Skip to main content
SpringerLink
Log in

A separation theorem for stochastic singular linear quadratic control problem with partial information

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we provide a separation theorem for the singular linear quadratic (LQ) control problem of Itô-type linear systems in the case of the state being partially observable. Above all, the Kalman-Bucy filtering of the dynamics is given by means of Girsanov transformation, by which the suboptimal feedback control of the LQ problem is determined. Furthermore, it is shown that the well-posedness of the LQ problem is equivalent to the solvability of a generalized differential Riccati equation (GDRE).

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. Ait Rami, M., Moore, J.B., Zhou, X.Y. Indefinite stochastic linear quadratic control and generalized differential Riccati equation. SIAM J. on Control Optimization, 40(4): 1296–1311 (2001)

    MathSciNet  Google Scholar 

  2. Athens, M. The matrix minimum principle. Information and Control, 11(5–6): 592–606 (1967)

    Article  MathSciNet  Google Scholar 

  3. Bensoussan, A. Stochastic control of partially observable systems. Cambridge University Press, New York, 1992

    Book  MATH  Google Scholar 

  4. Carravetta, F., Germani, A., Shuakayev, M.K. A New Suboptimal Approach to the Filtering Problem for Bilinear Stochastic Differential Systems. SIAM J. on Control Optimization, 38(4): 1171–1203 (2000)

    Article  MathSciNet  Google Scholar 

  5. Chen, B.S., Zhang, W. Stochastic H 2 /H ∞ control with state-dependent noise. IEEE Trans. on Automatic Control, 49(1): 45–57 (2004)

    Article  Google Scholar 

  6. Clements, D.J., Anderson, B.D.O. Singular Optimal Control: the Linear Quadratic Problem. Spinger, Berlin, 1978

    Book  Google Scholar 

  7. Dai, L. Singualr Control System. Springer-Verlag, New York, 1989

    Book  Google Scholar 

  8. Hassibi, B., Sayed, A.H., Kailath, T. Linear estimation in Krein spaces-Part I: Theory. IEEE Trans. on Automatic Control, 41(1): 18–33(1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, Y., Zhang, W. Study on stochastic linear quadratic optimal control with constraints. Acta Automatica Sinica, 32(3): 246–254 (2006)

    Google Scholar 

  10. Kalman, R.E. Contibution to the theory of optimal control. Bol. Soc. Mat. Mexican, 5: 102–119 (1960)

    MathSciNet  Google Scholar 

  11. Moore, J.B. The singular solutions to a singular quadratic minimization problem. International Journal of Control, 20(3): 383–393 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  12. Oksendal, B. Stochastic differential equations: An introduction with applications, 6th ed., Corrected 4th printing. Spinger-Verlag, New York, 2007

    Google Scholar 

  13. Peng, S. Stochastic Hamilton-Jacobi-Bellman equality. SIAM J. on Control Optimization, 30(2): 284–304 (1992)

    Article  MATH  Google Scholar 

  14. Penrose, R. A generalized inverse of matrices. Proc. Cambridge Philos. Soc., 51(3): 406–413 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wu, Z., Wang, X. FBSDE with Poisson process and its application to linear quadratic stochastic optimal control Problem with random jumps. Acta Automatica Sinica, 29(6): 821–826 (2003)

    MathSciNet  Google Scholar 

  16. Zhang, W., Chen, B.S. On stabilization and exact observability of stochastic systems with their applications. Automatica, 40(1): 87–94 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, W., Chen, B.S., Tseng, C.S. Robust H ∞ filtering for nonlinear stochastic systems. IEEE Transactions on Signal Processing, 53(2): 589–598 (2005)

    Article  MathSciNet  Google Scholar 

  18. Zhang, W., Zhang, H., Chen, B.S. Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion. IEEE Trans. Automatic Control, 53(7): 1630–1642 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Science, Shandong University of Science and Technology, Shandong, China

    Hong-ji Ma & Ting Hou

  2. Department of Systems and Control, Beijing University of Aeronautics and Astronautics, Beijing, China

    Hong-ji Ma

Authors
  1. Hong-ji Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ting Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hong-ji Ma.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 61174078), the Mathematical Tianyuan Youth Foundation of China (Grant No. 11126094), the Key Project of Natural Science Foundation of Shandong Province (Grant No. ZR2009GZ001) and the research project of “SDUST Spring Bud” (Grant No. 2009AZZ074).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ma, Hj., Hou, T. A separation theorem for stochastic singular linear quadratic control problem with partial information. Acta Math. Appl. Sin. Engl. Ser. 29, 303–314 (2013). https://doi.org/10.1007/s10255-013-0218-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-013-0218-2

Keywords

2000 MR Subject Classification

Search

Navigation