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Stabilization of stationary affine control systems with discrete time

  • Control Theory
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Abstract

We obtain new sufficient conditions for the local and global asymptotic stabilization of the zero solution of a nonlinear affine control system with discrete time and with constant coefficients by a continuous state feedback. We assume that the zero solution of the free system is Lyapunov stable. For systems with linear drift, we construct a bounded control in the problem of global asymptotic state and output stabilization. Corollaries for bilinear systems are obtained.

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References

  1. Jurdjevic, V. and Quinn, J.P., Controllability and Stability, J. Differential Equations, 1978, vol. 28, no. 3, pp. 381–389.

    Article  MathSciNet  MATH  Google Scholar 

  2. Lin, W., Global Asymptotic Stabilization of General Nonlinear Systems with Stable Free Dynamics via Passivity and Bounded Feedback, Automatica, 1996, vol. 32, no. 6, pp. 915–924.

    Article  MathSciNet  MATH  Google Scholar 

  3. Tsinias, J., Stabilizability of Discrete-Time Nonlinear Systems, IMA J. Math. Control Inform., 1989, vol. 6, no. 2, pp. 135–150.

    Article  MathSciNet  MATH  Google Scholar 

  4. Byrnes, C.I., Lin, W., and Ghosh, B.K., Stabilization of Discrete-Time Nonlinear Systems by Smooth State Feedback, Systems Control Lett., 1993, vol. 21, no. 3, pp. 255–263.

    Article  MathSciNet  MATH  Google Scholar 

  5. Lin, W. and Byrnes, C.I., KYP Lemma, State Feedback and Dynamic Output Feedback in Discrete-Time Bilinear Systems, Systems Control Lett., 1994, vol. 23, no. 2, pp. 127–136.

    Article  MathSciNet  MATH  Google Scholar 

  6. Byrnes, C.I. and Lin, W., Losslessness, Feedback Equivalence, and the Global Stabilization of Discrete-Time Nonlinear Systems, IEEE Trans. Automat. Control, 1994, vol. 39, no. 1, pp. 83–98.

    Article  MathSciNet  MATH  Google Scholar 

  7. Lin, W., Input Saturation and Global Stabilization of Nonlinear Systems via State and Output Feedback, IEEE Trans. Automat. Control, 1995, vol. 40, no. 4, pp. 776–782.

    Article  MathSciNet  MATH  Google Scholar 

  8. Bacciotti, A. and Biglio, A., Some Remarks about Stability of Nonlinear Discrete-Time Control Systems, Nonlinear Differ. Equ. Appl., 2001, vol. 8, no. 4, pp. 425–438.

    Article  MathSciNet  MATH  Google Scholar 

  9. Haddad, W.M. and Chellaboina, V., Nonlinear Dynamical Systems and Control: a Lyapunov-Based Approach, Princeton, 2008.

    Google Scholar 

  10. Outbib, R. and Sallet, G., Stabilizability of the Angular Velocity of a Rigid Body Revisited, Systems Control Lett., 1992, vol. 18, no. 2, pp. 93–98.

    Article  MathSciNet  MATH  Google Scholar 

  11. Sontag, E.D., Mathematical Control Theory. Deterministic Finite Dimensional Systems, New York, 1998.

    MATH  Google Scholar 

  12. Zaitsev, V.A., Generalization of the Jurdjevic–Quinn Theorem and Stabilization of Bilinear Control Systems with Periodic Coefficients: I, Differ. Equ., 2013, vol. 49, no. 1, pp. 101–111.

    Article  MathSciNet  MATH  Google Scholar 

  13. Zaitsev, V.A., Consistency and Eigenvalue Assignment for Discrete-Time Bilinear Systems: I, Differ. Equ., 2014, vol. 50, no. 11, pp. 1495–1507.

    Article  MathSciNet  Google Scholar 

  14. Popova, S.N. and Tonkov, E.L., Control over the Lyapunov Exponents of Consistent Systems. I, Differ. Equ., 1994, vol. 30, no. 10, pp. 1556–1564.

    MathSciNet  MATH  Google Scholar 

  15. Zaitsev, V.A. and Tonkov, E.L., Attainability, Consistency and the Rotation Method by V.M. Millionshchikov, Russian Math., 1999, vol. 43, no. 2, pp. 42–52.

    MathSciNet  MATH  Google Scholar 

  16. Zaitsev, V.A., Consistent Systems and Pole Assignment: I, Differ. Equ., 2012, vol. 48, no. 1, pp. 120–135.

    Article  MathSciNet  MATH  Google Scholar 

  17. Zaitsev, V.A., Consistent Systems and Pole Assignment: II, Differ. Equ., 2012, vol. 48, no. 6, pp. 857–866.

    Article  MathSciNet  MATH  Google Scholar 

  18. Zaitsev, V.A., Consistency and Eigenvalue Assignment for Discrete-Time Bilinear Systems: II, Differ. Equ., 2015, vol. 51, no. 4, pp. 510–522.

    Article  MathSciNet  Google Scholar 

  19. Zaitsev, V.A., Eigenvalue Spectrum Assignment in Systems with Discrete Time, in XII Vserossiiskoe soveshchanie po problemam upravleniya (VSPU–2014: 16–19 iyunya 2014): Trudy (Proc. XII All-Russia Conf. on Control Problems (RCCP–2014: June 16–19, 2014)), Moscow: Inst. of Control Problems, 2014, pp. 861–867.

    Google Scholar 

  20. Zaitsev, V.A., Generalization of the Jurdjevic–Quinn Theorem and Stabilization of Bilinear Control Systems with Periodic Coefficients: II, Differ. Equ., 2013, vol. 49, no. 3, pp. 336–345.

    Article  MathSciNet  MATH  Google Scholar 

  21. Zaitsev, V.A., Stabilization of Affine Control Systems with Periodic Coefficients, Differ. Equ., 2013, vol. 49, no. 12, pp. 1619–1628.

    Article  MathSciNet  MATH  Google Scholar 

  22. Grüne, L. and Wirth, F., Feedback Stabilization of Discrete-Time Homogeneous Semi-Linear Systems, Systems Control Lett., 1999, vol. 37, no. 1, pp. 19–30.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Udmurt State University, Izhevsk, Russia

    V. A. Zaitsev

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  1. V. A. Zaitsev
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Correspondence to V. A. Zaitsev.

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Original Russian Text © V.A. Zaitsev, 2015, published in Differentsial’nye Uravneniya, 2015, Vol. 51, No. 12, pp. 1658–1668.

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Zaitsev, V.A. Stabilization of stationary affine control systems with discrete time. Diff Equat 51, 1637–1648 (2015). https://doi.org/10.1134/S0012266115120113

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  • DOI: https://doi.org/10.1134/S0012266115120113

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