Skip to main content
SpringerLink
Log in

Practical stabilizability of a class of homogeneous bilinear systems

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

This note investigates stabilizability of a class of homogeneous bilinear systems in which the drift term (A matrix) is unstable and the B matrix is rank deficient. It considers feedback control laws in the form of variable structure. It is possible to drive the system to within a small ball which includes the origin. Such stability is called practical stability. Four specific systems in \(R^3\), \(R^4\), and \(R^5\) are considered. A number of numerical examples are used to study the performance of the system.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Data availability

No external data were used for this work.

Notes

  1. It can also be concluded using the theorem 3.7 in [16, p. 84], or theorem 3.26 in [17, p. 114].

References

  1. Lakshmikantham V, Leela S, Martynyuk AA (1990) Practical stability of nonlinear systems. World Scientific, Hackensack

  2. Mohler RR (1973) Bilinear control processes. Academic Press, New York

  3. Bruni C, DiPillo C, Koch G (1974) Bilinear systems: an appealing class of “nearly linear” systems in theory and applications. IEEE Trans Autom Control 19(4):334–348

  4. Elliott DL (2009) Bilinear control processes. Springer, New York

  5. Ito K, Kunisch K (2007) Optimal bilinear control of an abstract Schrödinger equation. SIAM J Control Optim 46(1):274–287

  6. Mohler RR, Kolodziej W (1980) An overview of bilinear system theory and applications. IEEE Trans Syst Man Cybern 10(10):683–688

  7. Longchamp R (1980) State feedback control of bilinear systems. IEEE Trans Ind Electron 25(2):302–306

  8. Chen MS (1998) Exponential stabilization of a constrained bilinear system. Automatica 34(8):989–992

  9. C’elikovský S (1993) On the stabilization of the homogeneous bilinear systems. Syst Control Lett 21:503–510

  10. Piechottka U, Frank PM (1992) Controllability of bilinear systems. Automatica 28(5):1043–1045

  11. Koditschek DE, Narendfu KS (1985) The controllability of planar bilinear systems. IEEE Trans Autom Control 30(1):87–89

  12. Hwang YR, Chen MS, Wu T (2003) Division controllers for homogeneous dyadic bilinear systems. IEEE Trans Autom Control 48(4):701–705

  13. Chen MS, Hwang YR, Kuo YJ (2003) Exponentially stabilizing division controllers for homogeneous dyadic bilinear systems. IEEE Trans Autom Control 48(1):106–109

  14. Pedrycz W (1980) Stabilization of bilinear systems by a linear feedback control. Kybernetika 16(1):48–53

  15. Amato F, Cosentino C, Merola A (2009) Stabilization of bilinear systems via linear state-feedback control. IEEE Trans Circuits Syst II Express Briefs 56(1):76–80. https://doi.org/10.1109/TCSII.2008.2008528

  16. Slotine JJ, Li W (1991) Applied nonlinear control. Prentice Hall, Hoboken

  17. Kelley WG, Peterson AC (2010) The theory of differential equation, 2nd edn. Springer, New York

  18. Jebri H (2001) Global feedback stabilization of new class of bilinear systems. Syst Control Lett 42:313–320

  19. Rahn CD (1996) Stabilizability condition for strictly bilinear systems with purely imaginary spectra. IEEE Trans Autom Control 41(9):1346–1347

Download references

Funding

There were no funding to report for this work.

Author information

Authors and Affiliations

  1. Department of Engineering, Central Connecticut State University, 1615 Stanley st., New Britain, CT, 06053, USA

    M. Tadi

  2. Departmentof Electrical Engineering, University of Colorado Denver, 1200 Larimer st, Denver, CO, 80203, USA

    Miloje Radenkovic

Authors
  1. M. Tadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miloje Radenkovic
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Tadi.

Ethics declarations

Conflict of interest

There are no conict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tadi, M., Radenkovic, M. Practical stabilizability of a class of homogeneous bilinear systems. Int. J. Dynam. Control (2024). https://doi.org/10.1007/s40435-024-01442-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40435-024-01442-3

Keywords

Mathematics Subject Classification

Search

Navigation