Skip to main content
SpringerLink
Log in

Linear Quadratic Regulation for Discrete-Time Systems with Input Delay: Spectral Factorization Approach

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

The infinite-horizon linear quadratic regulation (LQR) problem is settled for discrete-time systems with input delay. With the help of an autoregressive moving average (ARMA) innovation model, solutions to the underlying problem are obtained. The design of the optimal control law involves in resolving one polynomial equation and one spectral factorization. The latter is the major obstacle of the present problem, and the reorganized innovation approach is used to clear it up. The calculation of spectral factorization finally comes down to solving two Riccati equations with the same dimension as the original systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. H. Davis, Foundations of Deterministic and Stochastic Control, Birkhauser, Boston, 2002.

    Google Scholar 

  2. K. J. Astrom, Introduction to Stochastic Control Theory, Academic Press, New York, 1970.

    Google Scholar 

  3. L. Chisci and E. Mosca, MMSE deconvolution via polynomial methods and its dual LQG regulation, Automatica, 1994, 30(7): 1197–1201.

    Article  Google Scholar 

  4. U. Shaked, A general transfer-function approach to the steady-state linear quadratic Gaussian stochastic control problem, International Journal of Control, 1976, 24(6): 771–800.

    Article  Google Scholar 

  5. M. Sternad and A. Ahlen, A novel derivation methodology for polynomial-LQ controller design, IEEE Trans. Automatic Control, 1993, 38(1): 116–121.

    Article  Google Scholar 

  6. V. Kucera, Discrete Linear Control: the Polynomial Equation Approach, Wiley, Chienester, U.K., 1979.

    Google Scholar 

  7. D. Chyung, Discrete systems with delays in control, IEEE Trans. Automatic Control, 1969, 14(2): 196–197.

    Article  Google Scholar 

  8. G. Tadmor and L. Mirkin, H ∞ control and estimation with preview-part II: matrix ARE solutions in discrete-time, IEEE Trans. Automatic Control, 2005, 50(1): 29–40.

    Article  Google Scholar 

  9. L. Mariani and B. Nicoletti, Optimal discrete-time systems with pure delay, IEEE Trans. Automatic Control, 1973, 18(3): 311–313.

    Article  Google Scholar 

  10. G. Meinsma and L. Mirkin, H ∞ control of systems with multiple I/O delays via decomposition to adobe problems, IEEE Trans. Automatic Control, 2005, 50(2): 199–211.

    Article  Google Scholar 

  11. A. A. Moelja and G. Meinsma, H 2 optimal control of systems with multiple I/O delays: time domain approach, Automatica, 2005, 41(7): 1229–1238.

    Article  Google Scholar 

  12. M. C. Delfour, The linear quadratic optimal control problem with delays in the state and control variables: a state space approach, IEEE SIAM J. Control and Optimization, 1986, 24(10): 835–883.

    Article  Google Scholar 

  13. H. Zhang, L. Xie, D. Zhang, and Y. Soh, A reorganized innovation approach to linear estimation, IEEE Trans. Automatic Control, 2004, 49(10): 1810–1814.

    Article  Google Scholar 

  14. H. Zhang, L. Xie, Y. Soh, and D. Zhang, H ∞ fixed-lag smoothing for discrete linear time-varying systems, Automatica, 2005, 41(5): 839–846.

    Article  Google Scholar 

  15. H. Zhang, G. Duan, and L. Xie, Linear quadratic regulation for linear time-varying systems with multiple input delays, Automatica, 3006, 42(9): 1465–1476.

  16. K. J. Hunt, V. Kucera, M. Sebek, Optimal regulation using measurement feedback: a polynomial approach, IEEE Trans. Automatic Control, 1992, 37(5): 682–685.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Control Science and Engineering, Shandong University, Jinan, 250061, China

    Hongguo Zhao, Huanshui Zhang & Chenghui Zhang

  2. Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen University Town Xili, Shenzhen, 518055, China

    Hongxia Wang

Authors
  1. Hongguo Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huanshui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hongxia Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chenghui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongguo Zhao.

Additional information

The research is supported by the National Natural Science Foundation of China under Grant No. 60574016.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhao, H., Zhang, H., Wang, H. et al. Linear Quadratic Regulation for Discrete-Time Systems with Input Delay: Spectral Factorization Approach. J. Syst. Sci. Complex. 21, 46–59 (2008). https://doi.org/10.1007/s11424-008-9065-2

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-008-9065-2

Keywords

Search

Navigation