The restrictive conditions to solve LTI Systems by Ordinary Differential Equations

TL;DR

This paper clarifies the assumptions and limitations of solving linear ODEs in control theory, especially regarding input smoothness and the connection with state space representations.

Contribution

It formally delineates the boundaries of standard solution methods for linear ODEs and establishes the equivalence with state space representations.

Findings

Identifies gaps in didactic literature about input smoothness assumptions.

Provides a formal delimitation of solution applicability for linear ODEs.

Establishes a formal connection between ODEs and state space representations.

Abstract

Ordinary differential equations (ODE's) are a cornerstone of systems and control theory. Accordingly, they are standard material in undergraduate programs in engineering and there is abundant didactic literature about this topic. Yet, the solution methods and formulas prescribed in this didactic literature are unclear about the assumptions behind their derivation and thus about the limits of their applicability. Specifically, smoothness of the input is rarely discussed, even though it is a critical property to define the character of the solutions and the validity of the methods and formulas prescribed. On the other hand, the relationships with the state space representation (SSR) of linear systems is absent from this same literature and only marginally discussed in more advanced texts. In this paper we detail these gaps left behind in the didactic literature, then we provide a formal delimitation of the boundaries of the standard solutions and methods for linear ODE's. Our analysis relies on some key properties of state space representations, so we establish the formal connections between ODEs and SSR's, defining an equivalence between the two that is absent in the literature and is of conceptual interest by itself.