Decentralized adaptive output-feedback controller design for stochastic nonlinear interconnected systems

doi:10.1016/j.automatica.2012.08.022

Automatica

Volume 48, Issue 11, November 2012, Pages 2866-2873

https://doi.org/10.1016/j.automatica.2012.08.022 Get rights and content

Abstract

In this paper, the controller design for a class of stochastic interconnected systems with both parametric uncertainties and unknown nonlinear interactions is presented. The diffusion terms considered are dependent on the outputs of local subsystems and allowed to be unbounded. First, by employing decentralized state observers, totally decentralized adaptive tracking controllers with suitable parameter adaptive laws are designed to ensure the boundedness in probability of all the signals in the closed-loop system. It is shown that the tracking errors converge to small residual sets around the origin. Then, for systems with relaxed diffusion vector fields, a decentralized adaptive stabilizing scheme is proposed to ensure the boundedness in probability of all signals in the closed-loop system.

Introduction

The notion of interconnected systems is introduced to describe complex systems consisting of subsystems interacting on each other; see for example Siljak (1991). Inspired by the fruitful results on centralized stochastic nonlinear adaptive control, attention has been paid to the stochastic nonlinear interconnected systems (Fan and Xie, 2010, Liu and Xie, 2010, Liu et al., 2007, Wu and Deng, 2008). However, results for stochastic nonlinear interconnected systems are still very limited compared to their deterministic counterpart. The main challenges are: (1) in the Itô stochastic differentiation, there exists a second-order derivative term, which brings difficulties in controller design and analysis; (2) the structure and the fundamental theory for stochastic interconnected systems are quite different and in fact much more complex than those in the deterministic case.

It is noted that, in the results on the stabilization problem, the diffusion vector fields of the considered stochastic systems are generally dependent on states or outputs of the local systems. While for the tracking problem, due to complicated couplings of tracking errors among subsystems, the diffusion vector fields have to satisfy some restrictive assumptions or to be bounded (Arslan and Basar, 2002, Fan and Ge, 2004, Fan and Xie, 2010, Ji and Xi, 2006, Wu and Deng, 2008). Efforts towards risk sensitive tracking for a class of centralized stochastic nonlinear systems were made in Arslan and Basar (2002), in which the diffusion terms were assumed to vanish along the reference trajectory. Obviously, by the assumption, the diffusion vector fields are related to the reference trajectory, which limits the applicability of the result. Under a similar assumption, asymptotic tracking was investigated in Fan and Ge (2004) for a centralized controlled stochastic nonlinear system, and its main idea was extended to a class of stochastic interconnected systems in Wu and Deng (2008). In Ji and Xi (2006), for a class of centralized nonlinear systems, an adaptive tracking scheme was proposed to achieve global tracking for any given reference signal in the way of stochastic disturbance attenuation. The stochastic disturbances considered there are not related to the system outputs and have bounded covariance.

Recently, the methodology developed in Zhou and Wen (2008) on decentralized backstepping adaptive tracking of deterministic interconnected nonlinear systems, was generalized to its stochastic counterpart in Fan and Xie (2010). However, the considered stochastic exogenous disturbances are not related to system outputs and the diffusion terms are bounded by an unknown constant.

In this paper, we will first consider the tracking problem for an enlarged class of stochastic interconnected systems, in which the diffusion vector fields depend on the outputs of the local subsystems, and are not necessary to be bounded. A scheme to design totally decentralized adaptive controllers is proposed. Then, for such a class of systems with a relaxed requirement in the diffusion fields, the scheme is modified to achieve decentralized adaptive stabilization in probability. Simulation studies are conducted to illustrate the effectiveness of the proposed schemes.

Section snippets

Problem formulation

In this paper, we investigate a class of stochastic nonlinear large-scaled systems consisting of $N$ interconnected subsystems modelled as below $d x_{i} = [A_{0, i} x_{i} + Φ_{i} (y_{i}) a_{i} + [\begin{matrix} 0 \\ b_{i} \end{matrix}] u_{i} + \sum_{k = 1}^{N} f_{i, k} (t, y_{k})] d t + g_{i}^{T} (y_{i}) d ω_{i},$ $y_{i} = x_{i, 1}, for i = 1, \dots, N,$ where $A_{0, i} = [\begin{matrix} 0 \\ ⋮ & I_{n_{i} - 1} \\ 0 & \dots & 0 \end{matrix}], b_{i} = [\begin{matrix} b_{i, m_{i}} \\ ⋮ \\ b_{i, 0} \end{matrix}], Φ_{i} (y_{i}) = [\begin{matrix} Φ_{i, 1} (y_{i}) \\ ⋮ \\ Φ_{i, n_{i}} (y_{i}) \end{matrix}],$ $x_{i} = {[x_{i, 1}, \dots, x_{i, n_{i}}]}^{T} \in R^{n_{i}}, u_{i} and y_{i} \in R$ represent the state vector, control input to be designed and output of the $i$ th subsystem, respectively; $ω_{i}$ is an $l_{i}$ -dimensional standard Wiener process; $a_{i} = {[a_{i, 1}, \dots, a_{i, r_{i}}]}^{T} \in R^{r}$

Decentralized state estimation filter

In this section, the state estimation filter will be designed to estimate the unmeasurable states of each local system, which employs local input and output only.

First, choose $K_{i} = {[k_{i, 1}, \dots, k_{i, n_{i}}]}^{T} \in R^{n_{i}}$ such that the matrix $A_{i} = A_{0, i} - K_{i} {(e_{n_{i}, 1})}^{T}$ is Hurwitz, and $e_{n, k}$ denotes the $k$ th coordinate vector in $R^{n}$ .

Define ${\hat{x}}_{i} = η_{i} + Ξ_{i} a_{i} + \sum_{ι = 0}^{m_{i}} b_{i, ι} v_{i, ι},$ where ${\begin{cases} {\dot{η}}_{i} = A_{i} η_{i} + K_{i} y_{i}, & η_{i} \in R^{n_{i}}, \\ {\dot{Ξ}}_{i} = A_{i} Ξ_{i} + Φ_{i} (y_{i}), & Ξ \in R^{n_{i} \times r_{i}}, \\ {\dot{ξ}}_{i} = A_{i} ξ_{i} + e_{n_{i}, n_{i}} u_{i}, & ξ_{i} \in R^{n_{i}}, \\ v_{i, ι} = A_{i}^{ι} ξ_{i}, & ι = 0, 1, \dots, m_{i} . \end{cases}$ In light of equation $A_{i}^{ι} e_{n_{i}, n_{i}} = e_{n_{i}, n_{i} - ι}, 0 \leq ι \leq n_{i}$ , we obtain that $v$

Decentralized adaptive tracking

In the sequel, adaptive tracking controllers are designed by employing a backstepping scheme (Krstic & Deng, 1998).

For $i = 1, \dots, N$ and $j = 2, \dots, ρ_{i}$ , define $z_{i, 1} = y_{i} - y_{r i},$ $z_{i, j} = v_{i, m_{i}, j} - α_{i, j - 1} (y_{i}, ξ_{i, 1}, \dots, ξ_{i, m_{i} + j - 1}, η_{i}, Ξ_{i}, {\hat{θ}}_{i}, {\hat{ζ}}_{i}, {\bar{Y}}_{r i, j - 1}),$ where $α_{i, j - 1} (y_{i}, ξ_{i, 1}, \dots, ξ_{i, m_{i} + j - 1}, η_{i}, Ξ_{i}, {\hat{θ}}_{i}, {\hat{ζ}}_{i}, {\bar{Y}}_{r i, j - 1})$ is a stabilizing function at the $j$ th step of the $i$ th loop and will be determined step-by-step later, ${\bar{Y}}_{r i, j} ≜ {[y_{r i}, {\dot{y}}_{r i}, \dots, y_{r i}^{(j)}]}^{T}$ . For notation simplicity, here and hereafter the arguments of $α_{i, j - 1} (y_{i}, ξ_{i, 1}, \dots, ξ_{i, m_{i} + j - 1},$

Decentralized adaptive stabilization

In this section, we will consider the stabilization problem for system (1)–(2) with a relaxed assumption on the diffusion vector fields. A decentralized adaptive output-feedback controller is designed to ensure the boundedness in probability of the closed-loop system.

Assumption 6

In the diffusion vector field of (1), $g_{i} (y_{i}) \in R^{l_{i} \times n_{i}}$ is smooth and vanishing at $y_{i} = 0$ , which implies $g_{i} (y_{i}) = y_{i} {\bar{g}}_{i} (y_{i}),$ where ${\bar{g}}_{i} (y_{i})$ is a smooth matrix.

Remark 5

Note that in Assumption 6, ${\bar{g}}_{i} (y_{i})$ is no longer required to be bounded. This is

An illustrative example

In this section, we illustrate the effectiveness of our proposed decentralized control schemes by considering the following stochastic nonlinear interconnected system with $N = 2$ . $Σ_{1} : {\begin{cases} [\begin{matrix} d x_{11} \\ d x_{12} \end{matrix}] = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] [\begin{matrix} x_{11} \\ x_{12} \end{matrix}] d t + [\begin{matrix} 2 y_{1} & y_{1}^{2} \\ 0 & y_{1} \end{matrix}] a_{1} d t + [\begin{matrix} 0 \\ b_{1} \end{matrix}] u_{1} d t + [\begin{matrix} 0 \\ sin (y_{1}) + 0.2 y_{2} \end{matrix}] d t + g_{1}^{T} (y_{1}) d w_{1}, \\ y_{1} = x_{11}; \end{cases}$ $Σ_{2} : {\begin{cases} [\begin{matrix} d x_{21} \\ d x_{22} \end{matrix}] = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] [\begin{matrix} x_{21} \\ x_{22} \end{matrix}] d t + [\begin{matrix} 0 & 0 \\ y_{2} & 1 + y_{2} \end{matrix}] a_{2} d t + [\begin{matrix} 0 \\ b_{2} \end{matrix}] u_{2} d t + [\begin{matrix} 0.5 sin (y_{1}) + 0.1 y_{2} \\ 0 \end{matrix}] d t + g_{2}^{T} (y_{2}) d w_{2}, \\ y_{2} = x_{21} . \end{cases}$ The unknown parameters are $a_{1} = {[1.5 2]}^{T}, a_{2} = {[1 1]}^{T}, b_{1} = 1, b_{2} = 2$ . All initial values of the parameter estimations and the filtered

Conclusions

In this paper, we have considered decentralized adaptive output-feedback control for a class of stochastic nonlinear interconnected systems. Our main contributions can be summarized as follows: (1) for systems with diffusion vector fields depending on the outputs of subsystems and being not necessary bounded, the tracking problem is solved with our proposed scheme; (2) the methodology previously developed for deterministic interconnected systems in Zhou and Wen (2008) is generalized to

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Huijin Fan received the B.S. degree in Mathematics from the Central China Normal University, Wuhan, China, in 1995, the M.S. degree in Applied Mathematics from the Chinese Academy of Sciences, China, in 1998, and the Ph.D. degree in Electrical Engineering from Nanyang Technological University, Singapore, in 2002. In 2002, she was a Research Associate at the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. Then, she joined the National University of Singapore, as a Research Fellow in the Department of Electrical and Computer Engineering. From April 2004 to July 2005, she was a Visiting Research Fellow at the Department of Electronics and Information Systems, Akita Prefectural University, Akita, Japan. Since Aug. 2005, she has been an Associate Professor at the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China. Her research interests include nonlinear stochastic system theory and multidimensional system theory.

Lixia Han received the B.S. degree and the Master’s degree in Control Engineering from the Harbin Engineering University, China, in 2009 and the Huazhong University of Science and Technology, China, in 2011, respectively. Then, she has been an Engineer in Huawei Technologies Co., Ltd, China. Her research interests include stochastic control and nonlinear system theory.

Changyun Wen received the B.Eng. degree from Xi’an Jiaotong University, China, in July 1983 and the Ph.D. degree from the University of Newcastle, Australia, in February 1990. From August 1989 to August 1991, he was a Postdoctoral Fellow with the University of Adelaide, Australia. Since August 1991, he has been with the School of Electrical and Electronic Engineering, Nanyang Technological University, where he is currently a Professor. His main research activities are in the areas of adaptive control, development of battery management systems, ejector based air-conditioning systems, switching and impulsive systems, system identification, control and synchronization of chaotic systems, and biomedical signal processing.

Dr. Wen is an Associate Editor of a number of journals including AUTOMATICA and the IEEE CONTROL SYSTEMS MAGAZINE. He also served the IEEE TRANSACTIONS ON AUTOMATIC CONTROL as an Associate Editor from January 2000 to December 2002. He has been actively involved in organizing international conferences playing the roles of General Chair, General Co-Chair, Technical Program Committee Chair, Program Committee Member, General Advisor, Publicity Chair, and so on. He received the IES Prestigious Engineering Achievement Award 2005 from the Institution of Engineers, Singapore (IES) in 2005.

He is a Fellow of IEEE, a Member of the IEEE Fellow Committee and a Distinguished Lecturer of IEEE Control Systems Society.

Li Xu received the B.Eng. degree from Huazhong University of Science and Technology, Wuhan, China, in 1982, and the M.Eng. and Dr. Eng. degrees from Toyohashi University of Technology, Toyohashi, Japan, in 1990 and 1993, respectively. From April 1993 to March 1998, he was an Assistant Professor at the Department of Knowledge-Based Information Engineering, Toyohashi University of Technology. From April 1998 to March 2000, he was a Lecturer at the Department of Information Management, Asahi University, Gifu, Japan. Since April 2000, he has been with the Faculty of Systems Science and Technology, Akita Prefectural University, Akita, Japan, where he is currently a Professor at the Department of Electronics and Information Systems. His research interests include multidimensional system theory, signal processing and the applications of computer algebra to system theory. Dr. Xu has been an Associate Editor for the international journal of Multidimensional Systems and Signal Processing (MSSP) since 2000.

^☆: This work was supported by the National Natural Science Foundation of China (No. 61174079 and No. 61034006). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Shuzhi Sam Ge under the direction of Editor Miroslav Krstic.

¹: Tel.: +86 27 87557992; fax: +86 27 87543130.

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Brief paperDecentralized adaptive output-feedback controller design for stochastic nonlinear interconnected systems☆

Abstract

Introduction

Section snippets

Problem formulation

Decentralized state estimation filter

Decentralized adaptive tracking

Decentralized adaptive stabilization

An illustrative example

Conclusions

Automatica

Automatica

Risk sensitive adaptive trackers for strick-feedback systems with output measurements

IEEE Transactions on Automatic Control

Inequalities

Brief paper
Decentralized adaptive output-feedback controller design for stochastic nonlinear interconnected systems☆