Skip to main content
SpringerLink
Log in

On nearly-controllable subspaces of a class of discrete-time bilinear systems

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

Abstract

If a linear time-invariant system is uncontrollable, then the state space can be decomposed as a direct sum of a controllable subspace and an uncontrollable subspace. In this paper, for a class of discrete-time bilinear systems which are uncontrollable but can be nearly controllable, by studying the nearly-controllable subspaces and defining the near-controllability index, the controllability properties of the systems are fully characterized. Examples are provided to illustrate the conceptions and results of the paper.

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. Wonham W M, Linear Multivariable Control: A Geometric Approach, Springer-Verlag, 3rd ed, 1985.

    Book  MATH  Google Scholar 

  2. Bruni C, Pillo G D, and Koch G, Bilinear systems: An appealing class of “nearly linear” systems in theory and applications, IEEE Transactions on Automatic Control, 1974, 19(4): 334–348.

    Article  MATH  Google Scholar 

  3. Mohler R R and Kolodziej W J, An overview of bilinear system theory and applications, IEEE Transactions on System, Man, and Cybernetics, 1980, 10(10): 683–688.

    Article  MathSciNet  MATH  Google Scholar 

  4. Elliott D L, Bilinear Control Systems: Matrices in Action, Springer-Verlag, 2009.

    Book  Google Scholar 

  5. Goka T, Tarn T J, and Zaborszky J, On the controllability of a class of discrete bilinear systems, Automatica, 1973, 9(5): 615–622.

    Article  MathSciNet  MATH  Google Scholar 

  6. Tarn T J, Elliott D L, and Goka T, Controllability of discrete bilinear systems with bounded control, IEEE Transactions on Automatic Control, 1973, 18(3): 298–301.

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng G S J, Tarn T J, and Elliott D L, Controllability of bilinear systems, Variable Structure Systems with Application to Economics and Biology, eds. by Ruberti A and Mohler R R, Springer-Verlag, 1975, 83–100.

    Chapter  Google Scholar 

  8. Evans M E and Murthy D N P, Controllability of a class of discrete time bilinear systems, IEEE Transactions on Automatic Control, 1977, 22(1): 78–83.

    Article  MathSciNet  MATH  Google Scholar 

  9. Tie L and Cai K Y, On controllability of a class of discrete-time homogeneous bilinear systems with solvable controls, International Journal of Robust and Nonlinear Control, 2012, 22(6): 591–603.

    Article  MathSciNet  Google Scholar 

  10. Tie L and Cai K Y, On near-controllability and stabilizability of a class of discrete-time bilinear systems, Systems & Control Letters, 2011, 60(8): 650–657.

    Article  MathSciNet  MATH  Google Scholar 

  11. Tie L, Cai K Y, and Lin Y, On uncontrollable discrete-time bilinear systems which are “nearly” controllable, IEEE Transactions on Automatic Control, 2010, 55(12): 2853–2858.

    Article  MathSciNet  Google Scholar 

  12. Tie L and Cai K Y, On near-controllability of nonlinear control systems, The 30th Chinese Control Conference, Yantai, Shandong, 2011, 131–136.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Automation Science and Electrical Engineering, Beihang University, Beijing, 100191, China

    Lin Tie

Authors
  1. Lin Tie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lin Tie.

Additional information

This research was supported by the China Postdoctoral Science Foundation funded project under Grant Nos. 2011M500216, 2012T50035, and the National Nature Science Foundation of China under Grant Nos. 61203231, 61273141.

This paper was recommended for publication by Editor ZHANG Jifeng.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tie, L. On nearly-controllable subspaces of a class of discrete-time bilinear systems. J Syst Sci Complex 26, 512–526 (2013). https://doi.org/10.1007/s11424-013-2012-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-013-2012-x

Keywords

Search

Navigation