Continuous finite-time state feedback stabilizers for some nonlinear stochastic systems

Abstract

This paper is concerned with the problem of finite-time stabilization for some nonlinear stochastic systems. Based on the stochastic Lyapunov theorem on finite-time stability that has been established by the authors in the paper, it is proven that Euler-type stochastic nonlinear systems can be finite-time stabilized via a family of continuous feedback controllers. Using the technique of adding a power integrator, a continuous, global state feedback controller is constructed to stabilize in finite time a large class of two-dimensional lower-triangular stochastic nonlinear systems. Also, for a class of three-dimensional lower-triangular stochastic nonlinear systems, a recursive design scheme of finite-time stabilization is given by developing the technique of adding a power integrator and constructing a continuous feedback controller. Finally, a simulation example is given to illustrate the theoretical results. . 1582J. YIN AND S. KHOO been proposed for various types of nonlinear dynamical systems in the literature (see [2][3][4][5][6][7][8][9][10][11][12] and references therein). In particular, the theory of finite-time stability for continuous autonomous systems has been established, and a Lyapunov stability theorem that investigates finite-time stability has also been presented by Bhat and Bernstein [13]. With the help of this theorem, Bhat and Bernstein [5] presented a continuous finite-time state feedback stabilizer for the translational and rotational double integrators; Hong et al. [6] constructed a class of output feedback controllers that achieve finite-time stability for the double integrator system by designing a nonlinear observer and using some results of homogenous systems in [4,14]; Huang et al. [8] proved global finite-time stabilization for higher-dimensional uncertain nonlinear systems in a lower-triangular form. Because only nonsmooth or non-Lipschitz continuous systems can have a finite-time stable property, a finite-time stable system might have better insensitivity to perturbations [3].The purpose of this paper is to develop finite-time stabilization control methods that are applicable to stochastic nonlinear systems. It is well known that the common Lyapunov theory plays an important role in stabilizing dynamical systems. Because of the connection between stochastic systems and dynamical systems, it is natural to first consider adopting control techniques that have been successfully applied to stabilize dynamical systems. It should be noted that most of existing stabilization results for stochastic nonlinear systems are based on the stochastic Lyapunov theorems on asymptotic stability in probability (see [15][16][17][18][19][20]). In these papers, the coefficients of stochastic nonlinear systems are required to satisfy the local Lipschitz condition in order to ensure uniqueness and local existence of strong solutions. However, Lemma 5.2.3 in [19] implies that almost all the sample paths of a stochastic differential equation starting from a non-zero initial state will never reach the origin su...

Finite-time stabilization of switched stochastic nonlinear systems with mixed odd and even powers

“…For the deterministic case, the finite-time stabilization has been developed (Ding, Li, & Zheng, 2012;Hong, 2002;Huang, Lin, & Yang, 2005;Nersesov & Haddad, 2008;Shen & Huang, 2012;Zhang, Feng, & Sun, 2012). Ai, Zhai, and Fei (2013), Khoo, Yin, Man, and Yu (2013) and Yin and Khoo (2015) addressed finite-time stabilization for some stochastic nonlinear systems. Finite-time state feedback controller was constructed for highorder stochastic nonlinear systems in strict-feedback form in Wang and Zhu (2015).…”

Global output‐feedback stabilization for a class of stochastic nonlinear systems via sampled‐data control

“…It is well known to all that, to guarantee a stochastic system have a unique global solution for any given initial value, the coefficients of the system are generally needed to satisfy the linear growth condition and the local Lipschitz condition. However, it is proved in that these conditions can be further relaxed, which can be described by the following lemma.…”

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Stabilization of high order stochastic nonholonomic systems

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