Exponential stabilization controller design for interconnected time delay systems

doi:10.1016/j.automatica.2008.02.010

Automatica

Volume 44, Issue 10, October 2008, Pages 2600-2606

https://doi.org/10.1016/j.automatica.2008.02.010 Get rights and content

Abstract

Decentralized robust control problem is investigated for a class of large scale systems with time varying delays. The considered systems have mismatches in time delay functions. A state coordinate transformation is first employed to change the original system into a cascade system. Then the virtual linear state feedback controller is developed to stabilize the first subsystem. Based on the virtual controller, a memoryless state feedback controller is constructed for the second subsystem. By choosing new Lyapunov Krasovskii functional, we show that the designed decentralized continuous adaptive controller makes the solutions of the closed-loop system exponentially convergent to a ball, which can be rendered arbitrary small by adjusting design parameters. Finally, a numerical example is given to show the feasibility and effectiveness of the proposed design techniques.

Introduction

Many practical systems are of large scale systems and consist of a set of interconnected subsystems in the real world, such as power systems, digital communication networks, economic systems and urban traffic networks. Robust control for large-scale systems have been one of the focused study topics in the past years, and a lot of achievements have been made, see Jiang (2004), Siljak (1991), Wen (1994), Zhang, Wen, and Soh (2000), and the references therein. However, the systems investigated in the above quoted literature are free of the time delays.

It is well known that time delays are frequently encountered in various engineering systems and can be cause of instability (Gu, Kharitonov, & Chen, 2003). The information transmission among the subsystems often induces appearance of time delays in the interconnected systems. For large scale systems with linear interconnection, the stability analysis and control problem have been extensively investigated. Ikeda and Siljak (1980) first introduced time delay into decentralized control of large scale systems and investigated the exponential stabilization problem. In Almi and Derbel (1995) and Lee and Radovic (1988), the control problem was considered for a class of time invariant large scale interconnected systems free of uncertainties subject to multiple constant delays. In Mahmoud and Bingulac (1998), the robust control problem was considered for a class of interconnected systems with interconnections free of time delays. The stabilization problem of large-scale stochastic systems with time delays was studied in Xie and Xie (2000), while stabilization of a class of time-varying large scale systems subject to multiple time-varying delays in the interconnections was investigated in Oucheriah (2000). Shyua, Liu, and Hsu (2005) proposed variable structure control method for large scale systems with known linear time delay interconnection. Based on linear matrix inequality approach, the stability criteria were presented in De Souza (2001) and Lin, Wang, and Lee (2006). For the case that the uncertain interconnections are bounded by linear functions with unknown coefficients, Wu (2002) presented the adaptive controller design methodology. Oucheriah (2005) further considered the input nonlinearity case and the closed-loop system was shown to be exponentially stable. By analysis on the above cited literatures, there are the following restrictions: (i) The subsystems should be linearly interconnected and (ii) The systems considered often satisfy the matching condition. Obviously, these conditions will limit the application of the achievements of the former literatures. With the interconnections bounded by polynomial functions, Hua, Guan, and Shi (2005) proposed the decentralized control design method.

In this paper, the above restricted conditions are removed on the interconnected time delay systems. We consider a class of interconnected time delay systems with mismatched time delay functions and general nonlinear interconnections. By changing the subsystem into a cascade system, we successfully dispose of the mismatched function. With the help of proposed novel nonlinear Lyapunov Krasovskii functional, the uncertain nonlinear time delay interconnections are well dealt with. The decentralized memoryless state feedback controller is designed such that the solutions of the resulting closed-loop system are uniformly ultimately bounded and exponentially convergent towards a ball with adjustable radius. Finally, numerical simulation is presented to show the potential of the proposed techniques.

Section snippets

System formulation and preliminaries

Consider an interconnected system with the $i$ th subsystem described by $S_{i} : {\dot{x}}_{i} (t) = A_{i} x_{i} (t) + A_{d i} x_{i} (t - τ_{i} (t)) + B_{i} (u_{i} + H_{i} (t, x_{1} (t), x_{2} (t), \dots, x_{N} (t), x_{1} (t - d_{i 1} (t)), x_{2} (t - d_{i 2} (t)), \dots, x_{N} (t - d_{i N} (t)))),$ where $N$ is the total number of subsystems in the large scale system, $x_{i} \in ℜ^{n_{i}}$ and $u_{i} \in ℜ^{m_{i}}$ represent the state and control vectors of the $i$ -th subsystem, respectively. $A_{i}, A_{d i} \in ℜ^{n_{i} \times n_{i}}$ and $B_{i} \in ℜ^{n_{i} \times m_{i}}$ are known constant matrices. Without loss of generality, we assume $Rank (B_{i}) = m_{i}$ . $H_{i} (\cdot)$ are uncertain nonlinear interconnections,

Controller design

First, we show how to determine the virtual control law $K_{i} y_{i 1} (t)$ . For $y_{i 1}$ -subsystem, choose the following Lyapunov functional $V_{i} = y_{i 1}^{T} (t) P_{i} y_{i 1} (t) + W_{i},$ with $W_{i} = \frac{1}{1 - τ_{i}^{*}} \int_{t - τ_{i} (t)}^{t} e^{- γ_{i} (t - ξ)} y_{i 1}^{T} (ξ) Q_{i} y_{i 1} (ξ) d ξ + \frac{ε_{i 2}^{- 1} e^{γ_{i} {\bar{τ}}_{i}}}{1 - τ_{i}^{*}} \int_{t - τ_{i} (t)}^{t} e^{- γ_{i} (t - ξ)} {‖ y_{i 2} (ξ) ‖}^{2} d ξ,$ where $P_{i}$ and $Q_{i}$ are positive matrices, $γ_{i}$ and $ε_{i 2}$ are positive scalars.

With Lyapunov functional (8), we have the following preliminary result:

Lemma 1

For system(7), if there exist positive matrices $M_{i}, L_{i}$ and matrix $N_{i}$ such that the following LMI holds for $i = 1, 2, \dots, N$ $Ψ_{i} =$

Numerical example

In this section, we give a numerical example to show the validity of the controllers designed in previous section.

Consider an interconnected system (1) composed of two subsystems with $A_{1} = [\begin{matrix} - 1 & 1 & 0 \\ 0 & 1 & 2 \\ 1 & 1 & 1 \end{matrix}], A_{d 1} = [\begin{matrix} - 0.1 & 0 & 0 \\ 0 & - 0.1 & 0.1 \\ 1 & 2 & 1 \end{matrix}],$ $A_{2} = [\begin{matrix} - 0.1 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & 1 \end{matrix}], A_{d 2} = [\begin{matrix} 0.1 & 0 & 0 \\ 0.1 & 0.1 & 0.1 \\ 1 & 2 & 3 \end{matrix}],$ and $B_{1} = B_{2} = {[\begin{matrix} 0 & 0 & 1 \end{matrix}]}^{T}$ , the interconnections are $H_{1} = c_{11} (t) \sum_{l = 1}^{3} x_{2 l} (t - d_{12} (t)) + c_{12} (t) x_{1}^{T} (t) x_{2} (t - d_{12} (t))$ $H_{2} = c_{21} (t) \sum_{l = 1}^{3} x_{1 l} (t - d_{21} (t)) + c_{22} (t) x_{2}^{T} (t) x_{1} (t - d_{21} (t)),$ where $c_{i l} (t)$ are bounded time-varying parameters with bounds unknown.

Now, we

Conclusion

It is a challenging problem for designing a decentralized controller for interconnected time delay systems with mismatched function and uncertain nonlinear interconnections. A new methodology is proposed to deal with the controller design problem in this paper. With the developed new nonlinear Lyapunov Krasovskii functional, we show that the designed controller renders that the closed-loop system has good transient state performance (exponentially converge) and good steady state performance

Acknowledgements

The authors are grateful to the associate editor and the anonymous reviewers for their valuable comments and suggestions that have helped improve the presentation of the paper. The work was partially supported in part by the NSFC Grants 60525303, 60604004.

Chang-Chun Hua received the Ph.D. degree in electrical engineering from Yanshan University, Qinhuangdao, China, in 2005. He is currently a Associate Professor in the Institute of Electrical Engineering, Yanshan University. His research interests are in nonlinear control systems, control systems design over network, tele-operation systems and intelligent control.

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Qing-Guo Wang received, respectively, the B.Eng. in Chemical Engineering in 1982, the M. Eng. in 1984 and Ph.D. in 1987, both in Industrial Automation, all from Zhejiang University, China. Since 1992 he has been with the Department of Electrical and Computer Engineering, National University of Singapore, where he is currently a Full Professor. He has published four research monographs and more than one hundred international refereed journal papers and holds 4 patents. His present research interests are mainly in robust, adaptive and multivariable control and optimization with emphasis on their applications in process, chemical and environmental industries. He was an Associate Editor for J. of Process Control, Jan 2002–Dec 2004; is an Associate Editor for ISA Transactions since 2000. He was the general chair of the 4th Asian Control Conference, Singapore, 2002; and the fourth International Conference on Control and Automation, Montreal, Canada, 2003.

Xin-Ping Guan received the B.S. degree in mathematics from Harbin Normal University, Harbin, China, and the M.S. degree in applied mathematics and the Ph.D. degree in electrical engineering, both from Harbin Institute of Technology, in 1986, 1991, and 1999, respectively. He is currently a Professor and Dean of the Institute of Electrical Engineering, Yanshan University, Qinhuangdao, China. He is the (co)author of more than 100 papers in mathematical, technical journals, and conferences. As (a)an (co)-investigator, he has finished more than 17 projects supported by National Natural Science Foundation of China (NSFC), the National Education Committee Foundation of China, and other important foundations. He is “Cheung Kong Scholars Programme” Special appointment professor. His current research interests include functional differential and difference equations, robust control and intelligent control for time-delay systems, chaos control and synchronization, and congestion control of networks. Dr. Guan is serving as a Reviewer of Mathematic Review of America, a Member of the Council of Chinese Artificial Intelligence Committee, and Vice-Chairman of Automation Society of Hebei Province, China.

^☆: This paper was not presented at any IFAC meeting. 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 paperExponential stabilization controller design for interconnected time delay systems☆

Abstract

Introduction

Section snippets

System formulation and preliminaries

Controller design

Numerical example

Conclusion

Acknowledgements

Chaos, Solitons and Fractals

Automatica

Automatica

Automatica

New hierarchical control algorithm for large scale time-delay systems

Control and Computers

Decentralized control of interconnected linear systems with delayed states

Kybernetika

Stability of time-delay systems

Robust adaptive control for uncertain time-delay systems

International Journal of Adaptive Control and Signal Processing

Brief paper
Exponential stabilization controller design for interconnected time delay systems☆