Revisiting the Theory of Stability over a Finite Interval

Abstract

A method for analyzing contractive stability over a finite interval with fixed settling time is proposed. The method employs a Lyapunov-type function and analyzes its behavior along the trajectories of the system.Keywords: contractive stability, settling time, stability over finite interval, Lyapunov-type function Introduction. The theory of stability of motion proposed and developed by Lyapunov [1] and many other scientists deals with systems operating over an infinite time interval and assumes that that the initial and, hence, subsequent perturbations are rather small. However, dynamic problems for machines and many applied problems are solved considering that the corresponding mechanical systems operate over finite time intervals. Moreover, in analyzing finite-time stability, it is important not only to take into account the fact that there exists d for a given e, but also to estimate the initial and subsequent deviations because, by the theorem on continuous dependence of solutions on initial conditions, it is always possible to find a small perturbation to the initial conditions to satisfy the stability conditions on a prescribed time interval over which the system operates. It is also important whether the initial and subsequent deviations are suitable for the problem under consideration. Taking these issues into account leads to some new types of stability such as technical stability, finite-time stability, etc. [4][5][6]14]. These types of stability are addressed in [2,3,10]. These types of stability were also used in [13] to solve specific problems of robotics. Grujic [7,8] proposed the concept of finite-time stability with fixed settling time, which is important for many problems of automatic control.Contractive finite-time stability [3] with certain settling time is addressed below. 1. Problem Formulation. Consider a system of differential equations of perturbed motion: