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Attraction domain estimate for single-input affine systems with constrained control

  • Nonlinear Systems
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Nonlinear single-input affine systems represented in a canonical (normal) form are considered. The control resource is assumed to be constrained. For a closed-loop system obtained by applying a linearizing feedback, the problem of finding an estimate of the attraction domain is set. A method for constructing an ellipsoidal estimate that is based on results of absolute stability theory is suggested. Construction of the estimate reduces to solving a system of linear matrix inequalities. The discussion is illustrated by numerical examples.

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  1. Tarbouriech, S., Garcia, G., Gomes da Silva, J.M., and Queinnec, I., Stability and Stabilization of Linear Systems with Saturating Actuators, London: Springer, 2011.

    Book  MATH  Google Scholar 

  2. Tarbouriech, S. and Turner, M., Anti-Windup Design: An Overview of Some Recent Advances and Open Problems, IET Control Theor. Appl., 2009, vol. 3, no. 1, pp. 1–19.

    Article  MathSciNet  Google Scholar 

  3. Turner, M.C., Herrmann, G., and Postlethwaite, I., Anti-Windup Compensation and the Control of Input-Constrained Systems, in Mathematical Methods for Robust and Nonlinear Control, Turner, M.C. and Bates, D.G., Eds., Berlin: Springer, 2007, pp. 143–174.

    Chapter  Google Scholar 

  4. Blanchini, F. and Miani, S., Set-Theoretic Methods in Control, Boston: Birkhauser, 2008.

    MATH  Google Scholar 

  5. Formal’skii, A.M., Upravlyaemost’ i ustoichivost’ sistem s ogranichennymi resursami (Controllability and Stability of Systems with Constrained Resources), Moscow: Nauka, 1974.

    MATH  Google Scholar 

  6. Herrmann, G., Turner, M.C., Menon, P.P., Bates, D.G., and Postlethwaite, I., Anti-Windup Synthesis for Nonlinear Dynamic Inversion Controllers, Proc. 5th IFAC Symp. on Robust Control Design (ROCOND), Toulouse, 2006.

    Google Scholar 

  7. Kapoor, N. and Daoutidis, P., An Observer Based Anti-Windup Scheme for Nonlinear Systems with Input Constraints, Int. J. Control, 1999, vol. 72, no. 1, pp. 18–29.

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhevnin, F.F. and Krishchenko, A.P., Controllability of Nonlinear Systems and Synthesis of Control Algorithms, Dokl. Akad. Nauk SSSR, 1981, vol. 258, no. 4, pp. 805–809.

    MathSciNet  MATH  Google Scholar 

  9. Rapoport, L.B., Estimation of Attraction Domain in a Wheeled Robot Control Problem, Autom. Remote Control, 2006, vol. 67, no. 9, pp. 1416–1435.

    Article  MathSciNet  MATH  Google Scholar 

  10. Pesterev, A.V. and Rapoport, L.B., Construction of Invariant Ellipsoids in the Stabilization Problem for aWheeled Robot Following a Curvilinear Path, Autom. Remote Control, 2009, vol. 70, no. 2, pp. 219–232.

    Article  MathSciNet  MATH  Google Scholar 

  11. Pesterev, A.V., Algorithm to Construct Invariant Ellipsoids in the Problem of Stabilization of Wheeled Robot Motion, Autom. Remote Control, 2009, vol. 70, no. 9, pp. 1528–1539.

    Article  MathSciNet  MATH  Google Scholar 

  12. Pesterev, A.V., Maximum-Volume Ellipsoidal Approximation of Attraction Domain in Stabilization Problem for Wheeled Robot, Proc. 18th IFAC World Congress, Milan, 2011, CD ROM.

    Google Scholar 

  13. Isidori, A., Nonlinear Control Systems, London: Springer, 1995.

    Book  MATH  Google Scholar 

  14. Tkachev, S.B., Stabilization of Nonstationary Affine Systems by the Virtual Output Method, Differ. Eq., 2007, vol. 43, no. 11, pp. 1546–1557.

    Article  MATH  Google Scholar 

  15. Pesterev, A.V. and Rapoport, L.B., Canonical Representation of the Path Following Problem for Wheeled Robots, Autom. Remote Control, 2013, vol. 74, no. 5, pp. 785–801.

    Article  MathSciNet  MATH  Google Scholar 

  16. Aizerman, M.A. and Gantmacher, F.R., Absolute Stability of Regulation Systems, San Francisco: Holden Day, 1964.

    Google Scholar 

  17. Pyatnitskii, E.S., Absolute Stability of Nonstationary Nonlinear Systems, Autom. Remote Control, 1970, vol. 31, no. 1, pp. 1–10.

    MathSciNet  Google Scholar 

  18. Polyak, B.T. and Shcherbakov, P.S., Robust Stability and Controllability, Moscow: Nauka, 2002.

    Google Scholar 

  19. Boyd, S., Ghaoui, L.E., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.

    Book  MATH  Google Scholar 

  20. Pesterev, A.V., Construction of the Best Ellipsoidal Approximation of the Attraction Domain in Stabilization Problem for a Wheeled Robot, Autom. Remote Control, 2011, vol. 72, no. 3, pp. 512–528.

    Article  MathSciNet  MATH  Google Scholar 

  21. Pesterev, A.V., Absolute Stability Analysis for a Linear Time Varying System of Special Form, 2016 Int. Conf. “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskiy’s Conference), June 1–3, 2016, DOI: 10.1109/STAB.2016.7541213.

    Google Scholar 

  22. Andronov, A.A., Leontovich, E.A., Gordon, I.I., and Maier, A.G., Qualitative Theory of Second-Order Dynamic Systems, New York: Wiley, 1973.

    MATH  Google Scholar 

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Correspondence to A. V. Pesterev.

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Original Russian Text © A.V. Pesterev, 2017, published in Avtomatika i Telemekhanika, 2017, No. 4, pp. 3–20.

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

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Pesterev, A.V. Attraction domain estimate for single-input affine systems with constrained control. Autom Remote Control 78, 581–594 (2017).

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