On the prescribed-time attractivity and frozen-time eigenvalues of linear time-varying systems

TL;DR

This paper establishes necessary and sufficient conditions for prescribed-time attractivity in linear time-varying systems and shows how frozen-time eigenvalues relate to system stability near convergence.

Contribution

It provides a complete characterization of prescribed-time attractivity for LTV systems and links frozen-time eigenvalues to convergence properties, offering a new approach to avoid singularities.

Findings

Frozen-time eigenvalues have negative real parts near convergence time.

Switching to frozen-time systems can prevent singularities without causing instability.

Time-varying feedback gains approach constant stabilizing gains as convergence nears.

Abstract

A system is called prescribed-time attractive if its solution converges at an arbitrary user-defined finite time. In this note, necessary and sufficient conditions are developed for the prescribed-time attractivity of linear time-varying (LTV) systems. It is proved that the frozen-time eigenvalues of a prescribed-time attractive LTV system have negative real parts when the time is sufficiently close to the convergence moment. This result shows that the ubiquitous singularity problem of prescribed-time attractive LTV systems can be avoided without instability effects by switching to the corresponding frozen-time system at an appropriate time. Consequently, it is proved that the time-varying state-feedback gain of a prescribed-time controller, designed for an unknown linear time-invariant system, approaches the set of stabilizing constant state-feedback gains.