Reduced-Order Observer for Estimating Mixed Variables in Tracking Systems under Unmatched Exogenous Disturbances

Abstract

We consider the problem of designing a tracking system for nonlinear affine SISO plants under unmatched exogenous and parametric disturbances and incomplete measurements. In the framework of the block approach, we develop a discontinuous dynamic feedback control law ensuring a tracking with a given accuracy for the output variable of the command signal under the assumption of exogenous actions being smooth. The main result of the paper is a single-parameter procedure for designing a reduced-order observer with high-gain feedbacks for measurement-based estimation of the error of tracking the mixed variables used to form the feedback.

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REFERENCES

  1. 1

    Nikiforov, V.O., Adaptive nonlinear tracking with complete compensation of unknown disturbances, Eur. J. Control, 1998, vol. 4, no. 2, pp. 132–139.

    Article  Google Scholar 

  2. 2

    Almedia, D.I. and Alvarez, J., Robust synchronization of nonlinear SISO systems using sliding mode control, Nonlinear Dyn., 2006, vol. 46, no. 3, pp. 293–306.

    Article  Google Scholar 

  3. 3

    Bobtsov, A.A., Kolyubin, S.A., Kremlev, A.S., and Pyrkin, A.A., An iterative algorithm of adaptive output control with complete compensation for unknown sinusoidal disturbance, Autom. Remote Control, 2012, vol. 73, no. 8, pp. 1327–1336.

    MathSciNet  Article  Google Scholar 

  4. 4

    Utkin, V.A. and Utkin, A.V., Problem of tracking in linear systems with parametric uncertainties under unstable zero dynamics, Autom. Remote Control, 2014, vol. 75, no. 9, pp. 1577–1592.

    MathSciNet  Article  Google Scholar 

  5. 5

    Bestle, D. and Zeits, M., Canonical form observer design for non-linear observers with linearizable error dynamics, Int. J. Control, 1981, vol. 23, pp. 419–431.

    Google Scholar 

  6. 6

    Floquet, T. and Barbot, J.P., An observability form for linear system with unknown inputs, Int. J. Control, 2006, no. 79, pp. 132–139.

  7. 7

    Krasnova, S.A. and Mysik, N.S., Cascade synthesis of a state observer with nonlinear correcting influences, Autom. Remote Control, 2014, vol. 75, no. 2, pp. 263–280.

    MathSciNet  Article  Google Scholar 

  8. 8

    Akhobadze, A.G. and Krasnova, S.A., Solving the problem of tracking under uncertainty conditions based on joined block-canonical form of controllability and observability, Upr. Bol’shimi Sist., 2009, no. 24, pp. 34–80.

  9. 9

    Krasnova, S.A. and Utkin, A.V., Analysis and synthesis of minimum phase nonlinear SISO systems under external unmatched perturbations, Autom. Remote Control, 2016, vol. 77, no. 9, pp. 1665–1675.

    MathSciNet  Article  Google Scholar 

  10. 10

    Krasnov, D.V. and Utkin, A.V., Synthesis of a multifunctional tracking system in conditions of uncertainty, Autom. Remote Control, 2019, vol. 80, no. 9, pp. 1704–1716.

    MathSciNet  Article  Google Scholar 

  11. 11

    Krasnova, S.A., Utkin, V.A., and Utkin, A.V., Block approach to analysis and design of the invariant nonlinear tracking systems, Autom. Remote Control, 2017, vol. 78, no. 12, pp. 2120–2140.

    MathSciNet  Article  Google Scholar 

  12. 12

    Utkin, V.A., Method of separation of motions in observation problems, Avtom. Telemekh., 1990, no. 3, pp. 27–37.

  13. 13

    Menard, T., Moulay, E., and Perruquetti, W., A global high-gain finite-time observer, IEEE Trans. Autom. Control, 2010, vol. 55, no. 6, pp. 1500–1506.

    MathSciNet  Article  Google Scholar 

  14. 14

    Il’in, A.V., Goncharov, O.I., and Fomichev, V.V., Designing observers for bilinear systems of a special form, Dokl. Ross. Akad. Nauk, 2013, vol. 449, no. 2, pp. 149–153.

    Google Scholar 

  15. 15

    Khalil, H.K. and Praly, L., High-gain observers in nonlinear feedback control, Int. J. Robust Nonlinear Control, 2014, vol. 24, no. 6, pp. 993–1015.

    MathSciNet  Article  Google Scholar 

  16. 16

    Fridman, L., Levant, A., and Davila, J., Observation of linear systems with unknown inputs via high-order sliding-modes, Int. J. Syst. Sci., 2007, vol. 38, no. 10, pp. 773–791.

    MathSciNet  Article  Google Scholar 

  17. 17

    Luenberger, D.B., Observers of multivariable systems, IEEE Trans. Autom. Control, 1966, vol. 11, no. 2, pp. 190–197.

    MathSciNet  Article  Google Scholar 

  18. 18

    Afri, C., Andrieu, V., Bako, L., and Dufour, P., State and parameter estimation: A nonlinear Luenberger observer approach, IEEE Trans. Autom. Control, 2017, vol. 62, no. 2, pp. 973–980.

    MathSciNet  Article  Google Scholar 

  19. 19

    Fomichev, V.V., Sufficient conditions for the stabilization of linear dynamical systems, Differ. Equations, 2015, vol. 51, no. 11, pp. 1512–1517.

    MathSciNet  Article  Google Scholar 

  20. 20

    Krasnoshchechenko, V.I. and Krishchenko, A.P., Nelineinye sistemy: geometricheskie metody analiza i sinteza (Nonlinear Systems: Geometrical Methods of Analysis and Synthesis), Moscow: Izd. Mosk. Gos. Tekn. Univ. im. N.E. Baumana, 2005.

    Google Scholar 

  21. 21

    Fomichev, V.V., Kraev, A.V., and Rogovskii, A.I., On the zero dynamics equations of some nonlinear systems affine in control, Differ. Equations, 2018, vol. 54, no. 12, pp. 1654–1668.

    MathSciNet  Article  Google Scholar 

  22. 22

    Antipov, A.S., Krasnov, D.V., and Utkin, A.V., Decomposition synthesis of control system of electromechanical plants under conditions of incomplete information, Prikl. Mat. Mekh., 2019, vol. 83, no. 4, pp. 530–548.

    Google Scholar 

  23. 23

    Utkin, V.I., Guldner, J., and Shi, J., Sliding Mode Control in Electromechanical Systems, New York: Taylor & Francis, 2009.

    Google Scholar 

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Funding

This work was partly supported by the Russian Foundation for Basic Research, project no. 20-01-00363A.

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Correspondence to D. V. Krasnov or A. V. Utkin.

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Translated by V. Potapchouck

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Krasnov, D.V., Utkin, A.V. Reduced-Order Observer for Estimating Mixed Variables in Tracking Systems under Unmatched Exogenous Disturbances. Diff Equat 56, 1650–1663 (2020). https://doi.org/10.1134/S00122661200120149

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