State feedback stabilization for stochastic feedforward nonlinear systems with time-varying delay

doi:10.1016/j.automatica.2013.01.007

Automatica

Volume 49, Issue 4, April 2013, Pages 936-942

https://doi.org/10.1016/j.automatica.2013.01.007 Get rights and content

Abstract

This paper investigates a class of stochastic feedforward nonlinear systems with time-varying delay. By introducing the homogeneous domination approach to stochastic systems, a state feedback controller is constructed to render the closed-loop system globally asymptotically stable in probability.

Introduction

Ever since the stochastic stability theory was established and improved by Deng, Krstić, and Williams (2001), Has’minskii (1980), Krstić and Deng (1998) and Mao (2007) and other references, the design and analysis of a backstepping controller for stochastic lower-triangular nonlinear systems has achieved remarkable development in recent years, see, e.g., Chen, Jiao, Li, and Li (2010), Deng and Krstic, 1997a, Deng and Krstic, 1997b, Deng and Krstic, 1999, Deng et al. (2001), Duan and Xie (2011), Duan, Xie, and Yu (2011), Duan, Yu, and Xie (2011), Fu, Tian, and Shi (2005), Krstić and Deng (1998), Liu and Duan (2010), Liu, Ge, and Zhang (2008), Liu, Jiang, and Zhang (2008), Liu and Xie, 2011, Liu and Xie, 2012, Liu and Zhang (2006), Liu and Zhang (2008), Liu, Zhang, and Jiang (2007), Li and Xie (2009), Li, Xie, and Zhang (2011), Tian and Xie (2007), Wu, Xie, Shi, and Xia (2009),Wu et al., 2006, Wu et al., 2007, Xie and Duan (2010), Xie, Duan, and Yu (2011), Xie and Li (2009), Xie and Tian, 2007, Xie and Tian, 2009, Yu and Xie (2010), Yu, Xie, and Duan (2010), Yu, Xie, Wu et al., 2010, Yu et al., 2011, and the references therein.

In this paper, we will consider stochastic feedforward nonlinear systems. Feedforward systems, which are also called upper-triangular systems, are another important class of nonlinear systems. Many physical devices, such as the cart–pendulum system in Mazenc and Bowong (2003) and the ball–beam with a friction term in Sepulchre, Janković, and Kokotović (1997), can be described by equations with the feedforward structure. In the recent papers on feedforward systems, Bekiaris-Liberis and Krstić (2010) studied delay-adaptive feedback for linear systems. The input delay compensation for forward complete and strict-feedforward nonlinear systems was solved by Krstić (2010). Global output feedback stabilization of feedforward nonlinear systems using a homogeneous domination approach was addressed by Qian and Li (2006). Ye (2011) considered the adaptive stabilization problem for feedforward nonlinear systems with time-delays by taking a nested saturation feedback. Zhang, Baron, Liu, and Boukas (2011) investigated the design of stabilizing controllers with a dynamic gain for feedforward nonlinear time-delay systems. For high-order nonlinear feedforward systems (i.e., the high-order of system is greater or equal to one), Ding, Qian, Li, and Li (2010) considered global stabilization problem by using the generalized adding a power integrator method and a series of nested saturation functions, Zhang et al., 2010, Zhang, Liu et al., 2011 dealt with the state feedback control for this kind of systems with time-delay. However, all these results are limited to deterministic systems. To the best of the authors’ knowledge, there is no any related result on stochastic feedforward nonlinear systems until now.

Since time-delay phenomena are often encountered in various engineering systems such as biological reactors, rolling mills, etc., and the existence of time-delay is a significant cause of instability and deteriorative performance, so the control design of stochastic nonlinear time-delay systems has been received much attention in recent years, see, e.g., Chen et al. (2010), Fu et al. (2005), Liu, Ge et al. (2008), Liu and Xie (2011), Lu, Su, and Tsai (2005), Xie and Xie (2000), Xu and Chen (2002), Yue and Han (2005) and Zhang, Boukas, and Lam (2008), and the references therein.

Due to the special characteristics of stochastic feedforward nonlinear systems and the appearance of time-varying delay, to the best of the authors’ knowledge, there is no efficient method as yet to design the state feedback controller for stochastic feedforward nonlinear time-delay systems.

The purpose of this paper is to solve the above-mentioned problem. To solve this problem, we first generalize the homogeneous domination approach in Qian and Li (2006) to a stochastic system. The underlying idea of this approach is that the homogeneous controller is first developed without considering the drift and diffusion terms, and then a scaling gain is introduced to the state feedback controller to dominate the drift and diffusion terms.

In this paper, by introducing the homogeneous domination approach, skillfully choosing an appropriate Lyapunov–Krasovskii functional, and successfully solving several troublesome obstacles in the design and analysis procedure, a state feedback controller is constructed to render the closed-loop system globally asymptotically stable in probability. A simulation example demonstrates the effectiveness of the approach.

The paper is organized as follows. Section 2 provides some preliminary results. The design and analysis of state feedback controller is given in Sections 3 Design of state feedback controller, 4 Stability analysis, following a simulation example in Section 5. Section 6 concludes this paper.

Section snippets

Preliminary results

The following notations, definitions and lemmas are to be used throughout the paper.

$R_{+}$ denotes the set of all nonnegative real numbers, $R^{n}$ denotes the real $n$ -dimensional space. For a given vector or matrix $X$ , $X^{T}$ denotes its transpose, $Tr {X}$ denotes its trace when $X$ is square, and $| X |$ is the Euclidean norm of a vector $X$ . $C ([- d, 0]; R^{n})$ denotes the space of continuous $R^{n}$ -value functions on $[- d, 0]$ endowed with the norm $‖ \cdot ‖$ defined by $‖ f ‖ = {sup}_{x \in [- d, 0]} | f (x) |$ for $f \in C ([- d, 0]; R^{n})$ ; $C_{F_{0}}^{b} ([- d, 0]; R^{n})$ denotes

Stochastic feedforward systems and assumptions

In this paper, we consider stochastic nonlinear systems $d x_{i} (t) = (x_{i + 1} (t) + f_{i} (t, x (t), x (t - d (t)))) d t + g_{i}^{T} (t, x (t), x (t - d (t))) d ω,$ $d x_{n} (t) = u (t) d t, i = 1, \dots, n - 1,$ where $x (t) = {(x_{1} (t), \dots, x_{n} (t))}^{T} \in R^{n}$ and $u (t) \in R$ are the system state and input, respectively, $x (t - d (t)) = {(x_{1} (t - d (t)), \dots, x_{n} (t - d (t)))}^{T}$ is the time-delayed state vectors, $d (t) : R_{+} \to [0, d]$ is time-varying delay. $ω$ is an $m$ -dimensional standard Wiener process defined on the complete probability space $(Ω, F, {F_{t}}_{t \geq 0}, P)$ . $f_{i} : R_{+} \times R^{n} \times R^{n} \to R$ and $g_{i} : R_{+} \times R^{n} \times R^{n} \to R^{m}$ , $i = 1, \dots, n - 1$ , are

Stability analysis

We state the main result in this paper.

Theorem 1

If Assumptions 1–2 hold for the stochastic feedforward nonlinear systems with time-varying delay (2), under the state feedback controller $u = κ^{n} v$ in (3), (9), then

(i) The closed-loop system has a unique solution on $[- d, \infty)$ ;

(ii) The equilibrium at the origin of the closed-loop system is GAS in probability.

Proof

We prove Theorem 1 in four steps.

Step 1: Since $f_{i} (\cdot)$ , $g_{i} (\cdot)$ , $i = 1, \dots, n - 1$ , are assumed to be locally Lipschitz, the closed-loop system consisting of (4), (9)

A simulation example

Consider the following stochastic nonlinear system $d x_{1} = (x_{2} + \frac{1}{10} x_{3} sin x_{1}) d t + \frac{2}{5} sin x_{3} (t - d (t)) cos x_{2} d ω,$ $d x_{2} = x_{3} d t,$ $d x_{3} = u d t,$ where $d (t) = \frac{1}{5} (1 + sin t)$ . It is easy to verify that Assumption 1, Assumption 2 are satisfied with $a_{1} = \frac{1}{10}$ , $a_{2} = \frac{2}{5}$ and $\dot{d} (t) = \frac{1}{5} cos t < 1$ .

Design of controller: We introduce the coordinate transformation: $η_{1} = x_{1}, η_{2} = \frac{x_{2}}{κ}, η_{3} = \frac{x_{3}}{κ^{2}}, v = \frac{u}{κ^{3}},$ system (26) becomes $d η_{1} = (κ η_{2} + \frac{1}{10} κ^{2} η_{3} sin η_{1}) d t + \frac{2}{5} sin (κ^{2} η_{3} (t - d (t))) cos (κ η_{2}) d ω,$ $d η_{2} = κ η_{3} d t,$ $d η_{3} = κ v d t .$ Choosing $ξ_{1} = η_{1}$ and $V_{1} (η_{1}) = \frac{1}{4} ξ_{1}^{4}$ , we obtain $L V_{1} \leq - κ c_{11} ξ_{1}^{4} + κ ξ_{1}^{3} (η_{2} - η_{2}^{*}) +$

A concluding remark

In this paper, the homogeneous domination approach is introduced to solve the stabilization problem for the stochastic feedforward nonlinear time-delay systems.

It should be pointed out that this paper is only the initial attempt to solve stochastic feedforward nonlinear time-delay systems, there still exist some problems to be investigated: One is to consider more general systems with weaker assumptions. Another is output feedback control of system (2). The third is to find a practical example.

Acknowledgments

The authors would like to express sincere gratitude to the Editor and Reviewers for their helpful comments in improving this paper.

Liang Liu is a Doctoral student at the Institute of Automation, Qufu Normal University. His current research interests include decentralized adaptive control of complex systems and stochastic nonlinear control.

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Xue-Jun Xie received his Ph.D. from the Institute of Systems Science, Chinese Academy of Sciences in 1999. His current research interests include stochastic nonlinear control systems and adaptive control.

^☆: This work is supported by the National Natural Science Foundation of China (61273125, 61104222), the Specialized Research Fund for the Doctoral Program of Higher Education (20103705110002), the Shandong Provincial Natural Science Foundation of China (ZR2012FM018), the Project of Taishan Scholar of Shandong Province, and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Changyun Wen under the direction of Editor Miroslav Krstic.

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Brief paperState feedback stabilization for stochastic feedforward nonlinear systems with time-varying delay☆

Abstract

Introduction

Section snippets

Preliminary results

Stochastic feedforward systems and assumptions

Stability analysis

A simulation example

A concluding remark

Acknowledgments

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Systems and Control Letters

Systems and Control Letters

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Brief paper
State feedback stabilization for stochastic feedforward nonlinear systems with time-varying delay☆