Unified stability criteria for perturbed LTV systems with unstable instantaneous dynamics

TL;DR

This paper establishes unified stability criteria for perturbed linear time-varying systems with potentially unstable instantaneous dynamics, allowing for jumps and nonlinear perturbations, and provides conditions ensuring exponential stability.

Contribution

It introduces new stability conditions for perturbed LTV systems with discontinuous dynamics and unstable matrices, extending existing theories to more general scenarios.

Findings

Derived conditions for uniform global exponential stability.

Connected stability results with existing theories for LTV and switched systems.

Validated criteria through a numerical example.

Abstract

In this work the stability of perturbed linear time-varying systems is studied. The main features of the problem are threefold. Firstly, the time-varying dynamics is not required to be continuous but allowed to have jumps. Also the system matrix is not assumed to be always Hurwitz. In addition, there is nonlinear time-varying perturbation which may be persistent. We first propose several mild regularity assumptions, under which the total variations of the system matrix and its abscissa are well-defined over arbitrary time interval. We then state our main result of the work, which requires the combined assessment of the total variation of the system matrix, the measure when the system is not sufficiently "stable" and the estimate of the perturbation to be upper bounded by a function affine in time. When this condition is met, we prove that the neighborhood of the origin, whose size depends on the magnitude of the perturbation, is uniformly globally exponentially stable for the system. We make several remarks, connecting our results with the known stability theory from continuous linear time-varying systems and switched systems. Finally, a numerical example is included to further illustrate the application of the main result.