Comments on “Design of Controllers with Arbitrary Convergence Time” [Automatica 112 (2020) 108710]

doi:10.1016/j.automatica.2022.110429

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Volume 142, August 2022, 110429

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Abstract

The notion of “free-will arbitrary time stability” of (Pal et al. (2020), Definition 4) is ill-posed. The proof of the paper’s central result (Pal et al. (2020), Theorem 2) is incomplete. The arbitrary time stability concept (Pal et al. (2020), Definition 3) is implied by prescribed-time stabilization by Song et al. (2017). Additionally, Theorem 1 by Pal et al. (2020) is obtained by a logarithmic change of the Lyapunov function from (Song et al. (2017), Corollary 1).

Section snippets

On technical correctness

We examine first the meaning of the basic concepts, given by Pal et al. (2020) for the underlying system $\dot{x} = f (t, x; ϕ), x (t_{0}) = x_{0} .$ As in Pal et al. (2020), $x \in R^{n}$ is the state of system (1), $t_{0} \in R_{\geq 0}$ stands for the initial time, $ϕ \in R^{p}$ represents the constant parameters of the system, and $f : R_{\geq 0} \times R^{n} \to R^{n}$ is a nonlinear function such that $f (t, 0; ϕ) = 0$ .

Definition 3 of arbitrary time stability and Definition 4 of free-will arbitrary time stability, introduced in Pal et al. (2020), rely on the fixed-time stability

On novelty

Comment $#4.$ Definition 3 coins the term “arbitrary time stability”, which is just a re-branding of the pre-existing concept of “prescribed-time stabilization” in Song et al. (2017). Definition 3 trivially extends the prescribed-time stabilization concept to an infinite horizon by requiring the state to remain zero after the prescribed time.

Comment $#5.$ Using the logarithmic transformation $V \mapsto ln (1 + V)$ in (8) of Song et al. (2017), Theorem 1 of Pal et al. (2020) is a direct consequence of Corollary 1

References (6)

PalA.K. et al.
Design of controllers with arbitrary convergence time
Automatica
(2020)
PalA.K. et al.
Authors’ Reply To: (CI 20-0229) Comments on Design of controllers with arbitrary convergence time [Automatica 108710]
Automatica
(2020)
SongY. et al.
Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time
Automatica
(2017)

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Cited by (1)

A novel theorem for prescribed-time control of nonlinear uncertain time-delay systems
2023, Automatica
This paper considers the problem of prescribed-time control for a class of nonlinear uncertain time-delay systems. First, the definition of prescribed-time stability for nonlinear time-delay systems is given, and the necessary conditions to realize the finite-time (including fixed/prescribed-time) stabilization of the nonlinear time-delay system are illustrated. Then, the prescribed-time Lyapunov stability theorem is proposed by utilizing the prescribed-time adjustment function and Lyapunov–Krasovskii (L–K) functional method. The stability of the resulting closed-loop system is proved and the different situations for uncertain time-delay systems are discussed. For first-order fully actuated system, a continuous controller is designed. The effects of different prescribed-time adjustment functions on the controller are also studied. Finally, a simulation example is provided to show the effectiveness of the theoretical results.

^☆: The authors of the commented-on paper stand by the assertions and contributions of their paper.

^☆☆: The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Editor André L. Tits.

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CorrespondenceComments on “Design of Controllers with Arbitrary Convergence Time” [Automatica 112 (2020) 108710]☆,☆☆

Abstract

Section snippets

On technical correctness

On novelty

Design of controllers with arbitrary convergence time

Automatica

Authors’ Reply To: (CI 20-0229) Comments on Design of controllers with arbitrary convergence time [Automatica 108710]

Automatica

Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time

Automatica

A novel theorem for prescribed-time control of nonlinear uncertain time-delay systems

Correspondence
Comments on “Design of Controllers with Arbitrary Convergence Time” [Automatica 112 (2020) 108710]☆,☆☆