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Modal synthesis method for norm computation ofH ∞ decentralized control systems (I)

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Abstract

When using H∞ techniques to design decentralized controllers for large systems, the whole system is divided into subsystems, which are analysed using H∞ control theory before being recombined. An analogy was established with substructural analysis in structural mechanics, in which H∞ decentralized control theory corresponds to substructural modal synthesis theory so that the optimal H∞ norm of the whole system corresponds to the fundamental vibration frequency of the whole structure. Hence, modal synthesis methodology and the extended Wittrick-Williams algorithm were transplanted from structural mechanics to compute the optimal H∞ norm of the control system. The orthogonality and the expansion theorem of eigenfunctions of the subsystems H∞ control are presented in part (I) of the paper. The modal synthesis method for computation of the optimal H∞ norm of decentralized control systems and numerical examples are presented in part (II).

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References

  1. Ho Y C, Mitter S K.Directions in Large-Scale Systems: Many-Person Optimization and Decentralized Control[M]. New York: Plenum Press, 1976.

    Google Scholar 

  2. Jamshidi M.Large-Scale Systems—Modeling, Control and Fuzzy Logic [M]. New Jersey: Prentice-Hall, 1997.

    Google Scholar 

  3. Cheng C F. Disturbances attenuation for interconnected systems by de-centralized control[J].International Journal of Control, 1997,66(2):213–224.

    Article  MATH  MathSciNet  Google Scholar 

  4. Date R A, Chow J H. A parametrization approach to optimalH 2 andH ∞ decentralized control problems[J].Automatica, 1992,29(2):457–463.

    Article  MathSciNet  Google Scholar 

  5. Veillette R J, Medanic J V, Perkins W R. Design of reliable control systems [J].IEEE Transactions on Automatic Control, 1992,37(3):290–304.

    Article  MATH  MathSciNet  Google Scholar 

  6. ZHONG Wan-xie, Zhong Xiang-xiang. Computational structural mechanics, optimal control and semi-analytical method for PDE[J].Computers & Structures, 1990,37(6):993–1004.

    Article  MathSciNet  Google Scholar 

  7. ZHONG Wan-xie, Howson W P, Williams F W.H ∞ control state feedback and Rayleigh quotient [J].Computer Methods in Applied Mechanics and Engineering, 2001,191(3–5):489–501.

    MathSciNet  Google Scholar 

  8. ZHONG Wan-xie, Williams F W.H ∞ filtering with secure eigenvalue calculation and precise integration[J].International Journal for Numerical Methods in Engineering, 1999,46(7):1017–1030.

    Article  MathSciNet  Google Scholar 

  9. ZHONG Wan-xie. Variational method and computation forH ∞ control[J].Applied Mathematics and Mechanics (English Edition), 2000,21(12):1407–1416.

    MathSciNet  Google Scholar 

  10. ZHONG Wan-xie, YANG Zai-shi. On the computation of the main eigen-pairs of the continuous-time linear quadratic control problems[J].Applied Mathematics and Mechanics (English Edition), 1991,12(1):49–54. (in Chinese)

    MathSciNet  Google Scholar 

  11. ZHONG Wan-xie, OUYANG Hua-jiang, DENG Zi-chen.Computational Structural Mechanics and Optimal Control[M]. Dalian: Dalian University of Technology Press, 1993. (in Chinese)

    Google Scholar 

  12. ZHONG Wan-xie, Williams F W. A precise time step integration method[J].Proceedings of the Institution of Mechanical, Engineers Part C-Journal of ME, 1994,208(C6):427–430.

    Google Scholar 

  13. ZHONG Wan-xie. On the Precise integration of Riccati differential equation[J].Computational Structural Mechanics and Its Applications, 1994,11(2):113–119. (in Chinese)

    Google Scholar 

  14. ZHONG Wan-xie. The method of precise integration of finite strip and wave guide problems[A]. In: P K K Lee, L G Tham, Y K Cheung Eds.Proceedings of International Conference on Computational Methods in Structure and Geotechnical Engineering[C]. Vol1. Hong Kong: China Translation & Printing Service Ltd, 1994,51–59.

    Google Scholar 

  15. ZHONG Wan-xie. Precise integration of eigen-waves for layered media[A]. In: Arantese Oliveira, Joao Bento Eds.Proc EPMESC-5[C]. Vol2. Taejon, Korea: Techno-Press, 1995,1209–1220.

    Google Scholar 

  16. Leung A Y T.Dynamic Stiffness & Sub-Structures [M]. London: Springer, 1993.

    Google Scholar 

  17. WANG Wen-liang, DU Zuo-run.Structural Vibration and Dynamic Sub-Structural Analysis[M]. Shanghai: Fudan University Press, 1985. (in Chinese)

    Google Scholar 

  18. ZHONG Wan-xie, Williams F W, Bennett P N. Extension of the Wittrick-Williams algorithm to mixed variable systems[J].Journal of Vibration and Acoustics, Transactions of the ASME, 1997,119(3):334–340.

    Google Scholar 

  19. ZHONG Wan-xie.Duality Method in Applied Mechanics[M]. Beijing: Science Press, 2002. (in Chinese)

    Google Scholar 

  20. Courant R, Hilbert D.Methods of Mathematical Physics (Vol I) [M]. New York: Interscience Publishers Inc, 1953.

    Google Scholar 

  21. Arthurs A M.Complementary Variational Principles [M]. Oxford: Clarendon Press, 1980.

    Google Scholar 

  22. ZHONG Wan-xie, WU Zhi-gang, GAO Qiang,et al. Modal synthesis method for norm computation ofH ∞ decentralized control systems(II) [J].Applied Mathematics and Mechanics (English Edition), 2004,25(2):135–142.

    Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, 116023, Dalian, P. R. China

    Zhong Wan-xie Professor, Academician of Chinese Academy of Science, Wu Zhi-gang & Gao Qiang

  2. Visiting City University of Hong Kong, Hong Kong, P. R. China

    Zhong Wan-xie Professor, Academician of Chinese Academy of Science

  3. Department of Building and Construction, City University of Hong Kong, Hong Kong, P. R. China

    A. Y. T. Leung & F. W. Williams

Authors
  1. Zhong Wan-xie Professor, Academician of Chinese Academy of Science
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  2. Wu Zhi-gang
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  3. Gao Qiang
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  4. A. Y. T. Leung
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  5. F. W. Williams
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Additional information

Contributed by ZHONG Wan-xie

Foundation items: the National Key Basic Research Special Foundation of China (G1999032805); the National Natural Science Foundation of China (19732020, 10202004); the UK Engineering and Physical Sciences Research Council(GR/R05437/01)

Biography: ZHONG Wan-xie (1934 ∼)

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Wan-xie, Z., Zhi-gang, W., Qiang, G. et al. Modal synthesis method for norm computation ofH ∞ decentralized control systems (I). Appl Math Mech 25, 123–134 (2004). https://doi.org/10.1007/BF02437313

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  • DOI: https://doi.org/10.1007/BF02437313

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2000 Mathematics Subject Classification

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