Skip to main content
SpringerLink
Log in

State observer synthesis by measurement results for nonlinear Lipschitz systems with uncertain disturbances

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

Abstract

We propose ways to synthesize state observers that ensure that the estimation error is bounded on a finite interval with respect to given sets of initial states and admissible trajectories and also simultaneous H ∞-suppression at every time moment of initial deviations and uncertain deviations bounded in L 2-norm, external disturbances for non-autonomous continuous Lipschitz systems. Here the gain of the observers depend on the time and are defined based on a numerical solution of optimization problems with differential linear matrix inequalities or numerical solution of the corresponding matrix comparison system. With the example of a single-link manipulator we show that their application for the state estimating of autonomous systems proves to be more efficient (in terms of convergence time and accuracy of the resulting estimates) as compared to observers with constant coefficients obtained with numerical solutions of optimization problems with linear matrix inequalities.

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. Witsenhausen, H.S., Sets of Possible States of Linear Systems Given Perturbed Observaitons, IEEE Trans. Automat. Control, 1968, vol. AC-13, pp. 556–558.

    Article  Google Scholar 

  2. Schweppe, F.C., Uncertain Dynamic Systems, Englewood Cliffs: Prentice Hall, 1973.

    Google Scholar 

  3. Krasovskii, N.N., On the Theory of Controllability and Observability of Linear Dynamical Systems, Prikl. Mat. Mekh., 1964, vol. 28, no. 1, pp. 3–14.

    Google Scholar 

  4. Kurzhanskii, A.B., Differential Observation Games, Dokl. Akad. Nauk USSR, 1972, vol. 207, no. 3, pp. 527–530.

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  6. Chernous’ko, F.L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem (Estimating the Phase State in Dynamical Systems), Moscow: Nauka, 1988.

    MATH  Google Scholar 

  7. Kurzhanski, A.V. and Valyi, I., Ellipsoidal Calculus for Estimation and Control, Boston: Birkhauser, 1997.

    Book  MATH  Google Scholar 

  8. Milanese, M. and Vicino, A., Optimal Estimation Theory for Dynamic Systems with Set Membership Uncertainty: An Overview, Automatica, 1991, vol. 27, no. 6, pp. 997–1009.

    Article  MathSciNet  MATH  Google Scholar 

  9. Kuntsevich, B.M., Upravlenie v usloviyakh neopredelennosti: garantirovannye rezul’taty v zadachakh upravleniya i identifikatsii (Control under Uncertainty: Guaranteed Results in Control and Identification Problems), Kiev: Naukova Dumka, 2006.

    Google Scholar 

  10. Furasov, V.D., Zadachi garantirovannoi identifikatsii (Guaranteed Identification Problems), Moscow: Binom, 2005.

    Google Scholar 

  11. Matasov, A.I., Estimators for Uncertain Dynamic Systems, Boston: Kluwer, 1999.

    MATH  Google Scholar 

  12. Polyak, B.T. and Topunov, M.V., Filtration for Nonrandom Disturbances: Method of Invariant Ellipsoids, Dokl. Akad. Nauk, 2008, vol. 418, no. 6, pp. 1–5.

    MATH  Google Scholar 

  13. Khlebnikov, M.V., Robust Filtering under Nonrandom Disturbances: The Invariant Ellipsoid Approach, Autom. Remote Control, 2009, vol. 70, no. 1, pp. 133–146.

    Article  MathSciNet  MATH  Google Scholar 

  14. Khalil, H.K. and Saberi, A., Adaptive Stabilization of a Class of Nonlinear Systems Using High-Gain Feedback, IEEE Trans. Automat. Control, 1987, vol. AC-32, no. 11, pp. 1031–1035.

    Article  MathSciNet  MATH  Google Scholar 

  15. Esfandiari, F. and Khalil, H., Output Feedback Stabilization of Fully Linearizable Systems, Int. J. Control, 1992, vol. 56, no. 5, pp. 1007–1037.

    Article  MathSciNet  MATH  Google Scholar 

  16. Besancon, G., Bornard, G., and Hammouri, H., Observer Synthesis for a Class of Nonlinear Control Systems, Eur. J. Control, 1996, vol. 3, no. 1, pp. 176–193.

    Article  MATH  Google Scholar 

  17. Besancon, G. and Hammouri, H., On Uniform Observation of non Uniformly Observable Systems, Syst. Control Lett., 1996, vol. 29, pp. 9–19.

    Article  MATH  Google Scholar 

  18. Khalil, H.K., Nonlinear Systems, New Jersey: Prentice Hall, 2002, 3rd ed. Translated under the title Nelineinye sistemy, Moscow–Izhevsk: NITs “Regulyarnaya i Khaoticheskaya Dinamika,” Inst. Komp. Issled., 2009.

    MATH  Google Scholar 

  19. Gauthier, J-P. and Kupka, I., Observability and Observers for Nonlinear Systems, Cambridge: Cambridge Univ. Press, 2004.

    MATH  Google Scholar 

  20. Besancon, G., Ed., Nonlinear Observers and Applications, Lecture Notes in Control and Information Sciences, Heidelberg: Springer, 2007.

    Google Scholar 

  21. Kalman, R.E. and Bucy, R.S., New Results in Linear Filtering and Prediction Theory, J. Basic Eng., 1961, vol. 83, pp. 95–108.

    Article  MathSciNet  Google Scholar 

  22. Luenberger, D.G., Observers for Multivariable Systems, IEEE Trans. Autom. Control, 1966, vol. 11, no. 2, pp. 190–197.

    Article  MathSciNet  Google Scholar 

  23. Nagpal, K.M. and Khargonekar, P.P., Filtering and Smoothing in an H∞ Setting, IEEE Trans. Automat. Control, 1991, vol. 36, no. 2, pp. 152–166.

    Article  MathSciNet  MATH  Google Scholar 

  24. Khargonekar, P.P., Nagpal, K.M., and Poolla, K.R., H∞-control with Transients, SIAM J. Control Optim., 1991, vol. 29, no. 6, pp. 1373–1393.

    Article  MathSciNet  MATH  Google Scholar 

  25. Banavar, R.N. and Speyer, J.L., A Linear-Quadratic Game Approach to Estimation and Smoothing, Proc. Am. Control Conf., Boston: IEEE, 1991, vol. 3, pp. 2818–2822.

    Google Scholar 

  26. Doyle, J.C. and Stein, G., Robustness with Observers, IEEE Trans. Automat. Control, 1979, vol. AC-24, no. 4, pp. 607–611.

    Article  MathSciNet  MATH  Google Scholar 

  27. Petersen, I.R. and Holot, C.V., High-Gain Observers Applied to Problems in Disturbance Attenuation, H-infinity Optimization and the Stabilization of Uncertain Linear Systems, Proc. Am. Control Conf., Atlanta, June 1988, pp. 2490–2496.

    Google Scholar 

  28. Abbaszadeh, M. and Marquez, H.J., Robust H∞ Filtering for Lipschitz Nonlinear Systems via Multiobjective Optimization, J. Signal Inform. Process., 2010, no. 1, pp. 24–34.

    Article  Google Scholar 

  29. Malikov, A.I., Synthesis of State Estimation Algorithms for Nonlinear Controllable Systems with Matrix Comparison Systems, Vestn. KGTU im. A.N. Tupoleva, 1998, no. 3, pp. 54–59.

    Google Scholar 

  30. Ball, A.A. and Khalil, H.K., High-Gain Observers in the Presence of Measurement Noise: A Nonlinear Gain Approach, Proc. 47th IEEE Conf. on Decision and Control, Cancun, Mexico, Dec. 9–11, 2008, pp. 2288–2293.

    Google Scholar 

  31. Khalil, H.K. and Prally, L., High-Gain Observers in Nonlinear Feedback Control, Int. J. Robust. Nonlin. Control, 2014, vol. 24, no. 6, pp. 993–1015.

    Article  MathSciNet  MATH  Google Scholar 

  32. Khalil, H.K., Nonlinear Control, New York: Pearson, 2015.

    Google Scholar 

  33. Balandin, D.V. and Kogan, M.M., LMI Based H∞-optimal Control with Transients, Int. J. Control, 2010, vol. 83, no. 8, pp. 1664–1673.

    Article  MathSciNet  MATH  Google Scholar 

  34. Balandin, D.V. and Kogan, M.M., Minimax Filtering: γ0-Optimal Observers and Generalized H∞-Optimal Filters, Autom. Remote Control, 2013, vol. 74, no. 4, pp. 575–587.

    Article  MathSciNet  MATH  Google Scholar 

  35. Balandin, D.V., Kogan, M.M., Krivdina, L.N., et al., Design of Generalized Discrete-Time H∞-optimal Control over Finite and Infinite Intervals, Autom. Remote Control, 2014, vol. 75, no. 1, pp. 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  36. Kogan, M.M., Optimal Estimation and Filtration under Unknown Covariances of Random Factors, Autom. Remote Control, 2014, vol. 75, no. 11, pp. 1964–1981.

    Article  MathSciNet  MATH  Google Scholar 

  37. Kogan, M.M., Robust Estimation and Filtering in Uncertain Linear Systems under Unknown Covariations, Autom. Remote Control, 2015, vol. 76, no. 10, pp. 1751–1764.

    Article  MathSciNet  MATH  Google Scholar 

  38. Malikov, A.I., Matrix Systems of Differential Equations with Quasimonotonicity Condition, Izv. Vyssh. Uchebn. Zaved., Mat., 2000, no. 8, pp. 35–45.

    MathSciNet  MATH  Google Scholar 

  39. Malikov, A.I., State Estimation and Stabilization of Continuous Systems with Uncertain Nonlinearities and Disturbances, Autom. Remote Control, 2016, vol. 77, no. 5, pp. 764–778.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tupolev Kazan National Research Technical University, Kazan, Tatarstan, Russia

    A. I. Malikov

  2. Institute of Mechanics and Engineering, Kazan Science Center, Russian Academy of Sciences, Kazan, Tatarstan, Russia

    A. I. Malikov

Authors
  1. A. I. Malikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. I. Malikov.

Additional information

Original Russian Text © A.I. Malikov, 2017, published in Avtomatika i Telemekhanika, 2017, No. 5, pp. 16–35.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Malikov, A.I. State observer synthesis by measurement results for nonlinear Lipschitz systems with uncertain disturbances. Autom Remote Control 78, 782–797 (2017). https://doi.org/10.1134/S0005117917050022

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117917050022

Keywords

Search

Navigation