Decentralized adaptive stabilization for interconnected systems with dynamic input–output and nonlinear interactions

doi:10.1016/j.automatica.2010.03.003

Automatica

Volume 46, Issue 6, June 2010, Pages 1060-1067

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Abstract

This paper considers the problem of robust decentralized adaptive output feedback stabilization for a class of interconnected systems with dynamic input and output interactions and nonlinear interactions by using MT-filters and the backstepping design method. It is shown that the closed-loop decentralized system based on MT-filters is globally uniformly bounded, all the signals except for the parameter estimates can be regulated to zero asymptotically, and the $L_{2}$ and $L_{\infty}$ norms of the system outputs are also be bounded by functions of design parameters. The scheme is demonstrated by a simulation example.

Introduction

In recent years, research on decentralized adaptive control using the backstepping approach has received great attentions due to its advantages such as improving transient performance, etc. The first result without any requirement on subsystem relative degree was reported by Wen (1994). Since then, more general class of systems with the consideration of unmodeled dynamics or nonlinear interactions were studied by Jain and Khorrami (1997), Jiang (2000), Wen and Soh (1997), and Zhang, Wen, and Soh (2000). More recently, decentralized adaptive stabilization for nonlinear systems with dynamic interactions dependent upon subsystem outputs or unmodeled dynamics was studied in Jiang and Repperger (2001) and Liu and Li (2002). Liu, Zhang, and Jiang (2007) gave a result for stochastic nonlinear systems with parametric uncertainties, nonlinear uncertain interactions and stochastic inverse dynamics, and Wen and Zhou (2007) considered a class of nonlinear systems with nonsmooth hysteresis nonlinearities and higher order nonlinear interactions. In Zhou and Wen (2008), a new scheme to design decentralized backstepping adaptive tracking controllers for a class of nonlinear interconnected systems in the presence of external disturbances was solved by adding two new terms in the parameter update laws compared with the conventional backstepping method in Krstić, Kanellakopoulos, and Kokotović (1995), in which the interactions between subsystems are unknown and allowed to satisfy a high order nonlinear bound. However, except for Jiang and Repperger (2001), Wen and Soh (1997), and Zhang et al. (2000), all the results are only applicable to systems with interaction effects bounded by static functions of subsystem outputs. In Wen, Zhou, and Wang (2009), interconnected systems by considering both input and output dynamic interactions were firstly settled by using K-filters and backstepping design method; the $L_{2}$ and $L_{\infty}$ norms of the system outputs are also established.

In the widely cited in-depth monograph on the backstepping design method, Krstić et al. (1995) systematically studied two sets of filters, namely K-filters and MT-filters, with different merits and demerits, and applied them respectively to the design of output feedback adaptive controllers. The design with MT-filters, which was firstly proposed by Marino and Tomei (1992) and Marino and Tomei (1993), is motivated by the idea of using an adaptive observer for output feedback control.

Inspired by these results, the purpose of this paper is to further address the same problem as in Wen et al. (2009) by using MT-filters and the backstepping design method. Our main contributions are composed of three parts.

(i) This paper considers a class of interconnected systems with dynamic input, output and nonlinear interactions. These interactions are more general than those in Wen et al. (2009).

(ii) Due to the dynamic interactions and unmodeled dynamics appearing in the output of the subsystem, the adaptive laws obtained by adopting the MT-filtered transformation equation (8.156) in Krstić et al. (1995) are not available for measurement. Therefore how to construct a new filtered transformation such that the adaptive laws can be available for measurement constitutes one of the main difficulties in this paper.

(iii) By establishing a few key lemmas which play an important role in the analysis of stability and asymptotic convergence, we show that the closed-loop decentralized system based on MT-filters is globally uniformly bounded, all the signals except for the parameter estimates can be regulated to zero asymptotically, and the $L_{2}$ and $L_{\infty}$ norms of the system outputs are also be bounded by functions of design parameters. The effectiveness of the proposed scheme is demonstrated by a simulation example.

Section snippets

Problem formulation

Consider a class of interconnected systems consisting of $N$ subsystems, and the $i$ th subsystem is modeled by $y_{i} (t) = G_{i} (s) (1 + ν_{i i} {\bar{Δ}}_{i i} (s)) u_{i} (t) + H_{i} (s) (1 + ν_{i i} {\bar{Δ}}_{i i} (s)) \sum_{j = 1}^{N} f_{i j} (t, y_{j}) + \sum_{j = 1}^{N} ν_{i j} H_{i j} (s) u_{j} (t) + \sum_{j = 1}^{N} ν_{i j} H_{i j} (s) F_{j} (s) \sum_{k = 1}^{N} f_{j k} (t, y_{k}) + \sum_{j = 1}^{N} μ_{i j} Δ_{i j} (s) y_{j} (t), i = 1, \dots, N,$ where $u_{i}, y_{i} \in R$ are respectively the input and output of the $i$ th subsystem, $s$ denotes the differential operator $\frac{d}{d t}$ , $F_{i} (s) = \frac{D_{i} (s)}{B_{i} (s)}$ , $G_{i} (s) = \frac{B_{i} (s)}{A_{i} (s)}$ , $H_{i} (s) = \frac{D_{i} (s)}{A_{i} (s)}$ , $A_{i} (s) = s^{n_{i}} + a_{i, n_{i} - 1} s^{n_{i} - 1} + \dots + a_{i, 0}$ , $B_{i} (s) = b_{i, m_{i}} s^{m_{i}} + b_{i, m_{i} - 1} s^{m_{i} - 1} + \dots + b_{i, 0}$ , $D_{i} (s) = (s^{n}$

The design of the robust decentralized adaptive controller based on MT-filters

By Appendix, the $i$ th subsystem (1) can be transformed into the following state-space realization ${\dot{x}}_{i} = A_{i} x_{i} - a_{i} x_{i, 1} + {\bar{b}}_{i} u_{i} + f_{i}, = A_{i} x_{i} + F_{i} {(u_{i}, y_{i})}^{T} θ_{i} + a_{i} ℵ_{i} + f_{i},$ $y_{i} = x_{i, 1} + ℵ_{i} = c_{i}^{T} x_{i} + ℵ_{i}, i = 1, \dots, N,$ where $ℵ_{i} = ν_{i i} {\bar{Δ}}_{i i} x_{i, 1} + \sum_{j = 1}^{N} ν_{i j} \frac{H_{i j}}{G_{j}} x_{j, 1} + \sum_{j = 1}^{N} μ_{i j} Δ_{i j} y_{j},$ $f_{i} = \sum_{j = 1}^{N} f_{i j} (t, y_{j}), A_{i} = [\begin{matrix} 0_{(n_{i} - 1) \times 1} & I_{n_{i} - 1} \\ 0 & 0_{1 \times (n_{i} - 1)} \end{matrix}],$ $a_{i} = [\begin{matrix} a_{i, n_{i} - 1} \\ ⋮ \\ a_{i, 0} \end{matrix}], {\bar{b}}_{i} = [\begin{matrix} 0 \\ b_{i, m_{i}} \\ ⋮ \\ b_{i, 0} \end{matrix}] = [\begin{matrix} 0 \\ b_{i} \end{matrix}], c_{i} = [\begin{matrix} 1 \\ ⋮ \\ 0 \end{matrix}],$ $F_{i} {(u_{i}, y_{i})}^{T} = [[\begin{matrix} 0_{(ϱ_{i} - 1) \times (m_{i} + 1)} \\ I_{m_{i} + 1} \end{matrix}] u_{i}, - I_{n_{i}} y_{i}],$ $θ_{i} = {[\begin{matrix} b_{i}^{T}, a_{i}^{T} \end{matrix}]}^{T} .$ To estimate the state of each subsystem, a local MT-filter only using input and output is designed as ${\dot{ξ}}_{i} = A_{l i} ξ_{i}$

Main results

We first introduce the similarity transformations $[\begin{matrix} ε_{i, 1} \\ π_{i} \end{matrix}] ≜ [\begin{matrix} ε_{i, 1} \\ T_{i} ε_{i} \end{matrix}] = [\begin{matrix} e_{n_{i}, 1}^{T} \\ T_{i} \end{matrix}] ε_{i},$ $[\begin{matrix} {\hat{χ}}_{i, 1} \\ φ_{i} \end{matrix}] ≜ [\begin{matrix} {\hat{χ}}_{i, 1} \\ T_{i} {\hat{χ}}_{i} \end{matrix}] = [\begin{matrix} e_{n_{i}, 1}^{T} \\ T_{i} \end{matrix}] {\hat{χ}}_{i},$ for $i = 1, 2, \dots, N$ , where $T_{i} = [A_{l i} e_{n_{i} - 1, 1}, I_{n_{i} - 1}] = [A_{l i}, e_{n_{i} - 1, n_{i} - 1}]$ . From (5), the definitions of $K_{0 i}$ , $A_{0 i}$ and $T_{i}$ , it is easy to verify the following properties $T_{i} l_{i} = 0, T_{i} K_{0 i} = A_{l i} {\bar{l}}_{i},$ $T_{i} A_{0 i} = A_{l i} T_{i}, K_{0 i} = c_{0 i} l_{i} + [\begin{matrix} {\bar{l}}_{i} \\ 0 \end{matrix}] .$ Combining (15), (25) with (27), one obtains ${\dot{π}}_{i} = A_{l i} π_{i} + T_{i} [(a_{i} + s e_{n_{i}, 1}) ℵ_{i} + f_{i}] = A_{l i} π_{i} + T_{i} [{\bar{a}}_{i} ℵ_{i} + (s + a_{i, n_{i} - 1}) e_{n_{i}, 1} ℵ_{i} + f_{i}],$ where ${\bar{a}}_{i} = {(0, a_{i, n_{i} - 2}, \dots, a_{i, 0})}^{T}$ . From the definition of $T_{i}$ and

A simulation example

Considered the following interconnected systems $y_{i} (t) = \frac{1}{s (s + a_{i, 1})} (1 + \frac{ν_{i i}}{s + 2}) u_{i} (t) + \frac{[s, 1]}{s (s + a_{i, 1})} (1 + \frac{ν_{i i}}{s + 2}) [\begin{matrix} y_{1} + sin y_{2} \\ y_{2} sin y_{2} \end{matrix}] + \sum_{j = 1}^{2} \frac{ν_{i j}}{{(s + 1)}^{3}} u_{j} (t) + \sum_{j = 1}^{2} \frac{ν_{i j} [s, 1]}{{(s + 1)}^{3}} [\begin{matrix} y_{1} + sin y_{2} \\ y_{2} sin y_{2} \end{matrix}] + \sum_{j = 1}^{2} \frac{μ_{i j}}{s + 1} y_{j} (t), i = 1, 2,$ (57) can be transformed into the state-space realization ${\dot{x}}_{i} = A_{i} x_{i} - [\begin{matrix} a_{i, 1} \\ a_{i, 0} \end{matrix}] x_{i, 1} + [\begin{matrix} 0 \\ b_{i} \end{matrix}] u_{i} + f_{i}$ $y_{i} = x_{i, 1} + ℵ_{i}, i = 1, 2,$ where $x_{i} = [\begin{matrix} x_{i, 1} \\ x_{i, 2} \end{matrix}]$ , $A_{i} = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$ , $f_{i} = [\begin{matrix} y_{1} + sin y_{2} \\ y_{2} sin y_{2} \end{matrix}]$ , $ℵ_{i} = \frac{ν_{i i}}{s + 2} x_{i, 1} + \sum_{j = 1}^{2} ν_{i j} \frac{s (s + a_{j, 1})}{{(s + 1)}^{3}} x_{j, 1} + \sum_{j = 1}^{2} \frac{μ_{i j}}{s + 1} y_{j}$ .

MT-filters are chosen as $\begin{matrix} {\dot{ξ}}_{i} = - l_{i} ξ_{i}, {\dot{η}}_{i} = - l_{i} η_{i} + y_{i}, {\dot{λ}}_{i} = - l_{i} λ_{i} + u_{i} . \end{matrix}$ The change of

A concluding remark

Our future work is to extend the proposed methodology to more general systems, such as stochastic interconnected systems with SiISS inverse dynamics in Yu and Xie (2010), and $A_{i}, c_{i}$ in (2) with more general forms. Another issue is to apply the scheme to a practical example.

Liang Liu Master student at Qufu Normal University. His current research interests include decentralized adaptive control and stochastic nonlinear control.

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Cited by (0)

Liang Liu Master student at Qufu Normal University. His current research interests include decentralized adaptive control and stochastic nonlinear control.

Xue-Jun Xie received the Ph.D. degree from the Institute of Systems Science, Chinese Academy of Sciences, in 1999. He is currently Professor in Qufu Normal University, China. His current research interests include stochastic nonlinear control systems and adaptive control.

^☆: This work was supported by National Natural Science Foundation of China (No. 60774010, 10971256, 60974127), Natural Science Foundation of Jiangsu Province (No. BK2009083), Program for Fundamental Research of Natural Sciences in Universities of Jiangsu Province (No. 07KJB510114), Shandong Provincial Natural Science Foundation of China (No. ZR2009GM008, ZR2009AL014). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Zhihua Qu under the direction of Editor Andrew R. Teel.

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Automatica

Brief paperDecentralized adaptive stabilization for interconnected systems with dynamic input–output and nonlinear interactions☆

Abstract

Introduction

Section snippets

Problem formulation

The design of the robust decentralized adaptive controller based on MT-filters

Main results

A simulation example

A concluding remark

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Decentralized adaptive output feedback design for large-scale nonlinear systems

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Decentralized and adaptive nonlinear tracking of large-scale systems via output feedback

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New results in decentralized adaptive non-linear stabilization using output feedback

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Global adaptive observers for nonlinear systems via filtered transformations

IEEE Transactions on Automatic Control

Brief paper
Decentralized adaptive stabilization for interconnected systems with dynamic input–output and nonlinear interactions☆