Adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization

doi:10.1016/j.automatica.2008.10.006

Automatica

Volume 45, Issue 1, January 2009, Pages 126-133

https://doi.org/10.1016/j.automatica.2008.10.006 Get rights and content

Abstract

This paper investigates the adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization. By using the parameter separation lemma in [Lin, W., & Qian, C. (2002a). Adaptive control of nonlinearly parameterized systems: A nonsmooth feedback framework. IEEE Transactions on Automatic Control, 47, 757–774.] and some flexible algebraic techniques, and choosing an appropriate Lyapunov function, a smooth adaptive state-feedback controller is designed, which guarantees that the closed-loop system has an almost surely unique solution for any initial state, the equilibrium of interest is globally stable in probability, and the state can be regulated to the origin almost surely.

Introduction

Let us consider the following high-order stochastic systems with nonlinear parameterization: $d x_{i} = d_{i} ({\bar{x}}_{i}, θ) x_{i + 1}^{p_{i}} d t + φ_{i 1} ({\bar{x}}_{i}, θ) d t + φ_{i 2}^{T} ({\bar{x}}_{i}, θ) d ω, i = 1, \dots, n - 1,$ $d x_{n} = d_{n} ({\bar{x}}_{n}, θ) u^{p_{n}} d t + φ_{n 1} ({\bar{x}}_{n}, θ) d t + φ_{n 2}^{T} ({\bar{x}}_{n}, θ) d ω,$ where $x = {(x_{1}, \dots, x_{n})}^{T} \in R^{n}$ and $u \in R$ are the system state and the input, ${\bar{x}}_{i} = {(x_{1}, \dots, x_{i})}^{T}$ , ${\bar{x}}_{n} = x$ , the initial value $x_{0} \in R^{n}$ , $θ \in R^{m}$ is an unknown constant vector, $p_{i} \in N$ , $i = 1, \dots, n$ , are positive odd integers, $φ_{i 1}$ : $R^{i} \times R^{m} \to R$ and $φ_{i 2}$ : $R^{i} \times R^{m} \to R^{r}$ , $i = 1, \dots, n$ , are assumed to be smooth functions with $φ_{i 1} (0, θ) = 0$ and $φ_{i 2} (0, θ) = 0$ , $d_{i}$ : $R^{i} \times R^{m} \to R$ , $i = 1, \dots, n$ , are nonlinearly parameterized control coefficients with known sign, and $ω \in R^{r}$ is an $r$ -dimensional independent standard Wiener process defined on a complete probability space $(Ω, F, P)$ with $Ω$ being a sample space, $F$ being a filtration, and $P$ being a probability measure.

When $p_{i} = d_{i} (\cdot) = 1$ , $φ_{i j} ({\bar{x}}_{i}, θ) = θ^{T} ϕ_{i j} ({\bar{x}}_{i})$ for all $i = 1, \dots, n$ , $j = 1, 2$ , for this class of stochastic nonlinear systems, the design of a global stabilization controller has achieved remarkable development. Has’minskii (1980) and Kushner (1967) respectively presented the basic stability theory of stochastic control systems, which has now been widely applied and laid the mathematical foundation for the design and analysis of stochastic nonlinear controllers. According to the differences of selective Lyapunov functions, the existing literature on controller design can be mainly divided into two types. One type originated from the work of Pan and Basar, 1998, Pan and Basar, 1999 who derived a backstepping controller for strict-feedback systems by using a quadratic Lyapunov function and a risk-sensitive cost criterion. Subsequently, a series of problems have been solved (see, e.g., Liu, Pan, and Shi (2003), Liu and Zhang (2004), Pan, Ezal, Krener, and Kokotović (2001) and Pan, Liu, and Shi (2001)). Another essential improvement belongs to Krstić and Deng. By introducing the quartic Lyapunov function, Deng and Krstić, 1997a, Deng and Krstić, 1997b, Deng and Krstić, 2000 and Deng, Krstić, and Williams (2001) presented asymptotical stabilization control in the large under the assumption that the nonlinearities and disturbance equal zero at the equilibrium point of the open-loop system. Recently, for several classes of stochastic nonlinear systems with unmodeled dynamics and uncertain nonlinear functions, by using the long-term average tracking risk-sensitive cost criteria in Liu and Zhang (2006), the stochastic small-gain theorem in Wu, Xie, and Zhang (2007), and the dynamic signal and changing supply function in Liu, Zhang, and Jiang (2007) and Wu, Xie, and Zhang (2006), different adaptive output-feedback control schemes were studied. For high-order stochastic systems in which $p_{i} \geq 1$ , $d_{i} (\cdot) = 1$ , $φ_{i j} ({\bar{x}}_{i}, θ) = φ_{i j} ({\bar{x}}_{i})$ , Xie and Tian (2007) considered state-feedback stabilization, which may be the first investigation on this type of control problem.

The study of parameter estimation and adaptive control for nonlinearly parameterized systems has long been recognized as difficult due to the inherent nonlinearity, in the sense that the system may be neither feedback linearizable nor affine in the control input, and its linearization is uncontrollable. Very few results are available in the literature even in the case of feedback linearizable systems with nonlinear parameterization. In the past few years, several researchers started working on this difficult problem and obtained some interesting results. As we know, there are several avenues to deal with the obstacle caused by the unknown nonlinear parameters which are exceptionally difficult to estimate. One way is to impose certain conditions on the system parameters such as convex/concave parameterization (see, e.g., Annaswamy, Skantze, and Loh (1998), Cao and Annaswamy (2006), Kojić, Annaswamy, Loh, and Lozano (1999) and Loh, Annaswamy, and Skantze (1999)), or a priori bounds on the nonlinear parameters (see, e.g., Marino and Tomei, 1993, Marino and Tomei, 1995). The second method is to use a parameter separation lemma which transforms the high-order nonlinearly parameterized systems into systems with linear parameterization, see Lin and Pongvuthithum (2003) and Lin and Qian, 2000, Lin and Qian, 2002a. The third one is to design a switching type controller, in which the controller parameter is tuned in a switching manner via a switching logic, as in Ye, 2003, Ye, 2005.

This paper is motivated by, and aims at the study of adaptive state-feedback stabilization for general high-order stochastic nonlinear systems with nonlinear parameterization (1). By using the parameter separation lemma in Lin and Qian (2002b) and some flexible algebraic techniques, and choosing an appropriate Lyapunov function, a smooth adaptive state-feedback controller is designed, which guarantees that the closed-loop system has an almost surely unique solution for any initial value $x_{0}$ , the equilibrium of interest is globally stable in probability and the state can be regulated to the origin almost surely. A meticulous simulation example is given to show the systematic design and efficiency of the controller.

The remainder of the paper is organized as follows: Section 2 begins with some preliminary results. In Section 3, an adaptive state-feedback controller is designed and analyzed. A simulation example is given in Section 4. Finally, the paper is concluded in Section 5.

Section snippets

Notations and preliminary results

The following notations will 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 $| X |$ denotes the Euclidean norm of a vector $X$ . $C^{i}$ denotes the set of all functions with continuous $i$ th partial derivatives. $K$ denotes the set of all functions: $R^{+} \to R^{+}$ , which are continuous, strictly increasing and vanishing at zero; $K_{\infty}$ denotes

Controller design and stability analysis

We need the following assumptions for system (1).

Assumption 1

For each smooth function $φ_{i j}$ $(i = 1, \dots, n, j = 1, 2)$ , there exists a known nonnegative smooth function ${\bar{γ}}_{i j}$ : $R^{i} \times R^{m} \to R^{+}$ such that for any ${\bar{x}}_{i}$ and $θ$ , $| φ_{i j} ({\bar{x}}_{i}, θ) | \leq ({| x_{1} |}^{p_{i}} + \dots + {| x_{i} |}^{p_{i}}) {\bar{γ}}_{i j} ({\bar{x}}_{i}, θ) .$

Assumption 2

$p_{1} \geq p_{2} \dots \geq p_{n} \geq 1$ are odd integers.

Assumption 3

Without loss of generality, the sign of $d_{i} ({\bar{x}}_{i}, θ)$ ( $i = 1, \dots, n$ ) is assumed to be positive, and there exist smooth functions $λ_{i} ({\bar{x}}_{i}) > 0$ and ${\bar{μ}}_{i} ({\bar{x}}_{i}, θ) > 0$ such that $λ_{i} ({\bar{x}}_{i}) \leq d_{i} ({\bar{x}}_{i}, θ) \leq {\bar{μ}}_{i} ({\bar{x}}_{i}, θ), i = 1, \dots, n .$

Remark 1

Assumption 1, Assumption 2 are of the same form

A simulation example

Consider the following stochastic nonlinear system $d x_{1} = d_{1} (x_{1}, θ) x_{2}^{3} d t + θ x_{1}^{2} sin x_{1} d t + \frac{1}{3} x_{1}^{2} sin x_{1} d ω,$ $d x_{2} = d_{2} ({\bar{x}}_{2}, θ) u^{3} d t + \frac{1}{3} θ x_{1}^{3} sin x_{2} d ω .$ It is easy to obtain that $| φ_{11} | = | θ x_{1}^{2} sin x_{1} | \leq {| x_{1} |}^{3} \sqrt{1 + θ^{2}},$ $| φ_{12} | = \frac{1}{3} | x_{1}^{2} sin x_{1} | \leq \frac{1}{3} {| x_{1} |}^{3}, | φ_{21} | = 0,$ $| φ_{22} | = \frac{1}{3} | θ x_{1}^{3} sin x_{2} | \leq \frac{1}{3} {| x_{1} |}^{3} \sqrt{1 + θ^{2}},$ Assumption 1 is satisfied with ${\bar{γ}}_{11} = \sqrt{1 + θ^{2}}$ , ${\bar{γ}}_{12} = \frac{1}{3}$ , ${\bar{γ}}_{21} = 0$ and ${\bar{γ}}_{22} = \frac{1}{3} \sqrt{1 + θ^{2}}$ . $p_{1} = p_{2} = 3$ , Assumption 2 is satisfied. In simulation, choose $d_{1} (x_{1}, θ) = 2 + \frac{1}{4} sin θ x_{1}$ , $d_{2} ({\bar{x}}_{2}, θ) = \frac{1}{2} (1 + θ x_{1}^{2})$ and $θ = 1$ . Obviously, there exist positive functions $λ_{i}$ , ${\bar{μ}}_{i} (i = 1, 2)$ to satisfy

Conclusions

This paper investigates the adaptive state-feedback stabilization of high-order stochastic nonlinear systems with nonlinear parameterization for the first time. In this paper, a smooth adaptive state-feedback controller is designed, which guarantees that the closed-loop system has an almost surely unique solution for any initial value $x_{0}$ , the equilibrium of interest is globally stable in probability, and the states can be regulated to the origin almost surely.

There are some remaining problems

Acknowledgements

This work was supported by Program for New Century Excellent Talents in University of China (NCET-05-0607), National Natural Science Foundation of China (60774010), National Basic Research Program of China (2009CB724000), Program for Summit of Six Types of Talents of Jiangsu Province (07-A-020), and Program for Fundamental Research of Natural Sciences in Universities of Jiangsu Province (07KJB510114).

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

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Jie Tian received her Ph.D. degree from Qufu Normal University in 2008. Now she is a teacher of Shandong Agricultural University, China. Her research interest includes stochastic nonlinear control theory.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Shuzhi Sam Ge under the direction of Editor Miroslav Krstic.

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Brief paperAdaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization☆

Abstract

Introduction

Section snippets

Notations and preliminary results

Controller design and stability analysis

A simulation example

Conclusions

Acknowledgements

Automatica

Systems and Control Letters

Systems and Control Letters

Systems and Control Letters

IEEE Transactions on Automatic Control

Systems and Control Letters

Systems and Control Letters

Automatica

Automatica

Automatica

A hierarchical discretized-parameter polynomial adaptive estimator for non-linearly parameterized systems

International Journal of Control

Stabilization of stochastic nonlinear driven by noise of unknown covariance

IEEE Transactions on Automatic Control

Stochastic stability of diffrential equations

Stabilization of uncertain nonlinear systems

Stochastic stability and control

Brief paper
Adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization☆