On optimal predefined‐time stabilization

Abstract

Summary This paper addresses the problem of optimal predefined‐time stability. Predefined‐time stable systems are a class of fixed‐time stable dynamical systems for which the minimum bound of the settling‐time function can be defined a priori as an explicit parameter of the system. Sufficient conditions for a controller to solve the optimal predefined‐time stabilization problem for a given nonlinear system are provided. These conditions involve a Lyapunov function that satisfies a certain differential inequality for guaranteeing predefined‐time stability. It also satisfies the steady‐state Hamilton–Jacobi–Bellman equation for ensuring optimality. Furthermore, for nonlinear affine systems and a certain class of performance index, a family of optimal predefined‐time stabilizing controllers is derived. This class of controllers is applied to optimize the sliding manifold reaching phase in predefined time, considering both the unperturbed and perturbed cases. For the perturbed case, the idea of integral sliding mode control is jointly used to ensure robustness. Finally, as a study case, the predefined‐time optimization of the sliding manifold reaching phase in a pendulum system is performed using the developed methods, and numerical simulations are carried out to show their behavior. Copyright © 2017 John Wiley & Sons, Ltd.

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

“…State trajectory of the closed-loop system (17) to (19) with T c = 1s and the considered perturbation leading to the application of Theorem 2 results in the proper inequality (20). However, the estimation of the settling time using the proposed approach leads to a significantly lower slack between T c and the real settling time than with existing methods, such as the work of Polyakov, 16 as illustrated in Example 3.…”

“…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.…”

Time‐varying feedback for stabilization in prescribed finite time

“…1,25 In this work, we present a time-varying feedback alternative to achieving finite-time control, which is referred to as "prescribed-time control" 30 or predefined-time control. [31][32][33] We show that such prescribed-time control exhibits several superior features. (i) The time-varying gain-based prescribed-time control is built upon regular state feedback, not on fractional-power state feedback, thus resulting in smooth (C n ) control action everywhere during the entire operation of the system.…”

Semi-global predefined-time stable systems

“…Finally, a function ϕ satisfying (14), i.e., an odd continuous predefined-time stabilizing function for (8) is . Thus,…”