Stability Analysis of Continuous-Time Linear Time-Invariant Systems

TL;DR

This paper reviews mathematical methods for analyzing the stability of continuous-time linear systems, highlighting the advantages of the Laplace transform in stability assessment and system design.

Contribution

It compares three stability analysis methods and demonstrates the superiority of the Laplace transform for both stable and unstable systems, including relative stability insights.

Findings

Laplace transform allows analysis of both stable and unstable systems

It provides absolute and relative stability measures

Simplifies analysis of complex systems through algebraic properties

Abstract

This paper focuses on the mathematical approaches to the analysis of stability that is a crucial step in the design of dynamical systems. Three methods are presented, namely, absolutely integrable impulse response, Fourier integral, and Laplace transform. The superiority of Laplace transform over the other methods becomes clear for several reasons that include the following: 1) It allows for the analysis of the stable, as well as, the unstable systems. 2) It not only determines absolute stability (a yes/no answer), but also shines light on the relative stability (how stable/unstable the system is), allowing for a design with a good degree of stability. 3) Its algebraic and convolution properties significantly simplify the mathematical manipulations involved in the analysis, especially when tackling a complex system composed of several simpler ones. A brief relevant introduction to the subject of systems is presented for the unfamiliar reader. Additionally, appropriate physical concepts and examples are presented for better clarity.