Skip to main content
Log in

On the problem of optimal speed for the discrete linear system with bounded scalar control on the basis of 0-controllability sets

  • Linear Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

For a system with discrete time and bounded scalar control, consideration was given to the problem of optimal control in a minimal number of steps. The control is based on the 0-controllability sets.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Kwakernaak, H. and Sivan, R., Linear Optimal Control Systems, New York: Wiley, 1972. Translated under the title Lineinye optimal’nye sistemy upravleniya, Moscow: Mir, 1977.

    MATH  Google Scholar 

  2. Boltyanskii, V.G., Optimal’noe upravlenie diskretnymi sistemami (Optimal Control of Discrete Systems), Moscow: Nauka, 1973.

    Google Scholar 

  3. Tabak, D. and Kuo, B., Optimal Control by Mathematical Programming, New York: Prentice Hall, 1971. Translated under the title Optimal’noe upravlenie i matematicheskoe programmirovanie, Moscow: Nauka, 1975.

    Google Scholar 

  4. Propoi, A.I., Elementy teorii optimal’nykh diskretnykh protsessov (Elements of the Theory of Optimal Discrete Processes), Moscow: Nauka, 1973.

    Google Scholar 

  5. Combastel, C. and Raka, S.A., On Computing Envelopes for Discrete-time Linear Systems with Affine Parametric Uncertainties and Bounded Inputs, in Preprints 18 IFAC World Congr., Milano (Italy), August 28–September 2, 2011, pp. 4525–4533.

    Google Scholar 

  6. Kurzhanskiy, A.F. and Varaiya, P., Theory and Computational Techniques for Analysis of Discrete-Time Control Systems with Disturbances, Optim. Method Software, 2011, vol. 26, nos. 4–5, pp. 719–746.

    Article  MATH  MathSciNet  Google Scholar 

  7. Keerthi, S.S. and Gilbert, E.G., Computatuon of Minimum-time Feedback Control Laws for Discrete-Time Systems with State-Control, IEEE Trans. Autom. Control, 1987, vol. 32, no. 5, pp. 432–434.

    Article  MATH  Google Scholar 

  8. Lin, W.-S., Time-optimal Control Strategy for Saturating Linear Discrete Systems, Int. J. Control, 1986, vol. 43, no. 5, pp. 1343–1351.

    Article  MATH  Google Scholar 

  9. Eldem, V. and Selbuz, H., On the General Solution of the State Deadbeat Control Problem, IEEE Trans. Autom. Control, 1994, vol. 39, no. 5, pp. 1002–1006.

    Article  MATH  MathSciNet  Google Scholar 

  10. Benvenuti, L. and Farina, L., The Geometry of the Reachability Set for Linear Discrete-Time Systems with Positive Controls, SIAM J. Matrix Anal. Appl., 2006, vol. 28, no. 2, pp. 306–325.

    Article  MATH  MathSciNet  Google Scholar 

  11. de Leon-Canion, P. and Lunze, J., Dependable Control of Uncertain Linear Systems Based on Settheoretic Methods, Int. J. Control, 2010, vol. 83, no. 6, pp. 1248–1264.

    Article  Google Scholar 

  12. Fisher, M.E. and Gayek, J.E., Estimating Reachable Sets for Two-Dimensional Linear Discrete Systems, J. Optim. Theory Appl., 1988, vol. 56, no. 1, pp. 67–88.

    Article  MATH  MathSciNet  Google Scholar 

  13. Kostousova, E.K., On External Polyhedral Estimation of the Reachability Sets in the “Extended” Space for Linear Multistep Systems with Integral Control Constraints, Vychisl. Tekhnol., 2004, vol. 9, no. 4, pp. 54–72.

    MATH  MathSciNet  Google Scholar 

  14. Kostousova, E.K., External Polyhedral Estimates for Reachable Sets of Linear Discrete-Time Systems with Integral Bounds on Controls, Int. J. Pure Appl. Math., 2009, vol. 50, no. 2, pp. 187–194.

    MATH  MathSciNet  Google Scholar 

  15. Ding, M.-F., Liu, Y., and Gear, J.A., A Modified Centered Climbing Algorithm for Linear Programming, Appl. Math., 2012, vol. 3, pp. 1423–1429.

    Article  Google Scholar 

  16. Telgen, J., Minimal Representation of Convex Polyhedral Sets, J. Optim. Theory Appl., 1982, vol. 38, no. 1, pp. 1–24.

    Article  MATH  MathSciNet  Google Scholar 

  17. Ivanov, D.S., Ovchinnikov, M.Yu., and Tkachev, S.S., Orientation Control of Solid Body Suspended on String Using Fan Motors, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2011, no. 1, pp. 107–119.

    MathSciNet  Google Scholar 

  18. Polovinkin, E.S. and Balashov, M.V., Elementy vypuklogo i sil’no vypuklogo analiza (Elements of Convex and Strongly Convex Analysis), Moscow: Fizmatlit, 2004.

    Google Scholar 

  19. Schrijver, A., Theory of Linear and Integer Programming, New York: Wiley, 1986. Translated under the title Teoriya lineinogo i tselochislennogo programmirovaniya, Moscow: Mir, 1991, vols. 1, 2.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to D. N. Ibragimov.

Additional information

Original Russian Text © D.N. Ibragimov, A.N. Sirotin, 2015, published in Avtomatika i Telemekhanika, 2015, No. 9, pp. 3–30.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ibragimov, D.N., Sirotin, A.N. On the problem of optimal speed for the discrete linear system with bounded scalar control on the basis of 0-controllability sets. Autom Remote Control 76, 1517–1540 (2015). https://doi.org/10.1134/S0005117915090015

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117915090015

Keywords

Navigation