Predefined-time stability of dynamical systems with sliding modes

Abstract

This paper introduces a class of fixed-time stable dynamical systems with settling time as a explicit parameter, namely the inverse the gain. Those systems are defined as predefined-timed stable dynamical systems. Continuous and discontinuous are cases are presented. A detailed Lyapunov characterization of this class of systems is also shown. Finally, the application to the design of a class of first order sliding mode controllers is exposed.

Predefined-time stabilization of high order systems

“…To overcome the above, a class of first-order dynamical systems with the minimum upper bound of the fixed stabilization time equal to their only tuning gain has been studied [15], [16], similarly a class of second-order systems with similar features is presented in [17]. It is said that these systems exhibit the property of predefined-time stability.…”

Predefined-time tracking of a class of mechanical systems

“…Definition 2.3 (Settling-time set and its minimum bound [13], [14]): Let T be the set of all the bounds of the settling time function for the system (1), i.e.,…”

“…Definition 2.5 (Predefined-time stabilizing function [14]): For x ∈ R n , the predefined-time stabilizing function is defined as…”

“…In this sense, this paper introduces the concept of second-order predefined-time as a nested application of firstorder predefined-time stabilizing functions [14]. Furthermore, this idea is used to solve the problem of predefined-time exact tracking in fully actuated mechanical systems, assuming the availability of the state and the desired trajectory (as well as its two first derivatives) measurements.…”

“…they are much larger than the true fixed stabilization time (see for example [12], where the upper bound estimation is approximately 100 times larger than the actual true fixed stabilization time). To overcome the above, another class of dynamical systems which exhibit the property of predefined-time stability, have been studied [13], [14]. For this systems, the minimum upper bound of the fixed stabilization time appears explicitly in their tuning gains.…”