[ $H_{\infty}$ ] Multi-Step Prediction for Linear Discrete-Time Systems: A Distributed Algorithm

  • Wang, Hao-Qian (Department of Automation, Tsinghua University) ;
  • Zhang, Huan-Shui (School of Control Science and Engineering, Shandong University) ;
  • Hu, Hong (Shenzhen Graduate School of Harin Institute of Technology)
  • Published : 2008.02.28

Abstract

A new approach to $H_{\infty}$ multi-step prediction is developed by applying the innovation analysis theory. Although the predictor is derived by resorting to state augmentation, nevertheless, it is completely different from the previous works with state augmentation. The augmented state here is considered just as a theoretical mathematic tool for deriving the estimator. A distributed algorithm for the Riccati equation of the augmented system is presented. By using the reorganized innovation analysis, calculation of the estimator does not require any augmentation. A numerical example demonstrates the effect in reducing computing burden.

Keywords

References

  1. T. Kailath, A. Sayed, and B. Hassibi, Linear Estimation, Prentice-Hall, Englewood, Cliffs, NJ, 1999
  2. K. M. Nagpal and P. P. Khargonekar, "Filtering and smoothing in an $H_{\infty}$ setting," IEEE Trans. on Automatic Control, vol. 36, pp. 152-166, 1991 https://doi.org/10.1109/9.67291
  3. E. Fridman, U. Shaked, and L. H. Xie, "Robust $H_{\infty}$ filtering of linear systems with time-varying delay," IEEE Trans. on Automatic Control, vol. 48, no. 1, pp. 159-165, 2003 https://doi.org/10.1109/TAC.2002.806674
  4. U. Shaked, "$H_{\infty}$ optimal estimation-old and new result," Proc. of the 21st Brazilian Automatic Control Conference, Uberlandia, MG, Brasil, September 1998
  5. P. Colaneri and A. Ferrante, "A J-spectral factorization approach for $H_{\infty}$ estimation problem in discrete time," IEEE Trans. on Automatic Control, vol. 47, no. 12, pp. 2108-2113, 2002 https://doi.org/10.1109/TAC.2002.805666
  6. P. Colaneri and A. Ferrante, "Algebraic riccati equation and J-spectral factorization for $H_{\infty}$ estimation," System & Control Letters, vol. 51, no. 5, pp. 383-393, 2004 https://doi.org/10.1016/j.sysconle.2003.09.008
  7. P. Colaneri and A. Ferrante, "Algebraic riccati equation and J-spectral factorization for $H_{\infty}$ filtering and deconvolution," SIAM J. Contr. and Opt., vol. 45, no. 1, pp. 123-145, 2006 https://doi.org/10.1137/S0363012903434741
  8. H. Gao and C. Wang, "A delay-dependent approach to robust $H_{\infty}$ filtering for uncertain discretetime state-delayed systems," IEEE Trans. on Signal Processing, vol. 52, no. 6, pp. 1631-1640, 2004 https://doi.org/10.1109/TSP.2004.827188
  9. L. Mirkin and G. Tadmor, "Yet another $H_{\infty}$ disctetization," IEEE Trans. on Automatic Control, vol. 38, pp. 891-894, 2003
  10. G. Zames, "Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate," IEEE Trans. on Automatic Control, vol. 26, pp. 301-320, 1981 https://doi.org/10.1109/TAC.1981.1102603
  11. B. Hassibi, A. H. Sayed, and T. Kailath, Indefinite Quadratic Estimation and Control: A Unified Approach to H2 and $H_{\infty}$ Theories, SIAM Studies in Applied Mathematics Series, 1998
  12. Y. Theodor and U. Shaked, "Game theory approach to $H_{\infty}$ optimal discrete-time fixed-point and fixed-lag smoothing," IEEE Trans. on Automatic Control, vol. 39, pp. 1944-1948, 1994 https://doi.org/10.1109/9.317131
  13. H. Zhang, L. Xie, Y. C. Soh, and D. Zhang, "$H_{\infty}$ fixed-lag smoothing for linear time-varying discrete time systems," Automatica, vol. 41, no. 5, pp. 839-846, 2005 https://doi.org/10.1016/j.automatica.2004.11.028
  14. H. Zhang, L. Xie, D. Zhang, and Y. C. Soh, "A reorganized innovation approach to linear estimation," IEEE Trans. on Automatic Control, vol. 49, no. 10, pp. 1810-1814, 2004 https://doi.org/10.1109/TAC.2004.835599