Variable Structure Predefined‐Time Stabilization of Second‐Order Systems

Abstract

A controller that stabilizes second-order vector systems in predefined-time is introduced in this paper. That is, for second-order systems a controller is designed such that the trajectories reach the origin in a time defined in advance. The proposed controller is a variable structure controller that first drive the system trajectories to a linear manifold in predefined time and then drives the system trajectories to a non-smooth manifold with the predefined-time stability property, in predefined time also; this is done in order to avoid the differentiability problem that inherently appears when stabilizing high-order systems in finite time under the block control principle technique. The proposal is applied to the predefined-time exact tracking of fully actuated mechanical systems. As an example, the proposed solution is applied to a two-link planar manipulator, and numerical simulations are conducted to show its performance.

Enhancing the settling time estimation of a class of fixed‐time stable systems

“…16,19 However, it is still difficult to derive a relatively simple relationship between the system parameters and the upper bound of the settling time. 20,21 This drawback yields some difficulties in the tuning of the system parameters to achieve a prescribed-time stabilization (see the work of Cruz-Zavala et al, 7 for instance). The computation of the least upper bound of the settling time is usually not an easy task.…”

“…Indeed, it represents a wide class of systems, which present the fixed‐time stability property through homogeneity and Lyapunov analysis frameworks . However, it is still difficult to derive a relatively simple relationship between the system parameters and the upper bound of the settling time . This drawback yields some difficulties in the tuning of the system parameters to achieve a prescribed‐time stabilization (see the work of Cruz‐Zavala et al, for instance).…”

Predefined‐time backstepping control for a nonlinear strict‐feedback system

“…On the basis of Lyapunov‐like conditions for ensuring dynamic systems to exhibit predefined‐time stability or predefined‐time boundedness, many predefined‐time controllers were designed for various systems, for example, nonlinear second‐order systems, 6‐8 nonholonomic systems, 9 chaotic systems, 10,11 rigid spacecrafts, 12 robotic manipulators 13,14 as well as first‐order and second‐order multi‐agent systems 15,16 . However, the predefined‐time controllers derived on the basis of Lyapunov analysis 3‐14 may provide conservative estimate of upper bounds of system convergence time, especially for high‐order systems. The real convergence time may be much smaller than the estimated upper bound.…”

Predefined‐time control of cooperative manipulators

“…The predefined‐time stability concept and its application in sliding mode control are discussed in References 25‐31. It is mentioned that, for a predefined‐time stable system, an upper bound for the convergence time exists, and such a bound can be defined a priori during the control design.…”