTIs can be represented as ODE-LTI outputs.

Two ODE-LTIs representing the same DAE-LTI are feedback equivalent.

The framework facilitates solving LQ control problems for DAE-LTIs.

Abstract

In this paper we synthesize behavioral ideas with geometric control theory and propose a unified geometric framework for representing all solutions of a Linear Time Invariant Differential-Algebraic Equation (DAE-LTI) as outputs of classical Linear Time Invariant systems (ODE-LTI). An algorithm for computing an ODE-LTI that generates solutions of a given DAE-LTI is described. It is shown that two different ODE-LTIs which represent the same DAE-LTI are feedback equivalent. The proposed framework is then used to solve an LQ optimal control problem for DAE-LTIs with rectangular matrices.