Skip to main content
SpringerLink
Log in

Linear-quadratic regulator. I. a new solution

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

Abstract

In this paper, the classical linear-quadratic regulator problem is solved via use of the linear matrix inequality technique. This approach is shown to yield the optimal solution obtained by using the matrix Riccati equation. Various undesirable effects are discussed, which may appear when applying other solution methods known from the literature; numerical examples are presented.

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. Letov, A.M., Analytical Design of Controllers, Avtom. Telemekh., 1960, no. 4, pp. 436–443; no. 5, pp. 561–568; no. 6, pp. 661–665.

    Google Scholar 

  2. Kalman, R.E., Contributions to the Theory of Optimal Control, Bol. Soc. Mat. Mexicana, 2nd Series, 1960, vol. 5, no. 2, pp. 102–119.

    MathSciNet  Google Scholar 

  3. Anderson, B.D.O. and Moore, J.B., Linear Optimal Control, New York: Prentice-Hall, 1971.

    MATH  Google Scholar 

  4. Kwakernaak, H. and Sivan, R., Linear Optimal Control Systems, New York: Wiley, 1972.

    MATH  Google Scholar 

  5. Boyd, S., El Ghaoui, L., Feron, E., et. al., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.

    Book  MATH  Google Scholar 

  6. Balandin, D.V. and Kogan, M.M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh neravenstv (LMI-based Control System Design), Moscow: Fizmatlit, 2007.

    Google Scholar 

  7. Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.

    Google Scholar 

  8. Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh: tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems Subject to Exogenous Disturbances: The Linear Matrix Inequalitiy Technique), Moscow: LENAND, 2014.

    Google Scholar 

  9. Willems, J.S., Least Squares Stationary Optimal Control and the Algebraic Riccati Equation, IEEE Trans. Autom. Control, 1971, vol. 16, no. 6, pp. 621–634.

    Article  MathSciNet  Google Scholar 

  10. Petersen, I.R. and McFarlane, D.C., Optimal Guaranteed Cost Control and Filtering for Uncertain Linear Systems, IEEE Trans. Autom. Control, 1994, vol. 39, pp. 1971–1977.

    Article  MathSciNet  MATH  Google Scholar 

  11. Aleksandrov, A.G. and Nebaluev, N.A., Analytic Synthesis of Regulator Transfer Matrices on Basis of Frequency Quality Indices. I, Autom. Remote Control, 1971, vol. 32, no. 12, part 1, pp. 1871–1878.

    MATH  Google Scholar 

  12. Kalman, R.E., When is a Linear Control System Optimal?, Trans. ASME, Ser. D. J. Basic. Eng., 1964, vol. 86, pp. 51–60.

    Article  Google Scholar 

  13. Anderson, B.D.O. and Moore, J.B., Optimal Control: Linear Quadratic Methods, New York: Prentice Hall, 1989.

    Google Scholar 

  14. Zhou, K., Doyle, J.C., and Glover, K., Robust and Optimal Control, New Jersey: Prentice Hall, 1996.

    MATH  Google Scholar 

  15. Balandin, D.V. and Kogan, M.M., Synthesis of Linear Quadratic Control Laws on Basis of Linear Matrix Inequalities, Autom. Remote Control, 2007, vol. 68, no. 3, pp. 371–385.

    Article  MathSciNet  MATH  Google Scholar 

  16. Leibfritz, F. and Lipinski, W., Description of the Benchmark Examples in COMPleib 1.0, Technical Report, University of Trier, 2003, URL: wwwcomplibde.

    Google Scholar 

  17. Grant, M. and Boyd, S., CVX: Matlab Software for Disciplined Convex Programming (Web Page and Software), URL: http://stanfordedu/ boyd/cvx.

  18. Badawi, F.A., On a Quadratic Matrix Inequality and the Corresponding Algebraic Riccati Equation, Int. J. Control, 1982, vol. 36, no. 2, pp. 313–322.

    Article  MathSciNet  MATH  Google Scholar 

  19. Shaked, U. and Suplin, V., A New Bounded Real Lemma Representation for the Continuous-Time Case, IEEE Trans. Autom. Control, 2001, vol. 15, no. 1, pp. 44–48.

    MathSciNet  Google Scholar 

  20. Levine, W.S. and Athans, M., On the Determination of the Optimal Constant Output Feedback Gains for Linear Multivariable Systems, IEEE Trans. Autom. Control, 1970, vol. 46, no. 9, pp. 1420–1426.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Science, Moscow, Russia

    M. V. Khlebnikov, P. S. Shcherbakov & V. N. Chestnov

Authors
  1. M. V. Khlebnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. S. Shcherbakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. N. Chestnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. V. Khlebnikov.

Additional information

Original Russian Text © M.V. Khlebnikov, P.S. Shcherbakov, V.N. Chestnov, 2015, published in Avtomatika i Telemekhanika, 2015, No. 12, pp. 65–79.

This paper was recommended for publication by A.P. Kurdyukov, a member of the Editorial Board

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khlebnikov, M.V., Shcherbakov, P.S. & Chestnov, V.N. Linear-quadratic regulator. I. a new solution. Autom Remote Control 76, 2143–2155 (2015). https://doi.org/10.1134/S0005117915120048

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117915120048

Keywords

Search

Navigation