Consistent Initialization of the Laplace Transform

TL;DR

This paper introduces a novel, straightforward method for consistent initialization of the Laplace transform, addressing longstanding issues with the L+ approach by using singular-nonsingular decomposition to accurately compute initial conditions.

Contribution

It proposes a direct, first-principles approach for the L+ Laplace transform that ensures consistency and avoids the pitfalls of previous methods.

Findings

The method accurately computes initial conditions for the L+ transform.

It effectively handles discontinuities and impulses without inconsistency.

Physical principles validate the computed solutions.

Abstract

Consistent initialization of the Laplace transform has been a fundamental and long-standing issue. The consistency of the L- approach has been questioned, yet it is a popular approach since the L+ approach requires a priori computation of the 0+ initial conditions, which becomes a diligent task from the available methods. Also, the L+ Laplace transform of the impulse becomes zero, thus involving an inconsistency. In contrast to these direct approaches, some studies propose convoluted methods to address the issue. Here, a direct, facile, and first-principles novel approach for the L+ transform is proposed. It computes the consistent 0+ initial conditions and solution based on singular-nonsingular decomposition of the system model. The inconsistency associated with the L+ Laplace transform of discontinuous functions and impulse is not involved in the nonsingular part. The emergence of discontinuity is easily elucidated from the singular part, and the solution yields the same 0+ initial values as the initial conditions used. Further, physical first-principles can be used in the midst of the computation to validate the results to ensure error-free execution.