Finite-time stabilization and stabilizability of a class of controllable systems

doi:10.1016/S0167-6911(02)00119-6

Systems & Control Letters

Volume 46, Issue 4, 23 July 2002, Pages 231-236

https://doi.org/10.1016/S0167-6911(02)00119-6 Get rights and content

Abstract

In this paper, finite-time control problem of a class of controllable systems is considered. Explicit formulae are proposed for the finite-time stabilization of a chain of power-integrators, and then discussions about a generalized class of nonlinear systems are given.

Introduction

Finite-time stabilization problems have been studied mostly in the contexts of optimality, controllability, and deadbeat control for several decades. These control laws are usually time-varying, discontinuous, or even depending directly on the initial conditions of considered systems [1]. Recently, finite-time stability and finite-time stabilization via continuous time-invariant feedback have been studied and finite-time controllers involving terms containing fractional powers were constructed for second-order systems ([3], [4], [5], [9] and references therein). Furthermore, output feedback finite-time control design was also studied [12], [13].

In addition, many results were obtained for the stabilization of analytic small-time local controllable (STLC) nonlinear systems in a special triangular form, or the systems that can be approximated by the STLC systems in this triangular form [6], [7].

The work in this paper extends the previous results on asymptotic stabilization for the nonlinear systems discussed in some references such as [6], [7], [14] to continuous finite-time stabilization in some sense. The main result is given on the construction of continuous time-invariant finite-time controllers for a class of STLC systems.

Section snippets

Preliminaries

First of all, the concepts related to finite-time control are given (see [9]).

Definition 2.1

Consider a time-invariant system in the form of $x ̇ =f(x), f(0)=0, x∈R^{n},$ where $f : U ̂_{0} →R^{n}$ is continuous on an open neighborhood $U ̂_{0}$ of the origin. The equilibrium x=0 of the system is (locally) finite-time stable if (i) it is asymptotically stable, in Û, an open neighborhood of the origin, with $U ̂ ⊆ U ̂_{0}$ ; (ii) it is finite-time convergent in Û, that is, for any initial condition $x_{0} ∈ U ̂ ⧹{0}$ , there is a settling time T>0 such that

Finite-time stabilization

At first we focus on the construction of continuous finite-time stabilizing feedback laws for a class of STLC system (see [7]): $x ̇_{1} =x_{2}^{m_{1}},⋮ x ̇_{n−1} =x_{n}^{m_{n−1}}, x ̇_{n} =u,$ where m_i>0, i=1,…,n−1 are odd integers.

For convenience, as in [9], set $sig (y)^{α} =|y|^{α} sgn (y)$ for α>0, where |y| denotes the absolute value of real number y and sgn(·) the sign function. Clearly, sig(y)^α=y^α if α=q₁/q₂ where q_i>0, i=1,2 are odd integers. Note that $d d y |y|^{α+1} =(α+1) sig (y)^{α},$ and $d d y sig (y)^{α+1} =(α+1)|y|^{α} α>0.$

Theorem 3.1

Let r_i, β_i−1, i=1,…,n and k be

Concluding remarks

The paper focuses on continuous finite-time control (with fractional powers) for some classes of controllable nonlinear systems. The motivation of the research is to understand finite-time stabilizability via continuous time-invariant feedback. Moreover, we also wish to get ideas about how to employ nonsmoothness actively or skillfully for effective systems synthesis. In fact, from many numerical and experimental results, rather than only theoretic studies, it was observed that this class of

Acknowledgements

The author wishes to thank the reviewers and Mr. G. Yang very much for many helpful comments and constructive suggestions. This work was supported by National Natural Science Foundation and Project 973 of China.

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Finite-time stabilization and stabilizability of a class of controllable systems

Abstract

Introduction

Section snippets

Preliminaries

Finite-time stabilization

Concluding remarks

Acknowledgements

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IEEE Trans. Automat. Control