Prescribed-Time Stability Properties of Interconnected Systems

TL;DR

This paper investigates conditions under which interconnected systems achieve prescribed-time stability, ensuring convergence within a user-defined time, by analyzing properties of time-varying functions and extending results to multiple systems.

Contribution

It provides new conditions for prescribed-time stability in interconnected systems and generalizes the results to multiple interconnected systems.

Findings

Interconnected prescribed-time stable systems converge within the prescribed time.

Conditions relate to properties of time-varying blow-up functions.

Interconnection of two prescribed-time stabilized systems also achieves prescribed-time stability.

Abstract

Achieving control objectives (e.g., stabilization or convergence of tracking error to zero, input-to-state stabilization) in "prescribed time" has attracted significant research interest in recent years. The key property of prescribed-time results unlike traditional "asymptotic" results is that the convergence or other control objectives are achieved within an arbitrary designer-specified time interval instead of asymptotically as time goes to infinity. In this paper, we consider cascade and feedback interconnections of prescribed-time input-to-state stable (ISS) systems and study conditions under which the overall states of such interconnected systems also converge to the origin in the prescribed time interval. We show that these conditions are intrinsically related to properties of the time-varying "blow-up" functions that are central to prescribed-time control designs. We also generalize the results to interconnections of an arbitrary number of systems. As an illustrative example, we consider an interconnection of two uncertain systems that are prescribed-time stabilized using two different control design methods and show that the two separate controllers can be put together to achieve prescribed-time stability of the interconnected system.