Skip to main content
SpringerLink
Log in

Problems of Adaptive Optimal Control of Discrete-Time Systems under Bounded Disturbance and Linear Performance Indexes

  • Topical Issue
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

We present some problems of adaptive optimal robust control of linear discretetime systems under uncertainty and bounded external disturbance in which the optimal or guaranteed value of the performance index is a linear or linear-fractional function of unknown parameters of the system and set-membership estimation based on the method of recurrent objective inequalities reduces to recurrent updating of polyhedral estimates of unknown parameters. In such problems, computing current optimal estimates becomes a recurrent linear programming problem which is computationally tractable on modern computers for systems with a small number of estimated parameters.

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. Yakubovich, V.A., Recurrent Finite-Memory Convergent Algorithms for Solving Systems of Inequalities, Dokl. Akad. Nauk SSSR, 1966, vol. 166, no. 6, pp. 1308–1312.

    MathSciNet  Google Scholar 

  2. Yakubovich, V.A., On the Theory of Adaptive Systems, Dokl. Akad. Nauk SSSR, 1968, vol. 182, no. 3, pp. 518–522.

    MathSciNet  Google Scholar 

  3. Yakubovich, V.A., Adaptive Systems with Multi-Stage Objective Conditions, Dokl. Akad. Nauk SSSR, 1968, vol. 183, no. 2, pp. 303–306.

    MATH  Google Scholar 

  4. Yakubovich, V.A., Finite-Memory Convergent Algorithms for Solving Countable Systems of Inequalities and Their Application to Problems of Synthesis of Adaptive Systems, Dokl. Akad. Nauk SSSR, 1969, vol. 189, no. 3, pp. 495–498.

    Google Scholar 

  5. Fomin, V.N., Fradkov, A.L., and Yakubovich, V.A., Adaptivnoe upravlenie dinamicheskimi ob”ektami (Adaptive Control for Dynamical Objects), Moscow: Nauka, 1981.

    MATH  Google Scholar 

  6. Fradkov, A.L., Speed-Gradient Scheme and Its Application in Adaptive Control Problems, Autom. Remote Control, 1979, vol. 40, no. 9, pp. 1333–1342.

    MathSciNet  MATH  Google Scholar 

  7. Fradkov, A.L., Adaptivnoe upravlenie v slozhnykh sistemakh (Adaptive Control in Complex Systems), Moscow: Nauka, 1990.

    MATH  Google Scholar 

  8. Gusev, S.V., Finite-memory Convergent Algorithm for Estimating the Regression Function and its Application in Adaptive Control, Autom. Remote Control, 1989, vol. 50, no. 3, pp. 367–374.

    MATH  Google Scholar 

  9. Schweppe, F.C., Recursive State Estimation: Unknown but Bounded Errors and System Inputs, IEEE Trans. Automat. Control, 1968, vol. 13, no. 1, pp. 22–28.

    Article  Google Scholar 

  10. Kurzhanskii, A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti (Control and Observation under Uncertainty), Moscow: Nauka, 1977.

    MATH  Google Scholar 

  11. Walter, E., Ed., Special Issue on Parameter Identifications with Error Bounds, Math. Comput. Simulat., 1990, vol. 32, no. 5–6, pp. 447–607.

    Google Scholar 

  12. Norton, J.P., Ed., Special Issue on Bounded-Error Estimation, Int. J. Adaptive Control Signal Process., 1994, vol. 8, no. 1, pp. 1–118.

    Google Scholar 

  13. Norton, J.P., Ed., Special Issue on Bounded-Error Estimation, Int. J. Adaptive Control Signal Process., 1995, vol. 9, no. 1, pp. 1–132.

    Google Scholar 

  14. Sokolov, V.F., Adaptive Suboptimal Control in the Case of Constrained Noise, Autom. Remote Control, 1985, vol. 46, pp. 1, pp. 1131–1139.

    MATH  Google Scholar 

  15. Sokolov, V.F., Adaptive Suboptimal Control of a Linear System with Bounded Disturbances, Syst. Control Lett., 1985, vol. 6, pp. 93–98.

    Article  MathSciNet  MATH  Google Scholar 

  16. Barabanov, A.E. and Granichin, O.N., Optimal Controller for Linear Plant with Bounded Noise, Autom. Remote Control, 1984, vol. 45, pp. 1, pp. 578–584.

    MathSciNet  MATH  Google Scholar 

  17. Barabanov, A.E., Sintez minimaksnykh regulyatorov (Synthesis of Minimax Controllers), St. Petersburg: S.-Peterburg. Gos. Univ., 1996.

    Google Scholar 

  18. Dahleh, M.A. and Diaz-Bobillo, I.J., Control of Uncertain Systems: A Linear Programming Approach, Englewood Cliffs: Prentice Hall, 1995.

    MATH  Google Scholar 

  19. Sokolov, V.F., Closed-Loop Identification for the Best Asymptotic Performance of Adaptive Robust Control, Automatica, 1996, vol. 32, no. 8, pp. 1163–1176.

    Article  MathSciNet  MATH  Google Scholar 

  20. Sokolov, V.F., Adaptive Robust Control of a Discrete Scalar Object in the ℓ 4-Statement, Autom. Remote Control, 1998, vol. 59, no. 3, pp. 2, pp. 392–411.

    MATH  Google Scholar 

  21. Sokolov, V.F., Adaptive ℓ 4-Robust Control for SISO System, Syst. Control Lett., 2001, vol. 42(5), pp. 379–393.

    Article  MathSciNet  MATH  Google Scholar 

  22. Sokolov, V.F., Adaptive Suboptimal Tracking for a First-order Object under Lipschitz Uncertainty, Autom. Remote Control, 2003, vol. 64, no. 3, pp. 457–467.

    Article  MathSciNet  MATH  Google Scholar 

  23. Sokolov, V.F., Adaptive Suboptimal Tracking for the First-Order Plant with Lipschitz Uncertainty, IEEE Trans. Automat. Control, 2003, vol. 48, pp. 607–612.

    Article  MathSciNet  MATH  Google Scholar 

  24. Sokolov, V.F., Adaptive Suboptimal Tracking under Bounded Lipshitz Uncertainty in a Discrete Minimum-phase Object, Autom. Remote Control, 2017, vol. 78, no. 10, pp. 1775–1789.

    Article  MathSciNet  MATH  Google Scholar 

  25. Sokolov, V.F., ℓ 4 Robust Performance of Discrete-Time Systems with Structured Uncertainty, Syst. Control Lett., 2001, vol. 42(5), pp. 363–377.

    Article  MathSciNet  MATH  Google Scholar 

  26. Sokolov, V.F., Estimating Performance of the Robust Control System under Uunknown upper Disturbance Boundaries and Measurement Noise, Autom. Remote Control, 2010, vol. 71, no. 9, pp. 1741–1756.

    Article  MathSciNet  MATH  Google Scholar 

  27. Sokolov, V.F., ℓ 4-Suboptimal Robust Controller Design for Discrete Linear SISO Plants with Structured Uncertainty, Autom. Remote Control, 1999, vol. 60, no. 11, pp. 1587–1601.

    MATH  Google Scholar 

  28. Sokolov, V.F., ℓ 4 Optimal Robust Controller for SISO Plant under Coprime Factor Perturbations, IEEE Trans. Automat. Control, 2000, vol. 45, no. 12, pp. 2339–2345.

    Article  MathSciNet  MATH  Google Scholar 

  29. Walter, E. and Piet-Lahanier, H., Exact Recursive Polyhedral Description of the Feasible Parameter Set for Bounded Error, IEEE Trans. Automat. Control, 1989, vol. 34, pp. 911–915.

    Article  MathSciNet  MATH  Google Scholar 

  30. Sokolov, V.F., Adaptive Srobust Control of the First-Order Plant, Autom. Remote Control, 2008, vol. 69, no. 8, pp. 1357–1372.

    Article  MathSciNet  MATH  Google Scholar 

  31. Sokolov, V.F., Optimal Robust Steady-State Tracking for Unknown First-Order Plant, IEEE Trans. Automat. Control, 2009, vol. 54, no. 9, pp. 2199–2205.

    Article  MathSciNet  MATH  Google Scholar 

  32. Bondarko, V.A., Modifications of Finitely-Converging Algorithms of Adaptive Control, IFACPapersOnLine, 2015, vol. 48, no. 11, pp. 568–573.

    Google Scholar 

  33. Yakubovich, E.D., Solution of a Problem in the Optimal Control of a Discrete Linear Systems, Autom. Remote Control, 1975, vol. 36, no. 9, pp. 1447–1453.

    MathSciNet  MATH  Google Scholar 

  34. Yakubovich, V.A., Adaptive Suboptimal Control for a Linear Dynamical Object with Delay in the Control, Kibernetika, 1976, no. 1, pp. 26–43.

    MathSciNet  Google Scholar 

  35. Ponomarenko, V.I. and Yakubovich, V.A., Method of Recurrent Aim Inequalities in Adaptive Suboptimal Control Problems for Dynamical Objects, in Problems of Cybernetics. Adaptive Control Systems, Moscow: Nauchn. Sovet po Kibernetike, 1977, pp. 16–28.

    Google Scholar 

  36. Bondarko, V.A., Adaptive Suboptimal Control for Solutions of Linear Difference Equations, Dokl. Akad. Nauk SSSR, 1983, vol. 270, no. 2, pp. 301–303.

    Google Scholar 

  37. Bondarko, V.A. and Yakubovich, V.A., Method of Recursive Aim Inequalities in Adaptive Control Theory, Int. J. Adaptive Control Signal Proc., 1992, vol. 6, no. 3, pp. 141–160.

    Article  MATH  Google Scholar 

  38. Sokolov, V.F., Modeling the System of Suboptimal Robust Tracking under Unknown Upper Bounds on the Uncertainties and External Disturbances, Autom. Remote Control, 2014, vol. 75, no. 5, pp. 900–916.

    Article  MathSciNet  MATH  Google Scholar 

  39. Boyd, S. and Vandenberghe, L., Convex Optimization, Cambridge: Cambridge Univ. Press, 2004.

    Book  MATH  Google Scholar 

  40. Sokolov, V.F., Robust Tracking with Unknown Upper Bounds on the Perturbations and Measurement Noise, Autom. Remote Control, 2013, vol. 74, no. 1, pp. 76–89.

    Article  MathSciNet  MATH  Google Scholar 

  41. Xie, L.L. and Guo, L., How Much Uncertainty Can Be Dealt with by Feedback?, IEEE Trans. Automat. Control, 2000, vol. 45, no. 12, pp. 2203–2217.

    Article  MathSciNet  MATH  Google Scholar 

  42. Khammash, M. and Pearson, J.B., Performance Robustness of Discrete-Time Systems with Structured Uncertainty, IEEE Trans. Automat. Control, 1991, vol. 36, no. 4, pp. 398–412.

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhou, K., Doyle, J.C., and Glover, K., Robust and Optimal Control, Upper Saddle River: Prentice Hall, 1996.

    MATH  Google Scholar 

  44. Huang, C. and Guo, L., On Feedback Capability for a Class of Semiparametric Uncertain Systems, Automatica, 2012, vol. 48, no. 5, pp. 873–878.

    Article  MathSciNet  MATH  Google Scholar 

  45. Sokolov, V.F., Adaptive Stabilization of Parameter-Affine Minimum-Phase Plants under Lipschitz Uncertainty, Automatica, 2016, vol. 73, no. 11, pp. 64–70.

    Article  MathSciNet  MATH  Google Scholar 

  46. Sokolov, V.F., Adaptive Stabilization of Minimum Phase Plant under Lipschitz Uncertainty, Autom. Remote Control, 2016, vol. 77, no. 3, pp. 405–415.

    Article  MathSciNet  MATH  Google Scholar 

  47. Sokolov, V.F., Adaptive Minimax Control Based on Recursive Linear Programming, Autom. Remote Control, 1993, vol. 54, no. 12, pp. 1835–1845.

    MathSciNet  Google Scholar 

  48. Sokolov, V.F., Cone Algorithm of Identification for Adaptive Robust Control, IFAC Proc., 1994, vol. 27(8), pp. 575–580.

    Article  Google Scholar 

  49. Barabanov, A.E., Design of Adaptive H-Infinity-Optimal Controllers, Autom. Remote Control, 1999, vol. 60, no. 3, pp. 340–352.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Komi Research Center of the Ural Branch of the Russian Academy of Sciences, Syktyvkar, Russia

    V. F. Sokolov

Authors
  1. V. F. Sokolov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. F. Sokolov.

Additional information

Original Russian Text © V.F. Sokolov, 2018, published in Avtomatika i Telemekhanika, 2018, No. 6, pp. 155–171.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sokolov, V.F. Problems of Adaptive Optimal Control of Discrete-Time Systems under Bounded Disturbance and Linear Performance Indexes. Autom Remote Control 79, 1086–1099 (2018). https://doi.org/10.1134/S0005117918060085

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117918060085

Keywords

Search

Navigation